Research ArticleTarget Detection in Low Grazing Angle withAdaptive OFDM Radar

Yang Xia Zhiyong Song Zaiqi Lu Hao Wu and Qiang Fu

ATR Key Laboratory National University of Defense Technology Changsha Hunan 410073 China

Correspondence should be addressed to Yang Xia xiayang2020126com

Received 10 June 2015 Revised 31 August 2015 Accepted 7 October 2015

Academic Editor Kyeong Jin Kim

Copyright copy 2015 Yang Xia et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Multipath effect is the main factor of deteriorating target detection performance in low grazing angle scenario which results fromreflections on the groundsea surface Amplitudes of the received signals fluctuate acutely due to the random phase variations ofreflected signals along different paths thereby the performances of target detection and tracking are heavily influencedThis paperdeals with target detection in low grazing angle scenario with orthogonal frequency division multiplexing (OFDM) radar Realisticphysical and statistical effects are incorporated into themultipath propagationmodel By taking advantage ofmultipath propagationthat provides spatial diversity of radar system and frequency diversity of OFDMwaveform we derive a detection method based ongeneralized likelihood ratio test (GLRT) Then we propose an algorithm to optimally design the transmitted subcarrier weights toimprove the detection performance Simulation results show that the detection performance can be improved due to the multipatheffect and adaptive OFDM waveform design

1 Introduction

Target detection and tracking in low grazing angle scenario isone of themost challenging problems in radar community [1]Multipath effect is the main problem when detecting targetsin low grazing angle scenario Amplitudes of received signalsfluctuate acutely due to random phase (which is decidedby differential path length wavelength and characteristicsof reflected surface [2]) variations between different pathsthereby the detection performance is deteriorated Currentlyresearches on multipath effect are mainly focused on twoaspects suppressing multipath and utilizing it [3] Fromanother point of view multipath echoes also contain targetenergy and the detection performance may be enhanced ifthe energy from multipath reflections is accumulated

Orthogonal frequency division multiplexing (OFDM)was originally proposed as a digital modulation technique incommunication fields Later on it was introduced into radarcommunity [4] As a new broadband radar signal OFDMsignal advances in high spectral efficiency low probabilityof intercept and frequency diversity [5ndash7] An adaptive

technique to design the spectrum of OFDM was proposedin [8] by incorporating the scattering coefficients of thetarget at multiple frequencies and the results showed thatthe wideband ambiguity function (WAF) was improved dueto the adaptive waveform design Similar OFDM waveformdesign method can also be found in [9 10]

Focusing on the issue of target detection in urbanenvironment an optimized detection algorithm based onOFDM radar was proposed in [11 12] The results demon-strated that the detection performance was improved byutilizing multipath reflections However the proposed signalmodel was idealistic and only considered specular reflectionsThe problem of target detection in multipath scenarioswas reformulated as sparse spectrum estimation where thespectral parameters of OFDM radar signal are optimizedto improve the detection performance using multiobjectiveoptimization (MOO) technique [13 14]The performances ofgeneralized likelihood ratio test (GLRT) detectorwithOFDMradar in non-Gaussian clutter (log-normal Weibull and K-compound) were investigated in [15] where target fluctu-ations were also taken into consideration Consequently

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 147368 12 pageshttpdxdoiorg1011552015147368

2 International Journal of Antennas and Propagation

Radar

Sealand surface

R1

Rd

R2

Target

Image

Figure 1 Representative scenario of multipath propagation in lowgrazing angle

detection performance may be enhanced with OFDM radarin low grazing angle scenario by utilizing multipath reflec-tions

When detecting target in low grazing angle scenarioradar measurements are affected by many factors such asever-changing meteorological conditions in the troposphereEarthrsquos curvature and roughness of groundsea surface [16]All these factors will affect the detection performance andtherefore these factors should be taken into consideration tomake the signal model more realistic

This paper deals with target detection in low grazingangle scenario with OFDM radar To make the propagationmodel more accurate refraction of the lower atmosphere andthe Earthrsquos curvature are taken into consideration and themultipath propagation model is modified accordingly Basedon the characteristics of OFDM radar we derive a GLRTdetector in Gaussian clutter environment Then a waveformdesign method which optimizes the transmitted subcarrierweights is proposed to improve the detection performanceFinally the performance of the proposed detector is analyzedand discussed via simulation experiments

2 Modified Multipath Propagation Model

Multipath effect is one of the main problems when detectingand tracking target in low grazing angle scenario The echosignals received by radar receiver not only include directsignals but also include indirect signals Thus the receivedsignals are the sum of reflected signals along different pathsand the amplitudes fluctuate acutely due to the random phasevariations which deteriorates the performances of targetdetection and tracking Generally speaking signals reflectedmore than twice will be attenuated heavily which can alwaysbe neglected

The representative scenario of multipath propagation inlow grazing angle is shown in Figure 1 Target locates at adistance 119877119889 from the radar The source is assumed to be anarrowband signal which can be represented as [16]

119909 (119905) = 119886119890119895(120596119905+120585)

(1)

where 119886 120596 and 120585 denote the amplitude angular frequencyand initial phase respectively In the presence of multipath

the received signals consist of two components namely thedirect and indirect signals Direct signal is given by

119909119889 (119905) = 119909 (119905) 119890minus119895(2120587120582)(2119877119889) (2)

and the indirect signal is

119909119894 (119905) = 119909 (119905) 120588119890119895120593119890minus119895(2120587120582)119877119894 (3)

In (3) 120588119890119895120593 is the complex reflection coefficient and 119877119894

is the total length of indirect path For first-order reflectedsignals 119877119894 = 1198771 + 1198772 + 119877119889 and for second-order reflectedsignals 119877119894 = 2(1198771 + 1198772) Received signal can be representedas

119909119903 (119905) = 119909119889 (119905) + 119909119894 (119905)

= 119909 (119905) 119890119895(2120587120582)(2119877119889) (1 + 120588119890

119895(120593+(2120587120582)Δ119877))

(4)

whereΔ119877 = 119877119894minus2119877119889 In (4) the amplitudes of received signalsare dependent on the factor120588119890119895(120593+(2120587120582)Δ119877) which includes theeffects of complex reflection coefficient 120588119890119895120593 wavelength 120582and path difference Δ119877

In normal atmospheric conditions the pressure decreasesexponentially with height which causes a reduction in therefractivity with respect to height Under this condition aradio ray will diffract downward [17] Furthermore in mar-itime environment evaporation duct effect may be produceddue to the strong humidity gradients above (within firstfew meters) the air-sea boundary [18] which makes a radioray bend downward with a curvature more than the Earthrsquosradius The effect is dependent on a few factors such astemperature difference between the air and sea and the windspeed [19]

In addition to the atmospheric effects low grazing anglepropagation is also affected by the fact that the Earth iscurved The curvature of the Earth decreases the path lengthdifference between the direct and reflected waves and italso reduces the amplitudes of the reflected waves [19] Thisproblem is usually dealt with by replacing the Earth with animaginary flat Earth whose equivalent radius is

119877119890 = 1198770 (1 + 1198770

119889119873

119889ℎ10minus6)

minus1

(5)

where1198770 is the radius of actual Earth and119889119873119889ℎ is the refrac-tivity gradient For standard atmosphere minus79 le 119889119873119889ℎ le

0N-unitskm [19]Considering all the factors mentioned above and based

on ideal propagation model described in Figure 1 modifiedmultipath propagation model is shown in Figure 2 Theheight of radar and target is ℎ119903 and ℎ119905 respectively Theground distance separated by radar and target is 119903 ℎ1015840

119903and ℎ

1015840

119905

are the modified heights corresponding to radar and targetover flat-Earthmodel 1199031 is the grounddistance between radarand the reflected point and 120595 is the grazing angle

In order to calculate complex reflection coefficient in (4)we have to solve the grazing angle 120595 and Δ119877 first Thesevariables are all related to 1199031 which can be evaluated usingthe following cubic equation [20]

21199033

1minus 3119903119903

2

1+ [119903

2minus 2119877119890 (ℎ119903 + ℎ119905)] 1199031 + 2119877119890ℎ119903119903 = 0 (6)

International Journal of Antennas and Propagation 3

RadarTarget

hr

h998400r

r1 r2

ReRe120601

1206011 1206012

O

ht

h998400t

Tangent planeR1

R2

r

120595

Rd

Figure 2 Modified multipath propagation model

Then we solve ℎ1015840119903and ℎ1015840

119905by

ℎ1015840

119903≃ ℎ119903 minus

1199032

1

2119877119890

ℎ1015840

119905≃ ℎ119905 minus

(119903 minus 1199031)2

2119877119890

(7)

Meanwhile we can get 1206011 = 1199031119877119890 and 1206012 = (119903minus1199031)119877119890 Usinglaw of cosine yields

1198771 =radic1198772

119890+ (119877119890 + ℎ119903)

2minus 2119877119890 (119877119890 + ℎ119903) cos1206011

1198772 =radic1198772

119890+ (119877119890 + ℎ119905)

2minus 2119877119890 (119877119890 + ℎ119905) cos1206012

119877119889

= radic(119877119890 + ℎ119903)2+ (119877119890 + ℎ119905)

2minus 2 (119877119890 + ℎ119903) (119877119890 + ℎ119905) cos120601

(8)

Finally the grazing angle 120595 is given by

120595 =1

2(120587 minus arccos(

1198772

1+ 119877

2

2minus 119877

2

119889

211987711198772

)) (9)

The complex reflection coefficient can be calculated by[20]

120588119890119895120593

= Γ(]ℎ)119863119878 (10)

where Γ(]ℎ) is vertical polarization or horizontal polarizationreflection coefficient for a plane surface 119863 is the divergencefactor due to a curved surface and 119878 is root-mean-squared(RMS) specular scattering coefficient which represents theroughness of surface

Γ(]ℎ) is determined by frequency complex dielectricconstant and grazing angle120595 which can be calculated by [21]

ΓV ≃sin120595radic120576119888 minus 1

sin120595radic120576119888 + 1(11)

for vertical polarization and

Γℎ ≃sin120595 minus radic120576119888

sin120595 + radic120576119888

(12)

for horizontal polarization where 120576119888 is complex dielectricconstant which is given by 120576119888 = 1205761205760 minus 11989560120582120590 1205761205760 is relativedielectric constant of the reflecting medium and 120590 is itsconductivity

When the electromagnetic wave is incident on the surfaceof the Earth due to the slight differences in each incidentray the reflected wave is diverged and the reflected energyis defocusedWhen this happens radar power density will bereduced If the grazing angle is not large divergence factor119863can be approximated by [21]

119863 ≃ (1 +211990311199032

119877119890119903120595)

minus12

(13)

Due to the reflection of rough surface two componentswill be generated a diffuse component and a coherent com-ponent with reducedmagnitudeThe reduction in themagni-tude of the coherent component brought about by reflectionsfrom a rough surface is related to the grazing angle and thesignal wavelength which is [21]

119878 = 119890minus120583 (14)

where

120583 =

2 [2120587120578]2

120578 le 01 rad

0161205782+ 742120578 + 00468 otherwise

(15)

and 120578 is the surface roughness factor defined as 120578 = 120590119867120595120582120590119867 is the RMS of reflection surface and 120590119867120582 denotes theroughness of reflected surface The larger 120590119867120582 is the moreroughness the surface will be For smooth surface 120590119867120582 isapproximated to be zero

Substituting (11)ndash(15) into (10) we can get the complexreflection coefficient 120588119890119895120593 At the same time 120588119890119895(120593+(2120587120582)Δ119877)can be evaluated which is the impact of multipath propaga-tion on signal model

3 Measurement Model of OFDM Radar

We consider a monostatic radar employing an OFDM signal-ing system The transmitted signal can be described as

119904 (119905) =

119872minus1

sum

119898=0

119906 (119905 minus 119898119879119903) 11989011989521205871198910119905 (16)

where 1198910 119872 and 119879119903 represent the carrier frequency pulsenumber and pulse repetition interval (PRI) respectively 119906(119905)is the complex envelop of a single pulse which is given by

119906 (119905) =

119873minus1

sum

119899=0

119908119899rect (119905 119905119887) 1198901198952120587119899Δ119891119905

(17)

where

rect (119905 119905119887) =

1 0 le 119905 le 119905119887

0 otherwise(18)

4 International Journal of Antennas and Propagation

and 119873 is the subcarrier number w = [1199080 1199081 119908119873minus1]119879

represent the complex weights transmitted over differentsubcarriers and satisfying sum

119873minus1

119899=0|119908119899|

2= 1 For the sake

of keeping orthogonality between different subcarriers thesubcarrier spacing Δ119891 and time duration 119905119887 of a single pulseshould satisfy Δ119891119905119887 = 1 The total bandwidth is 119861 = 119873Δ119891

Assuming that the received signals contain 119875 differentpaths and the roundtrip delay corresponding to 119901th path is120591119901 119901 = 0 1 119875 minus 1 The received signal is

119910 (119905) =

119875minus1

sum

119901=0

119904 (120574119901 (119905 minus 120591119901)) (19)

where 120574119901 = 1+120573119901 accounts for the stretching or compressingin time of the reflected signal and 120573119901 = 2⟨k u119901⟩119888 representsthe Doppler spreading factor corresponding to 119901th path kand u119901 are the target velocity and unit direction of arrival(DOA) vector 119888 is the propagation speed Substituting (16)into (19) the received signal after demodulation is denotedby

119910 (119905)

=

119875minus1

sum

119901=0

119872minus1

sum

119898=0

119906 (120574119901 (119905 minus 120591119901) minus 119898119879119903) 11989011989521205871198910120573119901(119905minus1205910)119890

minus119895212058711989101205910

+ 119890 (119905)

(20)

Then substituting (17) into (20) we obtain

119910 (119905) =

119872minus1

sum

119898=0

119873minus1

sum

119899=0

119875minus1

sum

119901=0

119908119899119909119899119901rect (120574119901 (119905 minus 120591119901) minus 119898119879119903 119905119887)

sdot 1198901198952120587119899Δ119891(120574119901(119905minus120591119901)minus119898119879119903)119890

11989521205871198910120573119901(119905minus1205910)119890minus119895212058711989101205910 + 119890 (119905)

(21)

where 119909119899119901 represents the complex scattering coefficient oftarget corresponding to 119899th carrier and 119901th path 1205910 isthe roundtrip delay corresponding to the range cell underconsideration The relative time gaps between different pathsare very small compared to the actual roundtrip delay whichmeans 120591119901 asymp 1205910 for119901 = 0 1 119875minus1 Substituting 119905 = 1205910+119898119879119903

(119898 = 0 1 119872 minus 1) into (21) we obtain

119910 (119899119898) =

119875minus1

sum

119901=0

119908119899119909119899119901120601119899119901 (119898 V) + 119890 (119899119898) (22)

where

120601119899119901 (119898 V) = 119890minus119895212058711989101205910119890

1198952120587119891119899120573119901119898119879119903 (23)

and 119891119899 = 1198910 + 119899Δ119891 Equation (22) can be written into matrixform

y (119898) = WXΦ (119898 k) + e (119898) (24)

where

(i) y(119898) = [119910(0119898) 119910(1119898) 119910(119873minus1119898)]119879 is an119873times1

vector that represents output of the119898th pulse

(ii) W = diag(w) is an 119873 times 119873 complex diagonal matrixthat contains transmitted subcarrier weights

(iii) X = blkdiag (x1198790 x1198791 x119879

119873minus1) is an119873 times119873119875 complex

rectangular block-diagonal matrix x119899 = 119909119899 sdot 120588 119899 =

0 1 119873minus1 where 119909119899 is target scattering coefficienton the 119899th subcarrier and 120588 = [1205880 1205881 120588119875minus1]

119879

represents complex reflection coefficients over differ-ent paths for direct signals 120588119901 = 1 for first-orderreflected signals120588119901 = 120588119890

119895120593 for second-order reflectedsignals 120588119901 = (120588119890

119895120593)2 where 120588119890119895120593 is calculated by (10)

(iv) Φ(119898 k) = [1206010(119898 k) 1206011(119898 k) 120601119873minus1(119898 k)]119879 is an119873119875 times 1 complex vector and 120601119899(119898 k) = [1206011198990(119898 k)1206011198991(119898 k) 120601119899119875minus1(119898 k)]119879 119899 = 0 1 119873 minus 1

(v) e(119898) = [1198900(119898) 1198901(119898) 119890119873minus1(119898)]119879is an119873times 1 com-

plex vector of clutter

Then concatenating all temporal data into119873 times119872matrixthe OFDMmeasurement model in low grazing angle is

Y = WXΦ (k) + E (25)

where

Y = [y (0) y (1) y (119872 minus 1)] (26)

Φ(V) = [Φ(0 V)Φ(1 V) Φ(119872 minus 1 V)] is an 119873119875 times 119872

complex matrix that contains Doppler information over 119872pulses

E = [e(0) e(1) e(119872minus1)] is an119873times119872 complexmatrixthat consists of clutter returns

In low grazing angle scenario the clutter is usuallymodeled as compound-Gaussian model which is the productof speckle and texture [22] The speckle is fast-changing andmodeled as a complex Gaussian process while the texture isslow-changing and modeled as a nonnegative process [23]

In this paper the correlation length of the texture isassumed to be on the order of seconds and we only considerone coherent processing interval (CPI) which is assumedto be 60ms Due to the long correlation time of textureit is considered to be constant within each CPI changingaccording to a given probability density function (PDF) fromone CPI to the next Thus conditioned on a given value ofthe texture the clutter is simplified to Gaussian distributionHowever this simplification is not justified when predictingthe clutterrsquos behavior in time intervals larger than a CPI suchas clutter cancellation and constant false alarm rate (CFAR)detection

Thus we assume that the clutter is temporally whiteand circularly zero-mean complex Gaussian process withunknown positive definite covariance C The measurementsover different pulses are supposed to be independent whichmeans

vec (Y) sim CN119873119872 (vec (WXΦ (k)) C otimes I119872) (27)

whereotimes denotes Kronecker product Accordingly theOFDMmeasurements are distributed as

Y sim CN119873119872 (WXΦ (k) C otimes I119872) (28)

International Journal of Antennas and Propagation 5

4 Detection Test

In low grazing angle scenario received signals are the sumof signals along different paths which form the complicatedmeasurements The essence of detection is to judge whethera target is present or not in the range cell under test This isa classical two-hypothesis detection problem Therefore weconstruct a decision problem to choose between two possiblehypotheses the null hypothesis (target-free hypothesis) andthe alternate hypothesis (target-present hypothesis) whichcan be expressed as

H0 X = 0 C unknown

H1 X = 0 kC unknown(29)

ThemeasurementsY in two hypotheses are distributed as

Y

sim

H0 CN119873119872 (0C otimes I119872) C unknown

H1 CN119873119872 (WXΦ (k) C otimes I119872) kC unknown

(30)

According to the classical target detection theoryNeyman-Pearson (NP) detector is the optimal detector whichmaximizes the probability of detection at a constant prob-ability of false alarm However the target velocity k andclutter covarianceC are unknown andGLRT detector is usedinstead where the unknown parameters are replaced withtheir maximum likelihood estimates (MLEs) The formula-tion of GLRT is [24]

GLR (k) =119891 (Y | H1 k X C1)

119891 (Y | H0 C0)

H1

≶

H0

120574 (31)

where 119891(Y | H1) and 119891(Y | H0) are the likelihood functionsunder H1 and H0 respectively 120574 is the threshold that thedetector is compared with X and C1 are the MLE of X and CunderH1 while C0 is the MLE of C underH0

Definition 1 If the random matrix Y(119873 times 119872) is distributedas a matrix variate normal distribution (MVND) with meanM(119873 times 119872) and covariance Ω otimes Σ we use the notation Y sim

CN119873119872(MΩotimesΣ) and the probability density function (PDF)is given by [25]

119891 (Y)

=

exp (minus (12) tr [Ωminus1 (Y minusM)Σminus1(Y minusM)

119867])

(2120587)1198731198722

|Ω|1198732

|Σ|1198722

(32)

where tr(sdot) is the trace of matrixΩ and Σ are positive definitecovariance matrices over the column and row of Y Ω is an119873 times 119873 matrix and Σ is an 119872 times119872 matrix which are definedas [25]

Ω = 119864 [(Y minusM) (Y minusM)119867]

Σ = 119864 [(Y minusM)119867(Y minusM)]

(33)

According to the above descriptions we can derive PDFsof the measurements under two hypotheses

119891 (Y | H0) =

exp (minus (12) tr [Cminus10YY119867])

(2120587)1198731198722 1003816100381610038161003816C0

1003816100381610038161003816

1198732 (34)

119891 (Y | H1)

=

exp (minus (12) tr [Cminus11(Y minusWXΦ (k)) (Y minusWXΦ (k))119867])

(2120587)1198731198722 1003816100381610038161003816C1

1003816100381610038161003816

1198732

(35)

The log-likelihood function of (35) is

ln119891 (Y | H1)

= minus119872119873

2ln (2120587) minus 119873

2ln 1003816100381610038161003816C1

1003816100381610038161003816

minus1

2tr [Cminus1

1(Y minusWXΦ (k)) (Y minusWXΦ (k))119867]

(36)

Taking derivative of (36) with respect to C1 and makingit equal to zero the MLE of C1 is

C1 =1

119873(Y minusWXΦ (k)) (Y minusWXΦ (k))119867 (37)

Substituting (37) into (36) yields

XML

= argminX

ln 10038161003816100381610038161003816(Y minusWXΦ (k)) (Y minusWXΦ (k))11986710038161003816100381610038161003816 (38)

Usually the scattering matrix X does not yield a close-form MLE expression However noting that X has a block-diagonal structure it turns out to be a block-diagonal growthcurve (BDGC) problem Reference [26] derived the approx-imate maximum likelihood (AML) estimator for X which isgiven by

xAML = vecb (X) = (Γ119867Γ)minus1

Γ119867vec (Tminus12XΠΦ) (39)

where

Γ ≜ Φ119879⊛ (T12W)

T ≜ YΠperpΦY119867

ΠΦ ≜ Φ (k)119867 (Φ (k)Φ (k)119867)minus

Φ (k)

Πperp

Φ≜ I minusΠΦ

(40)

vecb(sdot) and ⊛ denote block-diagonal matrix vectorizationoperator and generalized Khatri-Rao product [27] respec-tively ΠΦ and Πperp

Φare orthogonal projection matrix of Φ(k)

and (sdot)minus represents generalized inverse of matrix (eg a

generalized inverse of matrix S is defined as Sminus such thatSSminusS = S) [27]

The log-likelihood function of (34) is

ln119891 (Y | H0) = minus119872119873

2ln (2120587) minus 119873

2ln 1003816100381610038161003816C0

1003816100381610038161003816

minus1

2tr [Cminus1

0YY119867]

(41)

6 International Journal of Antennas and Propagation

Taking derivative of (41) with respect toC0 andmaking itequal to zero we get MLE of C0

C0 =1

119873YY119867 (42)

Substituting (34) (35) (37) and (42) into (31) the GLRTdetector is

GLR (k) =10038161003816100381610038161003816C0

1003816100381610038161003816100381610038161003816100381610038161003816C1

10038161003816100381610038161003816

=

10038161003816100381610038161003816(1119873)YY11986710038161003816100381610038161003816

100381610038161003816100381610038161003816(1119873) (Y minusWXΦ (k)) (Y minusWXΦ (k))

119867100381610038161003816100381610038161003816

(43)

Since target velocity k is unknown it can be estimatedthrough k = argmaxkGLR(k) andtheGLRTdetector becomes

maxk

GLR (k) = GLR (k)H1

≶

H0

120574 (44)

5 Adaptive Waveform Design

In this section we develop an adaptive waveform designmethod based on maximizing the Mahalanobis-distanceto improve the detection performance Since the targetscattering coefficients vary with different subcarriers wemay change the transmitted weights W accordingly From(30) we know that the measurement Y is distributedas CN119873119872(WXΦ(k)C otimes I119872) under H1 hypothesis andCN119873119872(0C otimes I119872) under H0 hypothesis The squaredMahalanobis-distance is used to distinguish the above twodistributions It is noticeable that the column ofY is uncorre-lated which means that the measurements between differentpulses are independent The sum of squared Mahalanobis-distance over different pulses is given by [28]

1198892=

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898)) (45)

where120573119898 is a positive number and denotes theweight ofMaha-lanobis-distance over the119898th pulse satisfying sum119872minus1

119898=0120573119898 = 1

To maximize the detection performance we can formulatethe optimization as

wopt = arg maxWisinC119873

[

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))]

subject to w119867w = 1

(46)

The scattering coefficients are constant from pulse topulse and the measurement noise is uncorrelated betweendifferent pulses thus 120573119898 is a constant number that is 120573119898 =

1119872 for119898 = 0 1 119872 minus 1 Since

(WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))

= tr (Φ ( 119898)119867X119867W119867Cminus1WXΦ ( 119898))

= tr (Cminus1WXΦ ( 119898)Φ ( 119898)119867X119867W119867

)

(47)

the following theorem [11] is used to simplify the aboveequation

tr (S1AS2A119867) = a119867 [S119879

2⊙ S1] a (48)

where a isin C119873 and A = diag(a) ⊙ denotes element-wiseHadamard product Combining (47) and (48) we get

wopt = arg maxWisinC119873

(1

119872)w119867 [

119872minus1

sum

119898=0

(XΦ ( 119898)Φ ( 119898)119867X119867)

119879

⊙ Cminus1]w

subject to w119867w = 1

(49)

From (49) we know that the optimization problemis reduced to eigenvalue eigenvector problem and wopt isthe eigenvector corresponding to the largest eigenvalue of[sum

119872minus1

119898=0(XΦ( 119898)Φ( 119898)

119867X119867)119879 ⊙ Cminus1]

6 Numerical Results

In this section several numerical examples are presentedto illustrate the performance of proposed detector For

simplicity we consider 2D scenario as shown in Figure 2Thesimulation parameters are shown in Table 1

We use Monte Carlo simulations to realize the followingresults for the analytical expression between false alarm rateand threshold cannot be acquired The relation betweenprobability of false alarm rate and detection threshold corre-sponding to different carrier numbers is shown in Figure 3The curves show that when false alarm rate is kept fixed

International Journal of Antennas and Propagation 7

1

15

2

25

3

35

4

45

5

Det

ectio

n th

resh

old

(Th)

Probability of false alarm (Pfa)10minus3 10minus210minus5 10minus4 10minus1

N = 1

N = 3

N = 5

Figure 3 Detection threshold versus probability of false alarm at different carrier numbers

Table 1 Parameter settings of the simulations

Parameters ValuesHeight of target 200mVelocity of target [1414 1414]119879msScattering coefficient matrix of target X CN(0 1)

Height of radar platform 100mVelocity of radar platform [0 0]119879msRelative distance between radar and target 5 kmMultipath number 119875 3Carrier frequency 1198910 1 GHzBandwidth 119861 100MHzCarrier numbers119873 3Frequency weightsW 1radic119873I119873Pulse number119872 30Pulse repetition interval 119879119903 2ms

the higher the subcarrier number the higher the thresholdSignal-to-noise ratio (SNR) is defined as [11]

SNR =(1119873)sum

119872

119898=1(WX120601 (119898 k))119867WX120601 (119898 k)

tr (C) (50)

The effect of multipath number on the detection perfor-mance is shown in Figure 4 where the path number is setas 119875 = 1 and 119875 = 3 Figures 4(a) and 4(b) show 119875119889 versusSNR with different 119875fas and 119875119889 versus 119875fa when SNR is fixedto minus20 dB respectively The curves show that the detectionperformance is improved with the increasing of multipathnumber Consequently the derived detector with OFDM

waveform could exploit multipath reflections to enhance thedetection performance

In Figure 5 we analyze the effect of subcarrier number onthe performance of proposed detector where the subcarriernumber is set as 119873 = 1 3 and 5 respectively Figures 5(a)and 5(b) show 119875119889 versus SNR when 119875fa = 10

minus3 and 119875119889

versus 119875fa when SNR is fixed to minus20 dB respectively Thedetection performance is improved with the increasing ofsubcarrier number The results demonstrate that the detec-tion performance is improved due to the frequency diversityin an OFDM radar system

Figure 6 shows that the detection performance varieswith the target height which is set as 200m 400m and600m respectively Figure 6(a) shows 119875119889 versus SNR when119875fa = 10

minus3 while Figure 6(b) shows 119875119889 versus 119875fa when SNRis fixed to minus20 dB The results demonstrate that detectionperformance decreases with the increasing of target height inlow grazing angle scenario This can be explained that as thetarget height increases the grazing angle gets larger and thecomplex reflection coefficient over indirect paths becomessmaller which is shown in Figure 7 Thus the detectionperformance deteriorates

In Figure 8 we analyze the effect of varying the directionsof target velocity vector on the detection performance Herek119879 = [100 0] [717 717] and [0 100]ms respectivelyFigures 8(a) and 8(b) show 119875119889 versus SNR when 119875fa = 10

minus3

and 119875119889 versus 119875fa when SNR is fixed to minus20 dB respectivelyIn this paper we take advantage of multipath propagationby exploiting multiple Doppler shifts corresponding to theprojections of target velocity on each of the multipathcomponents When the angle between the target velocityvector and radar LOS decreases the Doppler frequency of

8 International Journal of Antennas and Propagation

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1 Pfa = 10minus3

P = 3 Pfa = 10minus3P = 1 Pfa = 10minus5

P = 3 Pfa = 10minus5

(a) 119875119889 versus SNR with different 119875fas

02

03

04

05

06

07

08

09

1

Probability of false alarm (Pfa)10minus3 10minus2 10minus1 100

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1

P = 3

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 4 Effect of the path number on the detection performance

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 1

N = 3

N = 5

(a) 119875119889 versus SNR when 119875fa = 10minus3

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 1

N = 3

N = 5

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 5 Effect of the subcarrier number on the detection performance

received signal will increaseThus the detection performanceis improved

In Figure 9 we show the detection performance improve-ment due to the adaptive waveform design Figure 9(a) shows119875119889 versus SNR when 119875fa = 10

minus3 while Figure 9(b) shows 119875119889versus 119875fa when SNR is fixed to minus20 dB We assume that thetransmitted waveform in the first119872 pulses has equal weightsthat is w = 1radic119873I Then based on (49) we compute woptfor the next 119872 pulses As is shown compared with fixed

waveform the detection performance is improved due to theadaptive waveform method proposed in this paper

Finally to validate our method in non-Gaussian scenar-ios we conduct additional experiments using compound K-distribution whose PDF is [29]

119891 (119911) =radic2V120583Γ (V)

(radic2V120583119911)

V

119870Vminus1(radic2V120583119911) (51)

International Journal of Antennas and Propagation 9

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

ht = 200mht = 400mht = 600m

(a) 119875119889 versus SNR when 119875fa = 10minus3

01

02

03

04

05

06

07

08

09

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

ht = 200mht = 400mht = 600m

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 6 Detection performance varies with target height

0 500 1000 1500 20000

02

04

06

08

1

Am

plitu

de o

f com

plex

refle

ctio

n co

effici

ent

Horizontal polarizationVertical polarization

Target height m

Figure 7 Amplitude of complex reflection coefficient varies with target height

and the texture follows a gamma distribution with PDF

119891 (119906) =1

Γ (V)(V120583)

V

119906Vminus1

119890minusV120583119906

120581 (119906) (52)

where Γ(sdot) is the Eulerian gamma function 120581(sdot) is the unitstep function and 119870Vminus1(sdot) is the modified Bessel function ofsecond kind with order V minus 1 120583 and V are the scale and shapeparameter respectively The speckle component is assumed

to be a Gaussian distribution with exponential correlationstructure covariance matrix [30]

[C]119894119895 = 120588|119894minus119895|

for 119894 119895 = 0 1 119873 minus 1 (53)

where 120588 is one-lag correlation coefficient The other parame-ters are the same as in Table 1

Figure 10 shows the detection performance with differentpath numbers (119875 = 1 3) in Gaussian and compound

10 International Journal of Antennas and Propagation

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 8 Effect of different directions of target velocity on detection performance

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 9 Comparison of detection performance with fixed waveform and adaptive waveform

K-distribution clutter and Figure 11 shows the detectionperformance with fixed and adaptive waveform design inGaussian and compound K-distribution clutter We find thatin compound K-distribution the detector derived underGaussian assumption also yields improved performance bytaking advantage of multipath effect and adaptive waveformdesign of OFDM radar However the performance gap existsdue to the model mismatch

7 Conclusions

In this paper we address the problem of target detectionin low grazing angle scenario with OFDM radar We con-sider the realistic physical and statistical effects to makethe propagation model more realistic Based on the char-acteristics of OFDM signal we develop a GLRT detectorand show that detection performance is improved due tothe multipath utilization and frequency diversity of OFDM

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

2 International Journal of Antennas and Propagation

Radar

Sealand surface

R1

Rd

R2

Target

Image

Figure 1 Representative scenario of multipath propagation in lowgrazing angle

detection performance may be enhanced with OFDM radarin low grazing angle scenario by utilizing multipath reflec-tions

When detecting target in low grazing angle scenarioradar measurements are affected by many factors such asever-changing meteorological conditions in the troposphereEarthrsquos curvature and roughness of groundsea surface [16]All these factors will affect the detection performance andtherefore these factors should be taken into consideration tomake the signal model more realistic

This paper deals with target detection in low grazingangle scenario with OFDM radar To make the propagationmodel more accurate refraction of the lower atmosphere andthe Earthrsquos curvature are taken into consideration and themultipath propagation model is modified accordingly Basedon the characteristics of OFDM radar we derive a GLRTdetector in Gaussian clutter environment Then a waveformdesign method which optimizes the transmitted subcarrierweights is proposed to improve the detection performanceFinally the performance of the proposed detector is analyzedand discussed via simulation experiments

2 Modified Multipath Propagation Model

Multipath effect is one of the main problems when detectingand tracking target in low grazing angle scenario The echosignals received by radar receiver not only include directsignals but also include indirect signals Thus the receivedsignals are the sum of reflected signals along different pathsand the amplitudes fluctuate acutely due to the random phasevariations which deteriorates the performances of targetdetection and tracking Generally speaking signals reflectedmore than twice will be attenuated heavily which can alwaysbe neglected

The representative scenario of multipath propagation inlow grazing angle is shown in Figure 1 Target locates at adistance 119877119889 from the radar The source is assumed to be anarrowband signal which can be represented as [16]

119909 (119905) = 119886119890119895(120596119905+120585)

(1)

where 119886 120596 and 120585 denote the amplitude angular frequencyand initial phase respectively In the presence of multipath

the received signals consist of two components namely thedirect and indirect signals Direct signal is given by

119909119889 (119905) = 119909 (119905) 119890minus119895(2120587120582)(2119877119889) (2)

and the indirect signal is

119909119894 (119905) = 119909 (119905) 120588119890119895120593119890minus119895(2120587120582)119877119894 (3)

In (3) 120588119890119895120593 is the complex reflection coefficient and 119877119894

is the total length of indirect path For first-order reflectedsignals 119877119894 = 1198771 + 1198772 + 119877119889 and for second-order reflectedsignals 119877119894 = 2(1198771 + 1198772) Received signal can be representedas

119909119903 (119905) = 119909119889 (119905) + 119909119894 (119905)

= 119909 (119905) 119890119895(2120587120582)(2119877119889) (1 + 120588119890

119895(120593+(2120587120582)Δ119877))

(4)

whereΔ119877 = 119877119894minus2119877119889 In (4) the amplitudes of received signalsare dependent on the factor120588119890119895(120593+(2120587120582)Δ119877) which includes theeffects of complex reflection coefficient 120588119890119895120593 wavelength 120582and path difference Δ119877

In normal atmospheric conditions the pressure decreasesexponentially with height which causes a reduction in therefractivity with respect to height Under this condition aradio ray will diffract downward [17] Furthermore in mar-itime environment evaporation duct effect may be produceddue to the strong humidity gradients above (within firstfew meters) the air-sea boundary [18] which makes a radioray bend downward with a curvature more than the Earthrsquosradius The effect is dependent on a few factors such astemperature difference between the air and sea and the windspeed [19]

In addition to the atmospheric effects low grazing anglepropagation is also affected by the fact that the Earth iscurved The curvature of the Earth decreases the path lengthdifference between the direct and reflected waves and italso reduces the amplitudes of the reflected waves [19] Thisproblem is usually dealt with by replacing the Earth with animaginary flat Earth whose equivalent radius is

119877119890 = 1198770 (1 + 1198770

119889119873

119889ℎ10minus6)

minus1

(5)

where1198770 is the radius of actual Earth and119889119873119889ℎ is the refrac-tivity gradient For standard atmosphere minus79 le 119889119873119889ℎ le

0N-unitskm [19]Considering all the factors mentioned above and based

on ideal propagation model described in Figure 1 modifiedmultipath propagation model is shown in Figure 2 Theheight of radar and target is ℎ119903 and ℎ119905 respectively Theground distance separated by radar and target is 119903 ℎ1015840

119903and ℎ

1015840

119905

are the modified heights corresponding to radar and targetover flat-Earthmodel 1199031 is the grounddistance between radarand the reflected point and 120595 is the grazing angle

In order to calculate complex reflection coefficient in (4)we have to solve the grazing angle 120595 and Δ119877 first Thesevariables are all related to 1199031 which can be evaluated usingthe following cubic equation [20]

21199033

1minus 3119903119903

2

1+ [119903

2minus 2119877119890 (ℎ119903 + ℎ119905)] 1199031 + 2119877119890ℎ119903119903 = 0 (6)

International Journal of Antennas and Propagation 3

RadarTarget

hr

h998400r

r1 r2

ReRe120601

1206011 1206012

O

ht

h998400t

Tangent planeR1

R2

r

120595

Rd

Figure 2 Modified multipath propagation model

Then we solve ℎ1015840119903and ℎ1015840

119905by

ℎ1015840

119903≃ ℎ119903 minus

1199032

1

2119877119890

ℎ1015840

119905≃ ℎ119905 minus

(119903 minus 1199031)2

2119877119890

(7)

Meanwhile we can get 1206011 = 1199031119877119890 and 1206012 = (119903minus1199031)119877119890 Usinglaw of cosine yields

1198771 =radic1198772

119890+ (119877119890 + ℎ119903)

2minus 2119877119890 (119877119890 + ℎ119903) cos1206011

1198772 =radic1198772

119890+ (119877119890 + ℎ119905)

2minus 2119877119890 (119877119890 + ℎ119905) cos1206012

119877119889

= radic(119877119890 + ℎ119903)2+ (119877119890 + ℎ119905)

2minus 2 (119877119890 + ℎ119903) (119877119890 + ℎ119905) cos120601

(8)

Finally the grazing angle 120595 is given by

120595 =1

2(120587 minus arccos(

1198772

1+ 119877

2

2minus 119877

2

119889

211987711198772

)) (9)

The complex reflection coefficient can be calculated by[20]

120588119890119895120593

= Γ(]ℎ)119863119878 (10)

where Γ(]ℎ) is vertical polarization or horizontal polarizationreflection coefficient for a plane surface 119863 is the divergencefactor due to a curved surface and 119878 is root-mean-squared(RMS) specular scattering coefficient which represents theroughness of surface

Γ(]ℎ) is determined by frequency complex dielectricconstant and grazing angle120595 which can be calculated by [21]

ΓV ≃sin120595radic120576119888 minus 1

sin120595radic120576119888 + 1(11)

for vertical polarization and

Γℎ ≃sin120595 minus radic120576119888

sin120595 + radic120576119888

(12)

for horizontal polarization where 120576119888 is complex dielectricconstant which is given by 120576119888 = 1205761205760 minus 11989560120582120590 1205761205760 is relativedielectric constant of the reflecting medium and 120590 is itsconductivity

When the electromagnetic wave is incident on the surfaceof the Earth due to the slight differences in each incidentray the reflected wave is diverged and the reflected energyis defocusedWhen this happens radar power density will bereduced If the grazing angle is not large divergence factor119863can be approximated by [21]

119863 ≃ (1 +211990311199032

119877119890119903120595)

minus12

(13)

Due to the reflection of rough surface two componentswill be generated a diffuse component and a coherent com-ponent with reducedmagnitudeThe reduction in themagni-tude of the coherent component brought about by reflectionsfrom a rough surface is related to the grazing angle and thesignal wavelength which is [21]

119878 = 119890minus120583 (14)

where

120583 =

2 [2120587120578]2

120578 le 01 rad

0161205782+ 742120578 + 00468 otherwise

(15)

and 120578 is the surface roughness factor defined as 120578 = 120590119867120595120582120590119867 is the RMS of reflection surface and 120590119867120582 denotes theroughness of reflected surface The larger 120590119867120582 is the moreroughness the surface will be For smooth surface 120590119867120582 isapproximated to be zero

Substituting (11)ndash(15) into (10) we can get the complexreflection coefficient 120588119890119895120593 At the same time 120588119890119895(120593+(2120587120582)Δ119877)can be evaluated which is the impact of multipath propaga-tion on signal model

3 Measurement Model of OFDM Radar

We consider a monostatic radar employing an OFDM signal-ing system The transmitted signal can be described as

119904 (119905) =

119872minus1

sum

119898=0

119906 (119905 minus 119898119879119903) 11989011989521205871198910119905 (16)

where 1198910 119872 and 119879119903 represent the carrier frequency pulsenumber and pulse repetition interval (PRI) respectively 119906(119905)is the complex envelop of a single pulse which is given by

119906 (119905) =

119873minus1

sum

119899=0

119908119899rect (119905 119905119887) 1198901198952120587119899Δ119891119905

(17)

where

rect (119905 119905119887) =

1 0 le 119905 le 119905119887

0 otherwise(18)

4 International Journal of Antennas and Propagation

and 119873 is the subcarrier number w = [1199080 1199081 119908119873minus1]119879

represent the complex weights transmitted over differentsubcarriers and satisfying sum

119873minus1

119899=0|119908119899|

2= 1 For the sake

of keeping orthogonality between different subcarriers thesubcarrier spacing Δ119891 and time duration 119905119887 of a single pulseshould satisfy Δ119891119905119887 = 1 The total bandwidth is 119861 = 119873Δ119891

Assuming that the received signals contain 119875 differentpaths and the roundtrip delay corresponding to 119901th path is120591119901 119901 = 0 1 119875 minus 1 The received signal is

119910 (119905) =

119875minus1

sum

119901=0

119904 (120574119901 (119905 minus 120591119901)) (19)

where 120574119901 = 1+120573119901 accounts for the stretching or compressingin time of the reflected signal and 120573119901 = 2⟨k u119901⟩119888 representsthe Doppler spreading factor corresponding to 119901th path kand u119901 are the target velocity and unit direction of arrival(DOA) vector 119888 is the propagation speed Substituting (16)into (19) the received signal after demodulation is denotedby

119910 (119905)

=

119875minus1

sum

119901=0

119872minus1

sum

119898=0

119906 (120574119901 (119905 minus 120591119901) minus 119898119879119903) 11989011989521205871198910120573119901(119905minus1205910)119890

minus119895212058711989101205910

+ 119890 (119905)

(20)

Then substituting (17) into (20) we obtain

119910 (119905) =

119872minus1

sum

119898=0

119873minus1

sum

119899=0

119875minus1

sum

119901=0

119908119899119909119899119901rect (120574119901 (119905 minus 120591119901) minus 119898119879119903 119905119887)

sdot 1198901198952120587119899Δ119891(120574119901(119905minus120591119901)minus119898119879119903)119890

11989521205871198910120573119901(119905minus1205910)119890minus119895212058711989101205910 + 119890 (119905)

(21)

where 119909119899119901 represents the complex scattering coefficient oftarget corresponding to 119899th carrier and 119901th path 1205910 isthe roundtrip delay corresponding to the range cell underconsideration The relative time gaps between different pathsare very small compared to the actual roundtrip delay whichmeans 120591119901 asymp 1205910 for119901 = 0 1 119875minus1 Substituting 119905 = 1205910+119898119879119903

(119898 = 0 1 119872 minus 1) into (21) we obtain

119910 (119899119898) =

119875minus1

sum

119901=0

119908119899119909119899119901120601119899119901 (119898 V) + 119890 (119899119898) (22)

where

120601119899119901 (119898 V) = 119890minus119895212058711989101205910119890

1198952120587119891119899120573119901119898119879119903 (23)

and 119891119899 = 1198910 + 119899Δ119891 Equation (22) can be written into matrixform

y (119898) = WXΦ (119898 k) + e (119898) (24)

where

(i) y(119898) = [119910(0119898) 119910(1119898) 119910(119873minus1119898)]119879 is an119873times1

vector that represents output of the119898th pulse

(ii) W = diag(w) is an 119873 times 119873 complex diagonal matrixthat contains transmitted subcarrier weights

(iii) X = blkdiag (x1198790 x1198791 x119879

119873minus1) is an119873 times119873119875 complex

rectangular block-diagonal matrix x119899 = 119909119899 sdot 120588 119899 =

0 1 119873minus1 where 119909119899 is target scattering coefficienton the 119899th subcarrier and 120588 = [1205880 1205881 120588119875minus1]

119879

represents complex reflection coefficients over differ-ent paths for direct signals 120588119901 = 1 for first-orderreflected signals120588119901 = 120588119890

119895120593 for second-order reflectedsignals 120588119901 = (120588119890

119895120593)2 where 120588119890119895120593 is calculated by (10)

(iv) Φ(119898 k) = [1206010(119898 k) 1206011(119898 k) 120601119873minus1(119898 k)]119879 is an119873119875 times 1 complex vector and 120601119899(119898 k) = [1206011198990(119898 k)1206011198991(119898 k) 120601119899119875minus1(119898 k)]119879 119899 = 0 1 119873 minus 1

(v) e(119898) = [1198900(119898) 1198901(119898) 119890119873minus1(119898)]119879is an119873times 1 com-

plex vector of clutter

Then concatenating all temporal data into119873 times119872matrixthe OFDMmeasurement model in low grazing angle is

Y = WXΦ (k) + E (25)

where

Y = [y (0) y (1) y (119872 minus 1)] (26)

Φ(V) = [Φ(0 V)Φ(1 V) Φ(119872 minus 1 V)] is an 119873119875 times 119872

complex matrix that contains Doppler information over 119872pulses

E = [e(0) e(1) e(119872minus1)] is an119873times119872 complexmatrixthat consists of clutter returns

In low grazing angle scenario the clutter is usuallymodeled as compound-Gaussian model which is the productof speckle and texture [22] The speckle is fast-changing andmodeled as a complex Gaussian process while the texture isslow-changing and modeled as a nonnegative process [23]

In this paper the correlation length of the texture isassumed to be on the order of seconds and we only considerone coherent processing interval (CPI) which is assumedto be 60ms Due to the long correlation time of textureit is considered to be constant within each CPI changingaccording to a given probability density function (PDF) fromone CPI to the next Thus conditioned on a given value ofthe texture the clutter is simplified to Gaussian distributionHowever this simplification is not justified when predictingthe clutterrsquos behavior in time intervals larger than a CPI suchas clutter cancellation and constant false alarm rate (CFAR)detection

Thus we assume that the clutter is temporally whiteand circularly zero-mean complex Gaussian process withunknown positive definite covariance C The measurementsover different pulses are supposed to be independent whichmeans

vec (Y) sim CN119873119872 (vec (WXΦ (k)) C otimes I119872) (27)

whereotimes denotes Kronecker product Accordingly theOFDMmeasurements are distributed as

Y sim CN119873119872 (WXΦ (k) C otimes I119872) (28)

International Journal of Antennas and Propagation 5

4 Detection Test

In low grazing angle scenario received signals are the sumof signals along different paths which form the complicatedmeasurements The essence of detection is to judge whethera target is present or not in the range cell under test This isa classical two-hypothesis detection problem Therefore weconstruct a decision problem to choose between two possiblehypotheses the null hypothesis (target-free hypothesis) andthe alternate hypothesis (target-present hypothesis) whichcan be expressed as

H0 X = 0 C unknown

H1 X = 0 kC unknown(29)

ThemeasurementsY in two hypotheses are distributed as

Y

sim

H0 CN119873119872 (0C otimes I119872) C unknown

H1 CN119873119872 (WXΦ (k) C otimes I119872) kC unknown

(30)

According to the classical target detection theoryNeyman-Pearson (NP) detector is the optimal detector whichmaximizes the probability of detection at a constant prob-ability of false alarm However the target velocity k andclutter covarianceC are unknown andGLRT detector is usedinstead where the unknown parameters are replaced withtheir maximum likelihood estimates (MLEs) The formula-tion of GLRT is [24]

GLR (k) =119891 (Y | H1 k X C1)

119891 (Y | H0 C0)

H1

≶

H0

120574 (31)

where 119891(Y | H1) and 119891(Y | H0) are the likelihood functionsunder H1 and H0 respectively 120574 is the threshold that thedetector is compared with X and C1 are the MLE of X and CunderH1 while C0 is the MLE of C underH0

Definition 1 If the random matrix Y(119873 times 119872) is distributedas a matrix variate normal distribution (MVND) with meanM(119873 times 119872) and covariance Ω otimes Σ we use the notation Y sim

CN119873119872(MΩotimesΣ) and the probability density function (PDF)is given by [25]

119891 (Y)

=

exp (minus (12) tr [Ωminus1 (Y minusM)Σminus1(Y minusM)

119867])

(2120587)1198731198722

|Ω|1198732

|Σ|1198722

(32)

where tr(sdot) is the trace of matrixΩ and Σ are positive definitecovariance matrices over the column and row of Y Ω is an119873 times 119873 matrix and Σ is an 119872 times119872 matrix which are definedas [25]

Ω = 119864 [(Y minusM) (Y minusM)119867]

Σ = 119864 [(Y minusM)119867(Y minusM)]

(33)

According to the above descriptions we can derive PDFsof the measurements under two hypotheses

119891 (Y | H0) =

exp (minus (12) tr [Cminus10YY119867])

(2120587)1198731198722 1003816100381610038161003816C0

1003816100381610038161003816

1198732 (34)

119891 (Y | H1)

=

exp (minus (12) tr [Cminus11(Y minusWXΦ (k)) (Y minusWXΦ (k))119867])

(2120587)1198731198722 1003816100381610038161003816C1

1003816100381610038161003816

1198732

(35)

The log-likelihood function of (35) is

ln119891 (Y | H1)

= minus119872119873

2ln (2120587) minus 119873

2ln 1003816100381610038161003816C1

1003816100381610038161003816

minus1

2tr [Cminus1

1(Y minusWXΦ (k)) (Y minusWXΦ (k))119867]

(36)

Taking derivative of (36) with respect to C1 and makingit equal to zero the MLE of C1 is

C1 =1

119873(Y minusWXΦ (k)) (Y minusWXΦ (k))119867 (37)

Substituting (37) into (36) yields

XML

= argminX

ln 10038161003816100381610038161003816(Y minusWXΦ (k)) (Y minusWXΦ (k))11986710038161003816100381610038161003816 (38)

Usually the scattering matrix X does not yield a close-form MLE expression However noting that X has a block-diagonal structure it turns out to be a block-diagonal growthcurve (BDGC) problem Reference [26] derived the approx-imate maximum likelihood (AML) estimator for X which isgiven by

xAML = vecb (X) = (Γ119867Γ)minus1

Γ119867vec (Tminus12XΠΦ) (39)

where

Γ ≜ Φ119879⊛ (T12W)

T ≜ YΠperpΦY119867

ΠΦ ≜ Φ (k)119867 (Φ (k)Φ (k)119867)minus

Φ (k)

Πperp

Φ≜ I minusΠΦ

(40)

vecb(sdot) and ⊛ denote block-diagonal matrix vectorizationoperator and generalized Khatri-Rao product [27] respec-tively ΠΦ and Πperp

Φare orthogonal projection matrix of Φ(k)

and (sdot)minus represents generalized inverse of matrix (eg a

generalized inverse of matrix S is defined as Sminus such thatSSminusS = S) [27]

The log-likelihood function of (34) is

ln119891 (Y | H0) = minus119872119873

2ln (2120587) minus 119873

2ln 1003816100381610038161003816C0

1003816100381610038161003816

minus1

2tr [Cminus1

0YY119867]

(41)

6 International Journal of Antennas and Propagation

Taking derivative of (41) with respect toC0 andmaking itequal to zero we get MLE of C0

C0 =1

119873YY119867 (42)

Substituting (34) (35) (37) and (42) into (31) the GLRTdetector is

GLR (k) =10038161003816100381610038161003816C0

1003816100381610038161003816100381610038161003816100381610038161003816C1

10038161003816100381610038161003816

=

10038161003816100381610038161003816(1119873)YY11986710038161003816100381610038161003816

100381610038161003816100381610038161003816(1119873) (Y minusWXΦ (k)) (Y minusWXΦ (k))

119867100381610038161003816100381610038161003816

(43)

Since target velocity k is unknown it can be estimatedthrough k = argmaxkGLR(k) andtheGLRTdetector becomes

maxk

GLR (k) = GLR (k)H1

≶

H0

120574 (44)

5 Adaptive Waveform Design

In this section we develop an adaptive waveform designmethod based on maximizing the Mahalanobis-distanceto improve the detection performance Since the targetscattering coefficients vary with different subcarriers wemay change the transmitted weights W accordingly From(30) we know that the measurement Y is distributedas CN119873119872(WXΦ(k)C otimes I119872) under H1 hypothesis andCN119873119872(0C otimes I119872) under H0 hypothesis The squaredMahalanobis-distance is used to distinguish the above twodistributions It is noticeable that the column ofY is uncorre-lated which means that the measurements between differentpulses are independent The sum of squared Mahalanobis-distance over different pulses is given by [28]

1198892=

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898)) (45)

where120573119898 is a positive number and denotes theweight ofMaha-lanobis-distance over the119898th pulse satisfying sum119872minus1

119898=0120573119898 = 1

To maximize the detection performance we can formulatethe optimization as

wopt = arg maxWisinC119873

[

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))]

subject to w119867w = 1

(46)

The scattering coefficients are constant from pulse topulse and the measurement noise is uncorrelated betweendifferent pulses thus 120573119898 is a constant number that is 120573119898 =

1119872 for119898 = 0 1 119872 minus 1 Since

(WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))

= tr (Φ ( 119898)119867X119867W119867Cminus1WXΦ ( 119898))

= tr (Cminus1WXΦ ( 119898)Φ ( 119898)119867X119867W119867

)

(47)

the following theorem [11] is used to simplify the aboveequation

tr (S1AS2A119867) = a119867 [S119879

2⊙ S1] a (48)

where a isin C119873 and A = diag(a) ⊙ denotes element-wiseHadamard product Combining (47) and (48) we get

wopt = arg maxWisinC119873

(1

119872)w119867 [

119872minus1

sum

119898=0

(XΦ ( 119898)Φ ( 119898)119867X119867)

119879

⊙ Cminus1]w

subject to w119867w = 1

(49)

From (49) we know that the optimization problemis reduced to eigenvalue eigenvector problem and wopt isthe eigenvector corresponding to the largest eigenvalue of[sum

119872minus1

119898=0(XΦ( 119898)Φ( 119898)

119867X119867)119879 ⊙ Cminus1]

6 Numerical Results

In this section several numerical examples are presentedto illustrate the performance of proposed detector For

simplicity we consider 2D scenario as shown in Figure 2Thesimulation parameters are shown in Table 1

We use Monte Carlo simulations to realize the followingresults for the analytical expression between false alarm rateand threshold cannot be acquired The relation betweenprobability of false alarm rate and detection threshold corre-sponding to different carrier numbers is shown in Figure 3The curves show that when false alarm rate is kept fixed

International Journal of Antennas and Propagation 7

1

15

2

25

3

35

4

45

5

Det

ectio

n th

resh

old

(Th)

Probability of false alarm (Pfa)10minus3 10minus210minus5 10minus4 10minus1

N = 1

N = 3

N = 5

Figure 3 Detection threshold versus probability of false alarm at different carrier numbers

Table 1 Parameter settings of the simulations

Parameters ValuesHeight of target 200mVelocity of target [1414 1414]119879msScattering coefficient matrix of target X CN(0 1)

Height of radar platform 100mVelocity of radar platform [0 0]119879msRelative distance between radar and target 5 kmMultipath number 119875 3Carrier frequency 1198910 1 GHzBandwidth 119861 100MHzCarrier numbers119873 3Frequency weightsW 1radic119873I119873Pulse number119872 30Pulse repetition interval 119879119903 2ms

the higher the subcarrier number the higher the thresholdSignal-to-noise ratio (SNR) is defined as [11]

SNR =(1119873)sum

119872

119898=1(WX120601 (119898 k))119867WX120601 (119898 k)

tr (C) (50)

The effect of multipath number on the detection perfor-mance is shown in Figure 4 where the path number is setas 119875 = 1 and 119875 = 3 Figures 4(a) and 4(b) show 119875119889 versusSNR with different 119875fas and 119875119889 versus 119875fa when SNR is fixedto minus20 dB respectively The curves show that the detectionperformance is improved with the increasing of multipathnumber Consequently the derived detector with OFDM

waveform could exploit multipath reflections to enhance thedetection performance

In Figure 5 we analyze the effect of subcarrier number onthe performance of proposed detector where the subcarriernumber is set as 119873 = 1 3 and 5 respectively Figures 5(a)and 5(b) show 119875119889 versus SNR when 119875fa = 10

minus3 and 119875119889

versus 119875fa when SNR is fixed to minus20 dB respectively Thedetection performance is improved with the increasing ofsubcarrier number The results demonstrate that the detec-tion performance is improved due to the frequency diversityin an OFDM radar system

Figure 6 shows that the detection performance varieswith the target height which is set as 200m 400m and600m respectively Figure 6(a) shows 119875119889 versus SNR when119875fa = 10

minus3 while Figure 6(b) shows 119875119889 versus 119875fa when SNRis fixed to minus20 dB The results demonstrate that detectionperformance decreases with the increasing of target height inlow grazing angle scenario This can be explained that as thetarget height increases the grazing angle gets larger and thecomplex reflection coefficient over indirect paths becomessmaller which is shown in Figure 7 Thus the detectionperformance deteriorates

In Figure 8 we analyze the effect of varying the directionsof target velocity vector on the detection performance Herek119879 = [100 0] [717 717] and [0 100]ms respectivelyFigures 8(a) and 8(b) show 119875119889 versus SNR when 119875fa = 10

minus3

and 119875119889 versus 119875fa when SNR is fixed to minus20 dB respectivelyIn this paper we take advantage of multipath propagationby exploiting multiple Doppler shifts corresponding to theprojections of target velocity on each of the multipathcomponents When the angle between the target velocityvector and radar LOS decreases the Doppler frequency of

8 International Journal of Antennas and Propagation

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1 Pfa = 10minus3

P = 3 Pfa = 10minus3P = 1 Pfa = 10minus5

P = 3 Pfa = 10minus5

(a) 119875119889 versus SNR with different 119875fas

02

03

04

05

06

07

08

09

1

Probability of false alarm (Pfa)10minus3 10minus2 10minus1 100

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1

P = 3

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 4 Effect of the path number on the detection performance

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 1

N = 3

N = 5

(a) 119875119889 versus SNR when 119875fa = 10minus3

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 1

N = 3

N = 5

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 5 Effect of the subcarrier number on the detection performance

received signal will increaseThus the detection performanceis improved

In Figure 9 we show the detection performance improve-ment due to the adaptive waveform design Figure 9(a) shows119875119889 versus SNR when 119875fa = 10

minus3 while Figure 9(b) shows 119875119889versus 119875fa when SNR is fixed to minus20 dB We assume that thetransmitted waveform in the first119872 pulses has equal weightsthat is w = 1radic119873I Then based on (49) we compute woptfor the next 119872 pulses As is shown compared with fixed

waveform the detection performance is improved due to theadaptive waveform method proposed in this paper

Finally to validate our method in non-Gaussian scenar-ios we conduct additional experiments using compound K-distribution whose PDF is [29]

119891 (119911) =radic2V120583Γ (V)

(radic2V120583119911)

V

119870Vminus1(radic2V120583119911) (51)

International Journal of Antennas and Propagation 9

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

ht = 200mht = 400mht = 600m

(a) 119875119889 versus SNR when 119875fa = 10minus3

01

02

03

04

05

06

07

08

09

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

ht = 200mht = 400mht = 600m

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 6 Detection performance varies with target height

0 500 1000 1500 20000

02

04

06

08

1

Am

plitu

de o

f com

plex

refle

ctio

n co

effici

ent

Horizontal polarizationVertical polarization

Target height m

Figure 7 Amplitude of complex reflection coefficient varies with target height

and the texture follows a gamma distribution with PDF

119891 (119906) =1

Γ (V)(V120583)

V

119906Vminus1

119890minusV120583119906

120581 (119906) (52)

where Γ(sdot) is the Eulerian gamma function 120581(sdot) is the unitstep function and 119870Vminus1(sdot) is the modified Bessel function ofsecond kind with order V minus 1 120583 and V are the scale and shapeparameter respectively The speckle component is assumed

to be a Gaussian distribution with exponential correlationstructure covariance matrix [30]

[C]119894119895 = 120588|119894minus119895|

for 119894 119895 = 0 1 119873 minus 1 (53)

where 120588 is one-lag correlation coefficient The other parame-ters are the same as in Table 1

Figure 10 shows the detection performance with differentpath numbers (119875 = 1 3) in Gaussian and compound

10 International Journal of Antennas and Propagation

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 8 Effect of different directions of target velocity on detection performance

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 9 Comparison of detection performance with fixed waveform and adaptive waveform

K-distribution clutter and Figure 11 shows the detectionperformance with fixed and adaptive waveform design inGaussian and compound K-distribution clutter We find thatin compound K-distribution the detector derived underGaussian assumption also yields improved performance bytaking advantage of multipath effect and adaptive waveformdesign of OFDM radar However the performance gap existsdue to the model mismatch

7 Conclusions

In this paper we address the problem of target detectionin low grazing angle scenario with OFDM radar We con-sider the realistic physical and statistical effects to makethe propagation model more realistic Based on the char-acteristics of OFDM signal we develop a GLRT detectorand show that detection performance is improved due tothe multipath utilization and frequency diversity of OFDM

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

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International Journal of

International Journal of Antennas and Propagation 3

RadarTarget

hr

h998400r

r1 r2

ReRe120601

1206011 1206012

O

ht

h998400t

Tangent planeR1

R2

r

120595

Rd

Figure 2 Modified multipath propagation model

Then we solve ℎ1015840119903and ℎ1015840

119905by

ℎ1015840

119903≃ ℎ119903 minus

1199032

1

2119877119890

ℎ1015840

119905≃ ℎ119905 minus

(119903 minus 1199031)2

2119877119890

(7)

Meanwhile we can get 1206011 = 1199031119877119890 and 1206012 = (119903minus1199031)119877119890 Usinglaw of cosine yields

1198771 =radic1198772

119890+ (119877119890 + ℎ119903)

2minus 2119877119890 (119877119890 + ℎ119903) cos1206011

1198772 =radic1198772

119890+ (119877119890 + ℎ119905)

2minus 2119877119890 (119877119890 + ℎ119905) cos1206012

119877119889

= radic(119877119890 + ℎ119903)2+ (119877119890 + ℎ119905)

2minus 2 (119877119890 + ℎ119903) (119877119890 + ℎ119905) cos120601

(8)

Finally the grazing angle 120595 is given by

120595 =1

2(120587 minus arccos(

1198772

1+ 119877

2

2minus 119877

2

119889

211987711198772

)) (9)

The complex reflection coefficient can be calculated by[20]

120588119890119895120593

= Γ(]ℎ)119863119878 (10)

where Γ(]ℎ) is vertical polarization or horizontal polarizationreflection coefficient for a plane surface 119863 is the divergencefactor due to a curved surface and 119878 is root-mean-squared(RMS) specular scattering coefficient which represents theroughness of surface

Γ(]ℎ) is determined by frequency complex dielectricconstant and grazing angle120595 which can be calculated by [21]

ΓV ≃sin120595radic120576119888 minus 1

sin120595radic120576119888 + 1(11)

for vertical polarization and

Γℎ ≃sin120595 minus radic120576119888

sin120595 + radic120576119888

(12)

for horizontal polarization where 120576119888 is complex dielectricconstant which is given by 120576119888 = 1205761205760 minus 11989560120582120590 1205761205760 is relativedielectric constant of the reflecting medium and 120590 is itsconductivity

When the electromagnetic wave is incident on the surfaceof the Earth due to the slight differences in each incidentray the reflected wave is diverged and the reflected energyis defocusedWhen this happens radar power density will bereduced If the grazing angle is not large divergence factor119863can be approximated by [21]

119863 ≃ (1 +211990311199032

119877119890119903120595)

minus12

(13)

Due to the reflection of rough surface two componentswill be generated a diffuse component and a coherent com-ponent with reducedmagnitudeThe reduction in themagni-tude of the coherent component brought about by reflectionsfrom a rough surface is related to the grazing angle and thesignal wavelength which is [21]

119878 = 119890minus120583 (14)

where

120583 =

2 [2120587120578]2

120578 le 01 rad

0161205782+ 742120578 + 00468 otherwise

(15)

and 120578 is the surface roughness factor defined as 120578 = 120590119867120595120582120590119867 is the RMS of reflection surface and 120590119867120582 denotes theroughness of reflected surface The larger 120590119867120582 is the moreroughness the surface will be For smooth surface 120590119867120582 isapproximated to be zero

Substituting (11)ndash(15) into (10) we can get the complexreflection coefficient 120588119890119895120593 At the same time 120588119890119895(120593+(2120587120582)Δ119877)can be evaluated which is the impact of multipath propaga-tion on signal model

3 Measurement Model of OFDM Radar

We consider a monostatic radar employing an OFDM signal-ing system The transmitted signal can be described as

119904 (119905) =

119872minus1

sum

119898=0

119906 (119905 minus 119898119879119903) 11989011989521205871198910119905 (16)

where 1198910 119872 and 119879119903 represent the carrier frequency pulsenumber and pulse repetition interval (PRI) respectively 119906(119905)is the complex envelop of a single pulse which is given by

119906 (119905) =

119873minus1

sum

119899=0

119908119899rect (119905 119905119887) 1198901198952120587119899Δ119891119905

(17)

where

rect (119905 119905119887) =

1 0 le 119905 le 119905119887

0 otherwise(18)

4 International Journal of Antennas and Propagation

and 119873 is the subcarrier number w = [1199080 1199081 119908119873minus1]119879

represent the complex weights transmitted over differentsubcarriers and satisfying sum

119873minus1

119899=0|119908119899|

2= 1 For the sake

of keeping orthogonality between different subcarriers thesubcarrier spacing Δ119891 and time duration 119905119887 of a single pulseshould satisfy Δ119891119905119887 = 1 The total bandwidth is 119861 = 119873Δ119891

Assuming that the received signals contain 119875 differentpaths and the roundtrip delay corresponding to 119901th path is120591119901 119901 = 0 1 119875 minus 1 The received signal is

119910 (119905) =

119875minus1

sum

119901=0

119904 (120574119901 (119905 minus 120591119901)) (19)

where 120574119901 = 1+120573119901 accounts for the stretching or compressingin time of the reflected signal and 120573119901 = 2⟨k u119901⟩119888 representsthe Doppler spreading factor corresponding to 119901th path kand u119901 are the target velocity and unit direction of arrival(DOA) vector 119888 is the propagation speed Substituting (16)into (19) the received signal after demodulation is denotedby

119910 (119905)

=

119875minus1

sum

119901=0

119872minus1

sum

119898=0

119906 (120574119901 (119905 minus 120591119901) minus 119898119879119903) 11989011989521205871198910120573119901(119905minus1205910)119890

minus119895212058711989101205910

+ 119890 (119905)

(20)

Then substituting (17) into (20) we obtain

119910 (119905) =

119872minus1

sum

119898=0

119873minus1

sum

119899=0

119875minus1

sum

119901=0

119908119899119909119899119901rect (120574119901 (119905 minus 120591119901) minus 119898119879119903 119905119887)

sdot 1198901198952120587119899Δ119891(120574119901(119905minus120591119901)minus119898119879119903)119890

11989521205871198910120573119901(119905minus1205910)119890minus119895212058711989101205910 + 119890 (119905)

(21)

where 119909119899119901 represents the complex scattering coefficient oftarget corresponding to 119899th carrier and 119901th path 1205910 isthe roundtrip delay corresponding to the range cell underconsideration The relative time gaps between different pathsare very small compared to the actual roundtrip delay whichmeans 120591119901 asymp 1205910 for119901 = 0 1 119875minus1 Substituting 119905 = 1205910+119898119879119903

(119898 = 0 1 119872 minus 1) into (21) we obtain

119910 (119899119898) =

119875minus1

sum

119901=0

119908119899119909119899119901120601119899119901 (119898 V) + 119890 (119899119898) (22)

where

120601119899119901 (119898 V) = 119890minus119895212058711989101205910119890

1198952120587119891119899120573119901119898119879119903 (23)

and 119891119899 = 1198910 + 119899Δ119891 Equation (22) can be written into matrixform

y (119898) = WXΦ (119898 k) + e (119898) (24)

where

(i) y(119898) = [119910(0119898) 119910(1119898) 119910(119873minus1119898)]119879 is an119873times1

vector that represents output of the119898th pulse

(ii) W = diag(w) is an 119873 times 119873 complex diagonal matrixthat contains transmitted subcarrier weights

(iii) X = blkdiag (x1198790 x1198791 x119879

119873minus1) is an119873 times119873119875 complex

rectangular block-diagonal matrix x119899 = 119909119899 sdot 120588 119899 =

0 1 119873minus1 where 119909119899 is target scattering coefficienton the 119899th subcarrier and 120588 = [1205880 1205881 120588119875minus1]

119879

represents complex reflection coefficients over differ-ent paths for direct signals 120588119901 = 1 for first-orderreflected signals120588119901 = 120588119890

119895120593 for second-order reflectedsignals 120588119901 = (120588119890

119895120593)2 where 120588119890119895120593 is calculated by (10)

(iv) Φ(119898 k) = [1206010(119898 k) 1206011(119898 k) 120601119873minus1(119898 k)]119879 is an119873119875 times 1 complex vector and 120601119899(119898 k) = [1206011198990(119898 k)1206011198991(119898 k) 120601119899119875minus1(119898 k)]119879 119899 = 0 1 119873 minus 1

(v) e(119898) = [1198900(119898) 1198901(119898) 119890119873minus1(119898)]119879is an119873times 1 com-

plex vector of clutter

Then concatenating all temporal data into119873 times119872matrixthe OFDMmeasurement model in low grazing angle is

Y = WXΦ (k) + E (25)

where

Y = [y (0) y (1) y (119872 minus 1)] (26)

Φ(V) = [Φ(0 V)Φ(1 V) Φ(119872 minus 1 V)] is an 119873119875 times 119872

complex matrix that contains Doppler information over 119872pulses

E = [e(0) e(1) e(119872minus1)] is an119873times119872 complexmatrixthat consists of clutter returns

In low grazing angle scenario the clutter is usuallymodeled as compound-Gaussian model which is the productof speckle and texture [22] The speckle is fast-changing andmodeled as a complex Gaussian process while the texture isslow-changing and modeled as a nonnegative process [23]

In this paper the correlation length of the texture isassumed to be on the order of seconds and we only considerone coherent processing interval (CPI) which is assumedto be 60ms Due to the long correlation time of textureit is considered to be constant within each CPI changingaccording to a given probability density function (PDF) fromone CPI to the next Thus conditioned on a given value ofthe texture the clutter is simplified to Gaussian distributionHowever this simplification is not justified when predictingthe clutterrsquos behavior in time intervals larger than a CPI suchas clutter cancellation and constant false alarm rate (CFAR)detection

Thus we assume that the clutter is temporally whiteand circularly zero-mean complex Gaussian process withunknown positive definite covariance C The measurementsover different pulses are supposed to be independent whichmeans

vec (Y) sim CN119873119872 (vec (WXΦ (k)) C otimes I119872) (27)

whereotimes denotes Kronecker product Accordingly theOFDMmeasurements are distributed as

Y sim CN119873119872 (WXΦ (k) C otimes I119872) (28)

International Journal of Antennas and Propagation 5

4 Detection Test

In low grazing angle scenario received signals are the sumof signals along different paths which form the complicatedmeasurements The essence of detection is to judge whethera target is present or not in the range cell under test This isa classical two-hypothesis detection problem Therefore weconstruct a decision problem to choose between two possiblehypotheses the null hypothesis (target-free hypothesis) andthe alternate hypothesis (target-present hypothesis) whichcan be expressed as

H0 X = 0 C unknown

H1 X = 0 kC unknown(29)

ThemeasurementsY in two hypotheses are distributed as

Y

sim

H0 CN119873119872 (0C otimes I119872) C unknown

H1 CN119873119872 (WXΦ (k) C otimes I119872) kC unknown

(30)

According to the classical target detection theoryNeyman-Pearson (NP) detector is the optimal detector whichmaximizes the probability of detection at a constant prob-ability of false alarm However the target velocity k andclutter covarianceC are unknown andGLRT detector is usedinstead where the unknown parameters are replaced withtheir maximum likelihood estimates (MLEs) The formula-tion of GLRT is [24]

GLR (k) =119891 (Y | H1 k X C1)

119891 (Y | H0 C0)

H1

≶

H0

120574 (31)

where 119891(Y | H1) and 119891(Y | H0) are the likelihood functionsunder H1 and H0 respectively 120574 is the threshold that thedetector is compared with X and C1 are the MLE of X and CunderH1 while C0 is the MLE of C underH0

Definition 1 If the random matrix Y(119873 times 119872) is distributedas a matrix variate normal distribution (MVND) with meanM(119873 times 119872) and covariance Ω otimes Σ we use the notation Y sim

CN119873119872(MΩotimesΣ) and the probability density function (PDF)is given by [25]

119891 (Y)

=

exp (minus (12) tr [Ωminus1 (Y minusM)Σminus1(Y minusM)

119867])

(2120587)1198731198722

|Ω|1198732

|Σ|1198722

(32)

where tr(sdot) is the trace of matrixΩ and Σ are positive definitecovariance matrices over the column and row of Y Ω is an119873 times 119873 matrix and Σ is an 119872 times119872 matrix which are definedas [25]

Ω = 119864 [(Y minusM) (Y minusM)119867]

Σ = 119864 [(Y minusM)119867(Y minusM)]

(33)

According to the above descriptions we can derive PDFsof the measurements under two hypotheses

119891 (Y | H0) =

exp (minus (12) tr [Cminus10YY119867])

(2120587)1198731198722 1003816100381610038161003816C0

1003816100381610038161003816

1198732 (34)

119891 (Y | H1)

=

exp (minus (12) tr [Cminus11(Y minusWXΦ (k)) (Y minusWXΦ (k))119867])

(2120587)1198731198722 1003816100381610038161003816C1

1003816100381610038161003816

1198732

(35)

The log-likelihood function of (35) is

ln119891 (Y | H1)

= minus119872119873

2ln (2120587) minus 119873

2ln 1003816100381610038161003816C1

1003816100381610038161003816

minus1

2tr [Cminus1

1(Y minusWXΦ (k)) (Y minusWXΦ (k))119867]

(36)

Taking derivative of (36) with respect to C1 and makingit equal to zero the MLE of C1 is

C1 =1

119873(Y minusWXΦ (k)) (Y minusWXΦ (k))119867 (37)

Substituting (37) into (36) yields

XML

= argminX

ln 10038161003816100381610038161003816(Y minusWXΦ (k)) (Y minusWXΦ (k))11986710038161003816100381610038161003816 (38)

Usually the scattering matrix X does not yield a close-form MLE expression However noting that X has a block-diagonal structure it turns out to be a block-diagonal growthcurve (BDGC) problem Reference [26] derived the approx-imate maximum likelihood (AML) estimator for X which isgiven by

xAML = vecb (X) = (Γ119867Γ)minus1

Γ119867vec (Tminus12XΠΦ) (39)

where

Γ ≜ Φ119879⊛ (T12W)

T ≜ YΠperpΦY119867

ΠΦ ≜ Φ (k)119867 (Φ (k)Φ (k)119867)minus

Φ (k)

Πperp

Φ≜ I minusΠΦ

(40)

vecb(sdot) and ⊛ denote block-diagonal matrix vectorizationoperator and generalized Khatri-Rao product [27] respec-tively ΠΦ and Πperp

Φare orthogonal projection matrix of Φ(k)

and (sdot)minus represents generalized inverse of matrix (eg a

generalized inverse of matrix S is defined as Sminus such thatSSminusS = S) [27]

The log-likelihood function of (34) is

ln119891 (Y | H0) = minus119872119873

2ln (2120587) minus 119873

2ln 1003816100381610038161003816C0

1003816100381610038161003816

minus1

2tr [Cminus1

0YY119867]

(41)

6 International Journal of Antennas and Propagation

Taking derivative of (41) with respect toC0 andmaking itequal to zero we get MLE of C0

C0 =1

119873YY119867 (42)

Substituting (34) (35) (37) and (42) into (31) the GLRTdetector is

GLR (k) =10038161003816100381610038161003816C0

1003816100381610038161003816100381610038161003816100381610038161003816C1

10038161003816100381610038161003816

=

10038161003816100381610038161003816(1119873)YY11986710038161003816100381610038161003816

100381610038161003816100381610038161003816(1119873) (Y minusWXΦ (k)) (Y minusWXΦ (k))

119867100381610038161003816100381610038161003816

(43)

Since target velocity k is unknown it can be estimatedthrough k = argmaxkGLR(k) andtheGLRTdetector becomes

maxk

GLR (k) = GLR (k)H1

≶

H0

120574 (44)

5 Adaptive Waveform Design

In this section we develop an adaptive waveform designmethod based on maximizing the Mahalanobis-distanceto improve the detection performance Since the targetscattering coefficients vary with different subcarriers wemay change the transmitted weights W accordingly From(30) we know that the measurement Y is distributedas CN119873119872(WXΦ(k)C otimes I119872) under H1 hypothesis andCN119873119872(0C otimes I119872) under H0 hypothesis The squaredMahalanobis-distance is used to distinguish the above twodistributions It is noticeable that the column ofY is uncorre-lated which means that the measurements between differentpulses are independent The sum of squared Mahalanobis-distance over different pulses is given by [28]

1198892=

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898)) (45)

where120573119898 is a positive number and denotes theweight ofMaha-lanobis-distance over the119898th pulse satisfying sum119872minus1

119898=0120573119898 = 1

To maximize the detection performance we can formulatethe optimization as

wopt = arg maxWisinC119873

[

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))]

subject to w119867w = 1

(46)

The scattering coefficients are constant from pulse topulse and the measurement noise is uncorrelated betweendifferent pulses thus 120573119898 is a constant number that is 120573119898 =

1119872 for119898 = 0 1 119872 minus 1 Since

(WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))

= tr (Φ ( 119898)119867X119867W119867Cminus1WXΦ ( 119898))

= tr (Cminus1WXΦ ( 119898)Φ ( 119898)119867X119867W119867

)

(47)

the following theorem [11] is used to simplify the aboveequation

tr (S1AS2A119867) = a119867 [S119879

2⊙ S1] a (48)

where a isin C119873 and A = diag(a) ⊙ denotes element-wiseHadamard product Combining (47) and (48) we get

wopt = arg maxWisinC119873

(1

119872)w119867 [

119872minus1

sum

119898=0

(XΦ ( 119898)Φ ( 119898)119867X119867)

119879

⊙ Cminus1]w

subject to w119867w = 1

(49)

From (49) we know that the optimization problemis reduced to eigenvalue eigenvector problem and wopt isthe eigenvector corresponding to the largest eigenvalue of[sum

119872minus1

119898=0(XΦ( 119898)Φ( 119898)

119867X119867)119879 ⊙ Cminus1]

6 Numerical Results

In this section several numerical examples are presentedto illustrate the performance of proposed detector For

simplicity we consider 2D scenario as shown in Figure 2Thesimulation parameters are shown in Table 1

We use Monte Carlo simulations to realize the followingresults for the analytical expression between false alarm rateand threshold cannot be acquired The relation betweenprobability of false alarm rate and detection threshold corre-sponding to different carrier numbers is shown in Figure 3The curves show that when false alarm rate is kept fixed

International Journal of Antennas and Propagation 7

1

15

2

25

3

35

4

45

5

Det

ectio

n th

resh

old

(Th)

Probability of false alarm (Pfa)10minus3 10minus210minus5 10minus4 10minus1

N = 1

N = 3

N = 5

Figure 3 Detection threshold versus probability of false alarm at different carrier numbers

Table 1 Parameter settings of the simulations

Parameters ValuesHeight of target 200mVelocity of target [1414 1414]119879msScattering coefficient matrix of target X CN(0 1)

Height of radar platform 100mVelocity of radar platform [0 0]119879msRelative distance between radar and target 5 kmMultipath number 119875 3Carrier frequency 1198910 1 GHzBandwidth 119861 100MHzCarrier numbers119873 3Frequency weightsW 1radic119873I119873Pulse number119872 30Pulse repetition interval 119879119903 2ms

the higher the subcarrier number the higher the thresholdSignal-to-noise ratio (SNR) is defined as [11]

SNR =(1119873)sum

119872

119898=1(WX120601 (119898 k))119867WX120601 (119898 k)

tr (C) (50)

The effect of multipath number on the detection perfor-mance is shown in Figure 4 where the path number is setas 119875 = 1 and 119875 = 3 Figures 4(a) and 4(b) show 119875119889 versusSNR with different 119875fas and 119875119889 versus 119875fa when SNR is fixedto minus20 dB respectively The curves show that the detectionperformance is improved with the increasing of multipathnumber Consequently the derived detector with OFDM

waveform could exploit multipath reflections to enhance thedetection performance

In Figure 5 we analyze the effect of subcarrier number onthe performance of proposed detector where the subcarriernumber is set as 119873 = 1 3 and 5 respectively Figures 5(a)and 5(b) show 119875119889 versus SNR when 119875fa = 10

minus3 and 119875119889

versus 119875fa when SNR is fixed to minus20 dB respectively Thedetection performance is improved with the increasing ofsubcarrier number The results demonstrate that the detec-tion performance is improved due to the frequency diversityin an OFDM radar system

Figure 6 shows that the detection performance varieswith the target height which is set as 200m 400m and600m respectively Figure 6(a) shows 119875119889 versus SNR when119875fa = 10

minus3 while Figure 6(b) shows 119875119889 versus 119875fa when SNRis fixed to minus20 dB The results demonstrate that detectionperformance decreases with the increasing of target height inlow grazing angle scenario This can be explained that as thetarget height increases the grazing angle gets larger and thecomplex reflection coefficient over indirect paths becomessmaller which is shown in Figure 7 Thus the detectionperformance deteriorates

In Figure 8 we analyze the effect of varying the directionsof target velocity vector on the detection performance Herek119879 = [100 0] [717 717] and [0 100]ms respectivelyFigures 8(a) and 8(b) show 119875119889 versus SNR when 119875fa = 10

minus3

and 119875119889 versus 119875fa when SNR is fixed to minus20 dB respectivelyIn this paper we take advantage of multipath propagationby exploiting multiple Doppler shifts corresponding to theprojections of target velocity on each of the multipathcomponents When the angle between the target velocityvector and radar LOS decreases the Doppler frequency of

8 International Journal of Antennas and Propagation

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1 Pfa = 10minus3

P = 3 Pfa = 10minus3P = 1 Pfa = 10minus5

P = 3 Pfa = 10minus5

(a) 119875119889 versus SNR with different 119875fas

02

03

04

05

06

07

08

09

1

Probability of false alarm (Pfa)10minus3 10minus2 10minus1 100

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1

P = 3

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 4 Effect of the path number on the detection performance

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 1

N = 3

N = 5

(a) 119875119889 versus SNR when 119875fa = 10minus3

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 1

N = 3

N = 5

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 5 Effect of the subcarrier number on the detection performance

received signal will increaseThus the detection performanceis improved

In Figure 9 we show the detection performance improve-ment due to the adaptive waveform design Figure 9(a) shows119875119889 versus SNR when 119875fa = 10

minus3 while Figure 9(b) shows 119875119889versus 119875fa when SNR is fixed to minus20 dB We assume that thetransmitted waveform in the first119872 pulses has equal weightsthat is w = 1radic119873I Then based on (49) we compute woptfor the next 119872 pulses As is shown compared with fixed

waveform the detection performance is improved due to theadaptive waveform method proposed in this paper

Finally to validate our method in non-Gaussian scenar-ios we conduct additional experiments using compound K-distribution whose PDF is [29]

119891 (119911) =radic2V120583Γ (V)

(radic2V120583119911)

V

119870Vminus1(radic2V120583119911) (51)

International Journal of Antennas and Propagation 9

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

ht = 200mht = 400mht = 600m

(a) 119875119889 versus SNR when 119875fa = 10minus3

01

02

03

04

05

06

07

08

09

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

ht = 200mht = 400mht = 600m

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 6 Detection performance varies with target height

0 500 1000 1500 20000

02

04

06

08

1

Am

plitu

de o

f com

plex

refle

ctio

n co

effici

ent

Horizontal polarizationVertical polarization

Target height m

Figure 7 Amplitude of complex reflection coefficient varies with target height

and the texture follows a gamma distribution with PDF

119891 (119906) =1

Γ (V)(V120583)

V

119906Vminus1

119890minusV120583119906

120581 (119906) (52)

where Γ(sdot) is the Eulerian gamma function 120581(sdot) is the unitstep function and 119870Vminus1(sdot) is the modified Bessel function ofsecond kind with order V minus 1 120583 and V are the scale and shapeparameter respectively The speckle component is assumed

to be a Gaussian distribution with exponential correlationstructure covariance matrix [30]

[C]119894119895 = 120588|119894minus119895|

for 119894 119895 = 0 1 119873 minus 1 (53)

where 120588 is one-lag correlation coefficient The other parame-ters are the same as in Table 1

Figure 10 shows the detection performance with differentpath numbers (119875 = 1 3) in Gaussian and compound

10 International Journal of Antennas and Propagation

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 8 Effect of different directions of target velocity on detection performance

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 9 Comparison of detection performance with fixed waveform and adaptive waveform

K-distribution clutter and Figure 11 shows the detectionperformance with fixed and adaptive waveform design inGaussian and compound K-distribution clutter We find thatin compound K-distribution the detector derived underGaussian assumption also yields improved performance bytaking advantage of multipath effect and adaptive waveformdesign of OFDM radar However the performance gap existsdue to the model mismatch

7 Conclusions

In this paper we address the problem of target detectionin low grazing angle scenario with OFDM radar We con-sider the realistic physical and statistical effects to makethe propagation model more realistic Based on the char-acteristics of OFDM signal we develop a GLRT detectorand show that detection performance is improved due tothe multipath utilization and frequency diversity of OFDM

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

International Journal of

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Active and Passive Electronic Components

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

4 International Journal of Antennas and Propagation

and 119873 is the subcarrier number w = [1199080 1199081 119908119873minus1]119879

represent the complex weights transmitted over differentsubcarriers and satisfying sum

119873minus1

119899=0|119908119899|

2= 1 For the sake

of keeping orthogonality between different subcarriers thesubcarrier spacing Δ119891 and time duration 119905119887 of a single pulseshould satisfy Δ119891119905119887 = 1 The total bandwidth is 119861 = 119873Δ119891

Assuming that the received signals contain 119875 differentpaths and the roundtrip delay corresponding to 119901th path is120591119901 119901 = 0 1 119875 minus 1 The received signal is

119910 (119905) =

119875minus1

sum

119901=0

119904 (120574119901 (119905 minus 120591119901)) (19)

where 120574119901 = 1+120573119901 accounts for the stretching or compressingin time of the reflected signal and 120573119901 = 2⟨k u119901⟩119888 representsthe Doppler spreading factor corresponding to 119901th path kand u119901 are the target velocity and unit direction of arrival(DOA) vector 119888 is the propagation speed Substituting (16)into (19) the received signal after demodulation is denotedby

119910 (119905)

=

119875minus1

sum

119901=0

119872minus1

sum

119898=0

119906 (120574119901 (119905 minus 120591119901) minus 119898119879119903) 11989011989521205871198910120573119901(119905minus1205910)119890

minus119895212058711989101205910

+ 119890 (119905)

(20)

Then substituting (17) into (20) we obtain

119910 (119905) =

119872minus1

sum

119898=0

119873minus1

sum

119899=0

119875minus1

sum

119901=0

119908119899119909119899119901rect (120574119901 (119905 minus 120591119901) minus 119898119879119903 119905119887)

sdot 1198901198952120587119899Δ119891(120574119901(119905minus120591119901)minus119898119879119903)119890

11989521205871198910120573119901(119905minus1205910)119890minus119895212058711989101205910 + 119890 (119905)

(21)

where 119909119899119901 represents the complex scattering coefficient oftarget corresponding to 119899th carrier and 119901th path 1205910 isthe roundtrip delay corresponding to the range cell underconsideration The relative time gaps between different pathsare very small compared to the actual roundtrip delay whichmeans 120591119901 asymp 1205910 for119901 = 0 1 119875minus1 Substituting 119905 = 1205910+119898119879119903

(119898 = 0 1 119872 minus 1) into (21) we obtain

119910 (119899119898) =

119875minus1

sum

119901=0

119908119899119909119899119901120601119899119901 (119898 V) + 119890 (119899119898) (22)

where

120601119899119901 (119898 V) = 119890minus119895212058711989101205910119890

1198952120587119891119899120573119901119898119879119903 (23)

and 119891119899 = 1198910 + 119899Δ119891 Equation (22) can be written into matrixform

y (119898) = WXΦ (119898 k) + e (119898) (24)

where

(i) y(119898) = [119910(0119898) 119910(1119898) 119910(119873minus1119898)]119879 is an119873times1

vector that represents output of the119898th pulse

(ii) W = diag(w) is an 119873 times 119873 complex diagonal matrixthat contains transmitted subcarrier weights

(iii) X = blkdiag (x1198790 x1198791 x119879

119873minus1) is an119873 times119873119875 complex

rectangular block-diagonal matrix x119899 = 119909119899 sdot 120588 119899 =

0 1 119873minus1 where 119909119899 is target scattering coefficienton the 119899th subcarrier and 120588 = [1205880 1205881 120588119875minus1]

119879

represents complex reflection coefficients over differ-ent paths for direct signals 120588119901 = 1 for first-orderreflected signals120588119901 = 120588119890

119895120593 for second-order reflectedsignals 120588119901 = (120588119890

119895120593)2 where 120588119890119895120593 is calculated by (10)

(iv) Φ(119898 k) = [1206010(119898 k) 1206011(119898 k) 120601119873minus1(119898 k)]119879 is an119873119875 times 1 complex vector and 120601119899(119898 k) = [1206011198990(119898 k)1206011198991(119898 k) 120601119899119875minus1(119898 k)]119879 119899 = 0 1 119873 minus 1

(v) e(119898) = [1198900(119898) 1198901(119898) 119890119873minus1(119898)]119879is an119873times 1 com-

plex vector of clutter

Then concatenating all temporal data into119873 times119872matrixthe OFDMmeasurement model in low grazing angle is

Y = WXΦ (k) + E (25)

where

Y = [y (0) y (1) y (119872 minus 1)] (26)

Φ(V) = [Φ(0 V)Φ(1 V) Φ(119872 minus 1 V)] is an 119873119875 times 119872

complex matrix that contains Doppler information over 119872pulses

E = [e(0) e(1) e(119872minus1)] is an119873times119872 complexmatrixthat consists of clutter returns

In low grazing angle scenario the clutter is usuallymodeled as compound-Gaussian model which is the productof speckle and texture [22] The speckle is fast-changing andmodeled as a complex Gaussian process while the texture isslow-changing and modeled as a nonnegative process [23]

In this paper the correlation length of the texture isassumed to be on the order of seconds and we only considerone coherent processing interval (CPI) which is assumedto be 60ms Due to the long correlation time of textureit is considered to be constant within each CPI changingaccording to a given probability density function (PDF) fromone CPI to the next Thus conditioned on a given value ofthe texture the clutter is simplified to Gaussian distributionHowever this simplification is not justified when predictingthe clutterrsquos behavior in time intervals larger than a CPI suchas clutter cancellation and constant false alarm rate (CFAR)detection

Thus we assume that the clutter is temporally whiteand circularly zero-mean complex Gaussian process withunknown positive definite covariance C The measurementsover different pulses are supposed to be independent whichmeans

vec (Y) sim CN119873119872 (vec (WXΦ (k)) C otimes I119872) (27)

whereotimes denotes Kronecker product Accordingly theOFDMmeasurements are distributed as

Y sim CN119873119872 (WXΦ (k) C otimes I119872) (28)

International Journal of Antennas and Propagation 5

4 Detection Test

In low grazing angle scenario received signals are the sumof signals along different paths which form the complicatedmeasurements The essence of detection is to judge whethera target is present or not in the range cell under test This isa classical two-hypothesis detection problem Therefore weconstruct a decision problem to choose between two possiblehypotheses the null hypothesis (target-free hypothesis) andthe alternate hypothesis (target-present hypothesis) whichcan be expressed as

H0 X = 0 C unknown

H1 X = 0 kC unknown(29)

ThemeasurementsY in two hypotheses are distributed as

Y

sim

H0 CN119873119872 (0C otimes I119872) C unknown

H1 CN119873119872 (WXΦ (k) C otimes I119872) kC unknown

(30)

According to the classical target detection theoryNeyman-Pearson (NP) detector is the optimal detector whichmaximizes the probability of detection at a constant prob-ability of false alarm However the target velocity k andclutter covarianceC are unknown andGLRT detector is usedinstead where the unknown parameters are replaced withtheir maximum likelihood estimates (MLEs) The formula-tion of GLRT is [24]

GLR (k) =119891 (Y | H1 k X C1)

119891 (Y | H0 C0)

H1

≶

H0

120574 (31)

where 119891(Y | H1) and 119891(Y | H0) are the likelihood functionsunder H1 and H0 respectively 120574 is the threshold that thedetector is compared with X and C1 are the MLE of X and CunderH1 while C0 is the MLE of C underH0

Definition 1 If the random matrix Y(119873 times 119872) is distributedas a matrix variate normal distribution (MVND) with meanM(119873 times 119872) and covariance Ω otimes Σ we use the notation Y sim

CN119873119872(MΩotimesΣ) and the probability density function (PDF)is given by [25]

119891 (Y)

=

exp (minus (12) tr [Ωminus1 (Y minusM)Σminus1(Y minusM)

119867])

(2120587)1198731198722

|Ω|1198732

|Σ|1198722

(32)

where tr(sdot) is the trace of matrixΩ and Σ are positive definitecovariance matrices over the column and row of Y Ω is an119873 times 119873 matrix and Σ is an 119872 times119872 matrix which are definedas [25]

Ω = 119864 [(Y minusM) (Y minusM)119867]

Σ = 119864 [(Y minusM)119867(Y minusM)]

(33)

According to the above descriptions we can derive PDFsof the measurements under two hypotheses

119891 (Y | H0) =

exp (minus (12) tr [Cminus10YY119867])

(2120587)1198731198722 1003816100381610038161003816C0

1003816100381610038161003816

1198732 (34)

119891 (Y | H1)

=

exp (minus (12) tr [Cminus11(Y minusWXΦ (k)) (Y minusWXΦ (k))119867])

(2120587)1198731198722 1003816100381610038161003816C1

1003816100381610038161003816

1198732

(35)

The log-likelihood function of (35) is

ln119891 (Y | H1)

= minus119872119873

2ln (2120587) minus 119873

2ln 1003816100381610038161003816C1

1003816100381610038161003816

minus1

2tr [Cminus1

1(Y minusWXΦ (k)) (Y minusWXΦ (k))119867]

(36)

Taking derivative of (36) with respect to C1 and makingit equal to zero the MLE of C1 is

C1 =1

119873(Y minusWXΦ (k)) (Y minusWXΦ (k))119867 (37)

Substituting (37) into (36) yields

XML

= argminX

ln 10038161003816100381610038161003816(Y minusWXΦ (k)) (Y minusWXΦ (k))11986710038161003816100381610038161003816 (38)

Usually the scattering matrix X does not yield a close-form MLE expression However noting that X has a block-diagonal structure it turns out to be a block-diagonal growthcurve (BDGC) problem Reference [26] derived the approx-imate maximum likelihood (AML) estimator for X which isgiven by

xAML = vecb (X) = (Γ119867Γ)minus1

Γ119867vec (Tminus12XΠΦ) (39)

where

Γ ≜ Φ119879⊛ (T12W)

T ≜ YΠperpΦY119867

ΠΦ ≜ Φ (k)119867 (Φ (k)Φ (k)119867)minus

Φ (k)

Πperp

Φ≜ I minusΠΦ

(40)

vecb(sdot) and ⊛ denote block-diagonal matrix vectorizationoperator and generalized Khatri-Rao product [27] respec-tively ΠΦ and Πperp

Φare orthogonal projection matrix of Φ(k)

and (sdot)minus represents generalized inverse of matrix (eg a

generalized inverse of matrix S is defined as Sminus such thatSSminusS = S) [27]

The log-likelihood function of (34) is

ln119891 (Y | H0) = minus119872119873

2ln (2120587) minus 119873

2ln 1003816100381610038161003816C0

1003816100381610038161003816

minus1

2tr [Cminus1

0YY119867]

(41)

6 International Journal of Antennas and Propagation

Taking derivative of (41) with respect toC0 andmaking itequal to zero we get MLE of C0

C0 =1

119873YY119867 (42)

Substituting (34) (35) (37) and (42) into (31) the GLRTdetector is

GLR (k) =10038161003816100381610038161003816C0

1003816100381610038161003816100381610038161003816100381610038161003816C1

10038161003816100381610038161003816

=

10038161003816100381610038161003816(1119873)YY11986710038161003816100381610038161003816

100381610038161003816100381610038161003816(1119873) (Y minusWXΦ (k)) (Y minusWXΦ (k))

119867100381610038161003816100381610038161003816

(43)

Since target velocity k is unknown it can be estimatedthrough k = argmaxkGLR(k) andtheGLRTdetector becomes

maxk

GLR (k) = GLR (k)H1

≶

H0

120574 (44)

5 Adaptive Waveform Design

In this section we develop an adaptive waveform designmethod based on maximizing the Mahalanobis-distanceto improve the detection performance Since the targetscattering coefficients vary with different subcarriers wemay change the transmitted weights W accordingly From(30) we know that the measurement Y is distributedas CN119873119872(WXΦ(k)C otimes I119872) under H1 hypothesis andCN119873119872(0C otimes I119872) under H0 hypothesis The squaredMahalanobis-distance is used to distinguish the above twodistributions It is noticeable that the column ofY is uncorre-lated which means that the measurements between differentpulses are independent The sum of squared Mahalanobis-distance over different pulses is given by [28]

1198892=

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898)) (45)

where120573119898 is a positive number and denotes theweight ofMaha-lanobis-distance over the119898th pulse satisfying sum119872minus1

119898=0120573119898 = 1

To maximize the detection performance we can formulatethe optimization as

wopt = arg maxWisinC119873

[

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))]

subject to w119867w = 1

(46)

The scattering coefficients are constant from pulse topulse and the measurement noise is uncorrelated betweendifferent pulses thus 120573119898 is a constant number that is 120573119898 =

1119872 for119898 = 0 1 119872 minus 1 Since

(WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))

= tr (Φ ( 119898)119867X119867W119867Cminus1WXΦ ( 119898))

= tr (Cminus1WXΦ ( 119898)Φ ( 119898)119867X119867W119867

)

(47)

the following theorem [11] is used to simplify the aboveequation

tr (S1AS2A119867) = a119867 [S119879

2⊙ S1] a (48)

where a isin C119873 and A = diag(a) ⊙ denotes element-wiseHadamard product Combining (47) and (48) we get

wopt = arg maxWisinC119873

(1

119872)w119867 [

119872minus1

sum

119898=0

(XΦ ( 119898)Φ ( 119898)119867X119867)

119879

⊙ Cminus1]w

subject to w119867w = 1

(49)

From (49) we know that the optimization problemis reduced to eigenvalue eigenvector problem and wopt isthe eigenvector corresponding to the largest eigenvalue of[sum

119872minus1

119898=0(XΦ( 119898)Φ( 119898)

119867X119867)119879 ⊙ Cminus1]

6 Numerical Results

In this section several numerical examples are presentedto illustrate the performance of proposed detector For

simplicity we consider 2D scenario as shown in Figure 2Thesimulation parameters are shown in Table 1

We use Monte Carlo simulations to realize the followingresults for the analytical expression between false alarm rateand threshold cannot be acquired The relation betweenprobability of false alarm rate and detection threshold corre-sponding to different carrier numbers is shown in Figure 3The curves show that when false alarm rate is kept fixed

International Journal of Antennas and Propagation 7

1

15

2

25

3

35

4

45

5

Det

ectio

n th

resh

old

(Th)

Probability of false alarm (Pfa)10minus3 10minus210minus5 10minus4 10minus1

N = 1

N = 3

N = 5

Figure 3 Detection threshold versus probability of false alarm at different carrier numbers

Table 1 Parameter settings of the simulations

Parameters ValuesHeight of target 200mVelocity of target [1414 1414]119879msScattering coefficient matrix of target X CN(0 1)

Height of radar platform 100mVelocity of radar platform [0 0]119879msRelative distance between radar and target 5 kmMultipath number 119875 3Carrier frequency 1198910 1 GHzBandwidth 119861 100MHzCarrier numbers119873 3Frequency weightsW 1radic119873I119873Pulse number119872 30Pulse repetition interval 119879119903 2ms

the higher the subcarrier number the higher the thresholdSignal-to-noise ratio (SNR) is defined as [11]

SNR =(1119873)sum

119872

119898=1(WX120601 (119898 k))119867WX120601 (119898 k)

tr (C) (50)

The effect of multipath number on the detection perfor-mance is shown in Figure 4 where the path number is setas 119875 = 1 and 119875 = 3 Figures 4(a) and 4(b) show 119875119889 versusSNR with different 119875fas and 119875119889 versus 119875fa when SNR is fixedto minus20 dB respectively The curves show that the detectionperformance is improved with the increasing of multipathnumber Consequently the derived detector with OFDM

waveform could exploit multipath reflections to enhance thedetection performance

In Figure 5 we analyze the effect of subcarrier number onthe performance of proposed detector where the subcarriernumber is set as 119873 = 1 3 and 5 respectively Figures 5(a)and 5(b) show 119875119889 versus SNR when 119875fa = 10

minus3 and 119875119889

versus 119875fa when SNR is fixed to minus20 dB respectively Thedetection performance is improved with the increasing ofsubcarrier number The results demonstrate that the detec-tion performance is improved due to the frequency diversityin an OFDM radar system

Figure 6 shows that the detection performance varieswith the target height which is set as 200m 400m and600m respectively Figure 6(a) shows 119875119889 versus SNR when119875fa = 10

minus3 while Figure 6(b) shows 119875119889 versus 119875fa when SNRis fixed to minus20 dB The results demonstrate that detectionperformance decreases with the increasing of target height inlow grazing angle scenario This can be explained that as thetarget height increases the grazing angle gets larger and thecomplex reflection coefficient over indirect paths becomessmaller which is shown in Figure 7 Thus the detectionperformance deteriorates

In Figure 8 we analyze the effect of varying the directionsof target velocity vector on the detection performance Herek119879 = [100 0] [717 717] and [0 100]ms respectivelyFigures 8(a) and 8(b) show 119875119889 versus SNR when 119875fa = 10

minus3

and 119875119889 versus 119875fa when SNR is fixed to minus20 dB respectivelyIn this paper we take advantage of multipath propagationby exploiting multiple Doppler shifts corresponding to theprojections of target velocity on each of the multipathcomponents When the angle between the target velocityvector and radar LOS decreases the Doppler frequency of

8 International Journal of Antennas and Propagation

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1 Pfa = 10minus3

P = 3 Pfa = 10minus3P = 1 Pfa = 10minus5

P = 3 Pfa = 10minus5

(a) 119875119889 versus SNR with different 119875fas

02

03

04

05

06

07

08

09

1

Probability of false alarm (Pfa)10minus3 10minus2 10minus1 100

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1

P = 3

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 4 Effect of the path number on the detection performance

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 1

N = 3

N = 5

(a) 119875119889 versus SNR when 119875fa = 10minus3

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 1

N = 3

N = 5

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 5 Effect of the subcarrier number on the detection performance

received signal will increaseThus the detection performanceis improved

In Figure 9 we show the detection performance improve-ment due to the adaptive waveform design Figure 9(a) shows119875119889 versus SNR when 119875fa = 10

minus3 while Figure 9(b) shows 119875119889versus 119875fa when SNR is fixed to minus20 dB We assume that thetransmitted waveform in the first119872 pulses has equal weightsthat is w = 1radic119873I Then based on (49) we compute woptfor the next 119872 pulses As is shown compared with fixed

waveform the detection performance is improved due to theadaptive waveform method proposed in this paper

Finally to validate our method in non-Gaussian scenar-ios we conduct additional experiments using compound K-distribution whose PDF is [29]

119891 (119911) =radic2V120583Γ (V)

(radic2V120583119911)

V

119870Vminus1(radic2V120583119911) (51)

International Journal of Antennas and Propagation 9

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

ht = 200mht = 400mht = 600m

(a) 119875119889 versus SNR when 119875fa = 10minus3

01

02

03

04

05

06

07

08

09

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

ht = 200mht = 400mht = 600m

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 6 Detection performance varies with target height

0 500 1000 1500 20000

02

04

06

08

1

Am

plitu

de o

f com

plex

refle

ctio

n co

effici

ent

Horizontal polarizationVertical polarization

Target height m

Figure 7 Amplitude of complex reflection coefficient varies with target height

and the texture follows a gamma distribution with PDF

119891 (119906) =1

Γ (V)(V120583)

V

119906Vminus1

119890minusV120583119906

120581 (119906) (52)

where Γ(sdot) is the Eulerian gamma function 120581(sdot) is the unitstep function and 119870Vminus1(sdot) is the modified Bessel function ofsecond kind with order V minus 1 120583 and V are the scale and shapeparameter respectively The speckle component is assumed

to be a Gaussian distribution with exponential correlationstructure covariance matrix [30]

[C]119894119895 = 120588|119894minus119895|

for 119894 119895 = 0 1 119873 minus 1 (53)

where 120588 is one-lag correlation coefficient The other parame-ters are the same as in Table 1

Figure 10 shows the detection performance with differentpath numbers (119875 = 1 3) in Gaussian and compound

10 International Journal of Antennas and Propagation

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 8 Effect of different directions of target velocity on detection performance

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 9 Comparison of detection performance with fixed waveform and adaptive waveform

K-distribution clutter and Figure 11 shows the detectionperformance with fixed and adaptive waveform design inGaussian and compound K-distribution clutter We find thatin compound K-distribution the detector derived underGaussian assumption also yields improved performance bytaking advantage of multipath effect and adaptive waveformdesign of OFDM radar However the performance gap existsdue to the model mismatch

7 Conclusions

In this paper we address the problem of target detectionin low grazing angle scenario with OFDM radar We con-sider the realistic physical and statistical effects to makethe propagation model more realistic Based on the char-acteristics of OFDM signal we develop a GLRT detectorand show that detection performance is improved due tothe multipath utilization and frequency diversity of OFDM

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 5

4 Detection Test

In low grazing angle scenario received signals are the sumof signals along different paths which form the complicatedmeasurements The essence of detection is to judge whethera target is present or not in the range cell under test This isa classical two-hypothesis detection problem Therefore weconstruct a decision problem to choose between two possiblehypotheses the null hypothesis (target-free hypothesis) andthe alternate hypothesis (target-present hypothesis) whichcan be expressed as

H0 X = 0 C unknown

H1 X = 0 kC unknown(29)

ThemeasurementsY in two hypotheses are distributed as

Y

sim

H0 CN119873119872 (0C otimes I119872) C unknown

H1 CN119873119872 (WXΦ (k) C otimes I119872) kC unknown

(30)

According to the classical target detection theoryNeyman-Pearson (NP) detector is the optimal detector whichmaximizes the probability of detection at a constant prob-ability of false alarm However the target velocity k andclutter covarianceC are unknown andGLRT detector is usedinstead where the unknown parameters are replaced withtheir maximum likelihood estimates (MLEs) The formula-tion of GLRT is [24]

GLR (k) =119891 (Y | H1 k X C1)

119891 (Y | H0 C0)

H1

≶

H0

120574 (31)

where 119891(Y | H1) and 119891(Y | H0) are the likelihood functionsunder H1 and H0 respectively 120574 is the threshold that thedetector is compared with X and C1 are the MLE of X and CunderH1 while C0 is the MLE of C underH0

Definition 1 If the random matrix Y(119873 times 119872) is distributedas a matrix variate normal distribution (MVND) with meanM(119873 times 119872) and covariance Ω otimes Σ we use the notation Y sim

CN119873119872(MΩotimesΣ) and the probability density function (PDF)is given by [25]

119891 (Y)

=

exp (minus (12) tr [Ωminus1 (Y minusM)Σminus1(Y minusM)

119867])

(2120587)1198731198722

|Ω|1198732

|Σ|1198722

(32)

where tr(sdot) is the trace of matrixΩ and Σ are positive definitecovariance matrices over the column and row of Y Ω is an119873 times 119873 matrix and Σ is an 119872 times119872 matrix which are definedas [25]

Ω = 119864 [(Y minusM) (Y minusM)119867]

Σ = 119864 [(Y minusM)119867(Y minusM)]

(33)

According to the above descriptions we can derive PDFsof the measurements under two hypotheses

119891 (Y | H0) =

exp (minus (12) tr [Cminus10YY119867])

(2120587)1198731198722 1003816100381610038161003816C0

1003816100381610038161003816

1198732 (34)

119891 (Y | H1)

=

exp (minus (12) tr [Cminus11(Y minusWXΦ (k)) (Y minusWXΦ (k))119867])

(2120587)1198731198722 1003816100381610038161003816C1

1003816100381610038161003816

1198732

(35)

The log-likelihood function of (35) is

ln119891 (Y | H1)

= minus119872119873

2ln (2120587) minus 119873

2ln 1003816100381610038161003816C1

1003816100381610038161003816

minus1

2tr [Cminus1

1(Y minusWXΦ (k)) (Y minusWXΦ (k))119867]

(36)

Taking derivative of (36) with respect to C1 and makingit equal to zero the MLE of C1 is

C1 =1

119873(Y minusWXΦ (k)) (Y minusWXΦ (k))119867 (37)

Substituting (37) into (36) yields

XML

= argminX

ln 10038161003816100381610038161003816(Y minusWXΦ (k)) (Y minusWXΦ (k))11986710038161003816100381610038161003816 (38)

Usually the scattering matrix X does not yield a close-form MLE expression However noting that X has a block-diagonal structure it turns out to be a block-diagonal growthcurve (BDGC) problem Reference [26] derived the approx-imate maximum likelihood (AML) estimator for X which isgiven by

xAML = vecb (X) = (Γ119867Γ)minus1

Γ119867vec (Tminus12XΠΦ) (39)

where

Γ ≜ Φ119879⊛ (T12W)

T ≜ YΠperpΦY119867

ΠΦ ≜ Φ (k)119867 (Φ (k)Φ (k)119867)minus

Φ (k)

Πperp

Φ≜ I minusΠΦ

(40)

vecb(sdot) and ⊛ denote block-diagonal matrix vectorizationoperator and generalized Khatri-Rao product [27] respec-tively ΠΦ and Πperp

Φare orthogonal projection matrix of Φ(k)

and (sdot)minus represents generalized inverse of matrix (eg a

generalized inverse of matrix S is defined as Sminus such thatSSminusS = S) [27]

The log-likelihood function of (34) is

ln119891 (Y | H0) = minus119872119873

2ln (2120587) minus 119873

2ln 1003816100381610038161003816C0

1003816100381610038161003816

minus1

2tr [Cminus1

0YY119867]

(41)

6 International Journal of Antennas and Propagation

Taking derivative of (41) with respect toC0 andmaking itequal to zero we get MLE of C0

C0 =1

119873YY119867 (42)

Substituting (34) (35) (37) and (42) into (31) the GLRTdetector is

GLR (k) =10038161003816100381610038161003816C0

1003816100381610038161003816100381610038161003816100381610038161003816C1

10038161003816100381610038161003816

=

10038161003816100381610038161003816(1119873)YY11986710038161003816100381610038161003816

100381610038161003816100381610038161003816(1119873) (Y minusWXΦ (k)) (Y minusWXΦ (k))

119867100381610038161003816100381610038161003816

(43)

Since target velocity k is unknown it can be estimatedthrough k = argmaxkGLR(k) andtheGLRTdetector becomes

maxk

GLR (k) = GLR (k)H1

≶

H0

120574 (44)

5 Adaptive Waveform Design

In this section we develop an adaptive waveform designmethod based on maximizing the Mahalanobis-distanceto improve the detection performance Since the targetscattering coefficients vary with different subcarriers wemay change the transmitted weights W accordingly From(30) we know that the measurement Y is distributedas CN119873119872(WXΦ(k)C otimes I119872) under H1 hypothesis andCN119873119872(0C otimes I119872) under H0 hypothesis The squaredMahalanobis-distance is used to distinguish the above twodistributions It is noticeable that the column ofY is uncorre-lated which means that the measurements between differentpulses are independent The sum of squared Mahalanobis-distance over different pulses is given by [28]

1198892=

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898)) (45)

where120573119898 is a positive number and denotes theweight ofMaha-lanobis-distance over the119898th pulse satisfying sum119872minus1

119898=0120573119898 = 1

To maximize the detection performance we can formulatethe optimization as

wopt = arg maxWisinC119873

[

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))]

subject to w119867w = 1

(46)

The scattering coefficients are constant from pulse topulse and the measurement noise is uncorrelated betweendifferent pulses thus 120573119898 is a constant number that is 120573119898 =

1119872 for119898 = 0 1 119872 minus 1 Since

(WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))

= tr (Φ ( 119898)119867X119867W119867Cminus1WXΦ ( 119898))

= tr (Cminus1WXΦ ( 119898)Φ ( 119898)119867X119867W119867

)

(47)

the following theorem [11] is used to simplify the aboveequation

tr (S1AS2A119867) = a119867 [S119879

2⊙ S1] a (48)

where a isin C119873 and A = diag(a) ⊙ denotes element-wiseHadamard product Combining (47) and (48) we get

wopt = arg maxWisinC119873

(1

119872)w119867 [

119872minus1

sum

119898=0

(XΦ ( 119898)Φ ( 119898)119867X119867)

119879

⊙ Cminus1]w

subject to w119867w = 1

(49)

From (49) we know that the optimization problemis reduced to eigenvalue eigenvector problem and wopt isthe eigenvector corresponding to the largest eigenvalue of[sum

119872minus1

119898=0(XΦ( 119898)Φ( 119898)

119867X119867)119879 ⊙ Cminus1]

6 Numerical Results

In this section several numerical examples are presentedto illustrate the performance of proposed detector For

simplicity we consider 2D scenario as shown in Figure 2Thesimulation parameters are shown in Table 1

We use Monte Carlo simulations to realize the followingresults for the analytical expression between false alarm rateand threshold cannot be acquired The relation betweenprobability of false alarm rate and detection threshold corre-sponding to different carrier numbers is shown in Figure 3The curves show that when false alarm rate is kept fixed

International Journal of Antennas and Propagation 7

1

15

2

25

3

35

4

45

5

Det

ectio

n th

resh

old

(Th)

Probability of false alarm (Pfa)10minus3 10minus210minus5 10minus4 10minus1

N = 1

N = 3

N = 5

Figure 3 Detection threshold versus probability of false alarm at different carrier numbers

Table 1 Parameter settings of the simulations

Parameters ValuesHeight of target 200mVelocity of target [1414 1414]119879msScattering coefficient matrix of target X CN(0 1)

Height of radar platform 100mVelocity of radar platform [0 0]119879msRelative distance between radar and target 5 kmMultipath number 119875 3Carrier frequency 1198910 1 GHzBandwidth 119861 100MHzCarrier numbers119873 3Frequency weightsW 1radic119873I119873Pulse number119872 30Pulse repetition interval 119879119903 2ms

the higher the subcarrier number the higher the thresholdSignal-to-noise ratio (SNR) is defined as [11]

SNR =(1119873)sum

119872

119898=1(WX120601 (119898 k))119867WX120601 (119898 k)

tr (C) (50)

The effect of multipath number on the detection perfor-mance is shown in Figure 4 where the path number is setas 119875 = 1 and 119875 = 3 Figures 4(a) and 4(b) show 119875119889 versusSNR with different 119875fas and 119875119889 versus 119875fa when SNR is fixedto minus20 dB respectively The curves show that the detectionperformance is improved with the increasing of multipathnumber Consequently the derived detector with OFDM

waveform could exploit multipath reflections to enhance thedetection performance

In Figure 5 we analyze the effect of subcarrier number onthe performance of proposed detector where the subcarriernumber is set as 119873 = 1 3 and 5 respectively Figures 5(a)and 5(b) show 119875119889 versus SNR when 119875fa = 10

minus3 and 119875119889

versus 119875fa when SNR is fixed to minus20 dB respectively Thedetection performance is improved with the increasing ofsubcarrier number The results demonstrate that the detec-tion performance is improved due to the frequency diversityin an OFDM radar system

Figure 6 shows that the detection performance varieswith the target height which is set as 200m 400m and600m respectively Figure 6(a) shows 119875119889 versus SNR when119875fa = 10

minus3 while Figure 6(b) shows 119875119889 versus 119875fa when SNRis fixed to minus20 dB The results demonstrate that detectionperformance decreases with the increasing of target height inlow grazing angle scenario This can be explained that as thetarget height increases the grazing angle gets larger and thecomplex reflection coefficient over indirect paths becomessmaller which is shown in Figure 7 Thus the detectionperformance deteriorates

In Figure 8 we analyze the effect of varying the directionsof target velocity vector on the detection performance Herek119879 = [100 0] [717 717] and [0 100]ms respectivelyFigures 8(a) and 8(b) show 119875119889 versus SNR when 119875fa = 10

minus3

and 119875119889 versus 119875fa when SNR is fixed to minus20 dB respectivelyIn this paper we take advantage of multipath propagationby exploiting multiple Doppler shifts corresponding to theprojections of target velocity on each of the multipathcomponents When the angle between the target velocityvector and radar LOS decreases the Doppler frequency of

8 International Journal of Antennas and Propagation

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1 Pfa = 10minus3

P = 3 Pfa = 10minus3P = 1 Pfa = 10minus5

P = 3 Pfa = 10minus5

(a) 119875119889 versus SNR with different 119875fas

02

03

04

05

06

07

08

09

1

Probability of false alarm (Pfa)10minus3 10minus2 10minus1 100

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1

P = 3

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 4 Effect of the path number on the detection performance

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 1

N = 3

N = 5

(a) 119875119889 versus SNR when 119875fa = 10minus3

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 1

N = 3

N = 5

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 5 Effect of the subcarrier number on the detection performance

received signal will increaseThus the detection performanceis improved

In Figure 9 we show the detection performance improve-ment due to the adaptive waveform design Figure 9(a) shows119875119889 versus SNR when 119875fa = 10

minus3 while Figure 9(b) shows 119875119889versus 119875fa when SNR is fixed to minus20 dB We assume that thetransmitted waveform in the first119872 pulses has equal weightsthat is w = 1radic119873I Then based on (49) we compute woptfor the next 119872 pulses As is shown compared with fixed

waveform the detection performance is improved due to theadaptive waveform method proposed in this paper

Finally to validate our method in non-Gaussian scenar-ios we conduct additional experiments using compound K-distribution whose PDF is [29]

119891 (119911) =radic2V120583Γ (V)

(radic2V120583119911)

V

119870Vminus1(radic2V120583119911) (51)

International Journal of Antennas and Propagation 9

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

ht = 200mht = 400mht = 600m

(a) 119875119889 versus SNR when 119875fa = 10minus3

01

02

03

04

05

06

07

08

09

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

ht = 200mht = 400mht = 600m

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 6 Detection performance varies with target height

0 500 1000 1500 20000

02

04

06

08

1

Am

plitu

de o

f com

plex

refle

ctio

n co

effici

ent

Horizontal polarizationVertical polarization

Target height m

Figure 7 Amplitude of complex reflection coefficient varies with target height

and the texture follows a gamma distribution with PDF

119891 (119906) =1

Γ (V)(V120583)

V

119906Vminus1

119890minusV120583119906

120581 (119906) (52)

where Γ(sdot) is the Eulerian gamma function 120581(sdot) is the unitstep function and 119870Vminus1(sdot) is the modified Bessel function ofsecond kind with order V minus 1 120583 and V are the scale and shapeparameter respectively The speckle component is assumed

to be a Gaussian distribution with exponential correlationstructure covariance matrix [30]

[C]119894119895 = 120588|119894minus119895|

for 119894 119895 = 0 1 119873 minus 1 (53)

where 120588 is one-lag correlation coefficient The other parame-ters are the same as in Table 1

Figure 10 shows the detection performance with differentpath numbers (119875 = 1 3) in Gaussian and compound

10 International Journal of Antennas and Propagation

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 8 Effect of different directions of target velocity on detection performance

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 9 Comparison of detection performance with fixed waveform and adaptive waveform

K-distribution clutter and Figure 11 shows the detectionperformance with fixed and adaptive waveform design inGaussian and compound K-distribution clutter We find thatin compound K-distribution the detector derived underGaussian assumption also yields improved performance bytaking advantage of multipath effect and adaptive waveformdesign of OFDM radar However the performance gap existsdue to the model mismatch

7 Conclusions

In this paper we address the problem of target detectionin low grazing angle scenario with OFDM radar We con-sider the realistic physical and statistical effects to makethe propagation model more realistic Based on the char-acteristics of OFDM signal we develop a GLRT detectorand show that detection performance is improved due tothe multipath utilization and frequency diversity of OFDM

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

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DistributedSensor Networks

International Journal of

6 International Journal of Antennas and Propagation

Taking derivative of (41) with respect toC0 andmaking itequal to zero we get MLE of C0

C0 =1

119873YY119867 (42)

Substituting (34) (35) (37) and (42) into (31) the GLRTdetector is

GLR (k) =10038161003816100381610038161003816C0

1003816100381610038161003816100381610038161003816100381610038161003816C1

10038161003816100381610038161003816

=

10038161003816100381610038161003816(1119873)YY11986710038161003816100381610038161003816

100381610038161003816100381610038161003816(1119873) (Y minusWXΦ (k)) (Y minusWXΦ (k))

119867100381610038161003816100381610038161003816

(43)

Since target velocity k is unknown it can be estimatedthrough k = argmaxkGLR(k) andtheGLRTdetector becomes

maxk

GLR (k) = GLR (k)H1

≶

H0

120574 (44)

5 Adaptive Waveform Design

In this section we develop an adaptive waveform designmethod based on maximizing the Mahalanobis-distanceto improve the detection performance Since the targetscattering coefficients vary with different subcarriers wemay change the transmitted weights W accordingly From(30) we know that the measurement Y is distributedas CN119873119872(WXΦ(k)C otimes I119872) under H1 hypothesis andCN119873119872(0C otimes I119872) under H0 hypothesis The squaredMahalanobis-distance is used to distinguish the above twodistributions It is noticeable that the column ofY is uncorre-lated which means that the measurements between differentpulses are independent The sum of squared Mahalanobis-distance over different pulses is given by [28]

1198892=

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898)) (45)

where120573119898 is a positive number and denotes theweight ofMaha-lanobis-distance over the119898th pulse satisfying sum119872minus1

119898=0120573119898 = 1

To maximize the detection performance we can formulatethe optimization as

wopt = arg maxWisinC119873

[

119872minus1

sum

119898=0

120573119898 (WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))]

subject to w119867w = 1

(46)

The scattering coefficients are constant from pulse topulse and the measurement noise is uncorrelated betweendifferent pulses thus 120573119898 is a constant number that is 120573119898 =

1119872 for119898 = 0 1 119872 minus 1 Since

(WXΦ ( 119898))119867Cminus1 (WXΦ ( 119898))

= tr (Φ ( 119898)119867X119867W119867Cminus1WXΦ ( 119898))

= tr (Cminus1WXΦ ( 119898)Φ ( 119898)119867X119867W119867

)

(47)

the following theorem [11] is used to simplify the aboveequation

tr (S1AS2A119867) = a119867 [S119879

2⊙ S1] a (48)

where a isin C119873 and A = diag(a) ⊙ denotes element-wiseHadamard product Combining (47) and (48) we get

wopt = arg maxWisinC119873

(1

119872)w119867 [

119872minus1

sum

119898=0

(XΦ ( 119898)Φ ( 119898)119867X119867)

119879

⊙ Cminus1]w

subject to w119867w = 1

(49)

From (49) we know that the optimization problemis reduced to eigenvalue eigenvector problem and wopt isthe eigenvector corresponding to the largest eigenvalue of[sum

119872minus1

119898=0(XΦ( 119898)Φ( 119898)

119867X119867)119879 ⊙ Cminus1]

6 Numerical Results

In this section several numerical examples are presentedto illustrate the performance of proposed detector For

simplicity we consider 2D scenario as shown in Figure 2Thesimulation parameters are shown in Table 1

We use Monte Carlo simulations to realize the followingresults for the analytical expression between false alarm rateand threshold cannot be acquired The relation betweenprobability of false alarm rate and detection threshold corre-sponding to different carrier numbers is shown in Figure 3The curves show that when false alarm rate is kept fixed

International Journal of Antennas and Propagation 7

1

15

2

25

3

35

4

45

5

Det

ectio

n th

resh

old

(Th)

Probability of false alarm (Pfa)10minus3 10minus210minus5 10minus4 10minus1

N = 1

N = 3

N = 5

Figure 3 Detection threshold versus probability of false alarm at different carrier numbers

Table 1 Parameter settings of the simulations

Parameters ValuesHeight of target 200mVelocity of target [1414 1414]119879msScattering coefficient matrix of target X CN(0 1)

Height of radar platform 100mVelocity of radar platform [0 0]119879msRelative distance between radar and target 5 kmMultipath number 119875 3Carrier frequency 1198910 1 GHzBandwidth 119861 100MHzCarrier numbers119873 3Frequency weightsW 1radic119873I119873Pulse number119872 30Pulse repetition interval 119879119903 2ms

the higher the subcarrier number the higher the thresholdSignal-to-noise ratio (SNR) is defined as [11]

SNR =(1119873)sum

119872

119898=1(WX120601 (119898 k))119867WX120601 (119898 k)

tr (C) (50)

The effect of multipath number on the detection perfor-mance is shown in Figure 4 where the path number is setas 119875 = 1 and 119875 = 3 Figures 4(a) and 4(b) show 119875119889 versusSNR with different 119875fas and 119875119889 versus 119875fa when SNR is fixedto minus20 dB respectively The curves show that the detectionperformance is improved with the increasing of multipathnumber Consequently the derived detector with OFDM

waveform could exploit multipath reflections to enhance thedetection performance

In Figure 5 we analyze the effect of subcarrier number onthe performance of proposed detector where the subcarriernumber is set as 119873 = 1 3 and 5 respectively Figures 5(a)and 5(b) show 119875119889 versus SNR when 119875fa = 10

minus3 and 119875119889

versus 119875fa when SNR is fixed to minus20 dB respectively Thedetection performance is improved with the increasing ofsubcarrier number The results demonstrate that the detec-tion performance is improved due to the frequency diversityin an OFDM radar system

Figure 6 shows that the detection performance varieswith the target height which is set as 200m 400m and600m respectively Figure 6(a) shows 119875119889 versus SNR when119875fa = 10

minus3 while Figure 6(b) shows 119875119889 versus 119875fa when SNRis fixed to minus20 dB The results demonstrate that detectionperformance decreases with the increasing of target height inlow grazing angle scenario This can be explained that as thetarget height increases the grazing angle gets larger and thecomplex reflection coefficient over indirect paths becomessmaller which is shown in Figure 7 Thus the detectionperformance deteriorates

In Figure 8 we analyze the effect of varying the directionsof target velocity vector on the detection performance Herek119879 = [100 0] [717 717] and [0 100]ms respectivelyFigures 8(a) and 8(b) show 119875119889 versus SNR when 119875fa = 10

minus3

and 119875119889 versus 119875fa when SNR is fixed to minus20 dB respectivelyIn this paper we take advantage of multipath propagationby exploiting multiple Doppler shifts corresponding to theprojections of target velocity on each of the multipathcomponents When the angle between the target velocityvector and radar LOS decreases the Doppler frequency of

8 International Journal of Antennas and Propagation

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1 Pfa = 10minus3

P = 3 Pfa = 10minus3P = 1 Pfa = 10minus5

P = 3 Pfa = 10minus5

(a) 119875119889 versus SNR with different 119875fas

02

03

04

05

06

07

08

09

1

Probability of false alarm (Pfa)10minus3 10minus2 10minus1 100

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1

P = 3

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 4 Effect of the path number on the detection performance

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 1

N = 3

N = 5

(a) 119875119889 versus SNR when 119875fa = 10minus3

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 1

N = 3

N = 5

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 5 Effect of the subcarrier number on the detection performance

received signal will increaseThus the detection performanceis improved

In Figure 9 we show the detection performance improve-ment due to the adaptive waveform design Figure 9(a) shows119875119889 versus SNR when 119875fa = 10

minus3 while Figure 9(b) shows 119875119889versus 119875fa when SNR is fixed to minus20 dB We assume that thetransmitted waveform in the first119872 pulses has equal weightsthat is w = 1radic119873I Then based on (49) we compute woptfor the next 119872 pulses As is shown compared with fixed

waveform the detection performance is improved due to theadaptive waveform method proposed in this paper

Finally to validate our method in non-Gaussian scenar-ios we conduct additional experiments using compound K-distribution whose PDF is [29]

119891 (119911) =radic2V120583Γ (V)

(radic2V120583119911)

V

119870Vminus1(radic2V120583119911) (51)

International Journal of Antennas and Propagation 9

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

ht = 200mht = 400mht = 600m

(a) 119875119889 versus SNR when 119875fa = 10minus3

01

02

03

04

05

06

07

08

09

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

ht = 200mht = 400mht = 600m

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 6 Detection performance varies with target height

0 500 1000 1500 20000

02

04

06

08

1

Am

plitu

de o

f com

plex

refle

ctio

n co

effici

ent

Horizontal polarizationVertical polarization

Target height m

Figure 7 Amplitude of complex reflection coefficient varies with target height

and the texture follows a gamma distribution with PDF

119891 (119906) =1

Γ (V)(V120583)

V

119906Vminus1

119890minusV120583119906

120581 (119906) (52)

where Γ(sdot) is the Eulerian gamma function 120581(sdot) is the unitstep function and 119870Vminus1(sdot) is the modified Bessel function ofsecond kind with order V minus 1 120583 and V are the scale and shapeparameter respectively The speckle component is assumed

to be a Gaussian distribution with exponential correlationstructure covariance matrix [30]

[C]119894119895 = 120588|119894minus119895|

for 119894 119895 = 0 1 119873 minus 1 (53)

where 120588 is one-lag correlation coefficient The other parame-ters are the same as in Table 1

Figure 10 shows the detection performance with differentpath numbers (119875 = 1 3) in Gaussian and compound

10 International Journal of Antennas and Propagation

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 8 Effect of different directions of target velocity on detection performance

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 9 Comparison of detection performance with fixed waveform and adaptive waveform

K-distribution clutter and Figure 11 shows the detectionperformance with fixed and adaptive waveform design inGaussian and compound K-distribution clutter We find thatin compound K-distribution the detector derived underGaussian assumption also yields improved performance bytaking advantage of multipath effect and adaptive waveformdesign of OFDM radar However the performance gap existsdue to the model mismatch

7 Conclusions

In this paper we address the problem of target detectionin low grazing angle scenario with OFDM radar We con-sider the realistic physical and statistical effects to makethe propagation model more realistic Based on the char-acteristics of OFDM signal we develop a GLRT detectorand show that detection performance is improved due tothe multipath utilization and frequency diversity of OFDM

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 7

1

15

2

25

3

35

4

45

5

Det

ectio

n th

resh

old

(Th)

Probability of false alarm (Pfa)10minus3 10minus210minus5 10minus4 10minus1

N = 1

N = 3

N = 5

Figure 3 Detection threshold versus probability of false alarm at different carrier numbers

Table 1 Parameter settings of the simulations

Parameters ValuesHeight of target 200mVelocity of target [1414 1414]119879msScattering coefficient matrix of target X CN(0 1)

Height of radar platform 100mVelocity of radar platform [0 0]119879msRelative distance between radar and target 5 kmMultipath number 119875 3Carrier frequency 1198910 1 GHzBandwidth 119861 100MHzCarrier numbers119873 3Frequency weightsW 1radic119873I119873Pulse number119872 30Pulse repetition interval 119879119903 2ms

the higher the subcarrier number the higher the thresholdSignal-to-noise ratio (SNR) is defined as [11]

SNR =(1119873)sum

119872

119898=1(WX120601 (119898 k))119867WX120601 (119898 k)

tr (C) (50)

The effect of multipath number on the detection perfor-mance is shown in Figure 4 where the path number is setas 119875 = 1 and 119875 = 3 Figures 4(a) and 4(b) show 119875119889 versusSNR with different 119875fas and 119875119889 versus 119875fa when SNR is fixedto minus20 dB respectively The curves show that the detectionperformance is improved with the increasing of multipathnumber Consequently the derived detector with OFDM

waveform could exploit multipath reflections to enhance thedetection performance

In Figure 5 we analyze the effect of subcarrier number onthe performance of proposed detector where the subcarriernumber is set as 119873 = 1 3 and 5 respectively Figures 5(a)and 5(b) show 119875119889 versus SNR when 119875fa = 10

minus3 and 119875119889

versus 119875fa when SNR is fixed to minus20 dB respectively Thedetection performance is improved with the increasing ofsubcarrier number The results demonstrate that the detec-tion performance is improved due to the frequency diversityin an OFDM radar system

Figure 6 shows that the detection performance varieswith the target height which is set as 200m 400m and600m respectively Figure 6(a) shows 119875119889 versus SNR when119875fa = 10

minus3 while Figure 6(b) shows 119875119889 versus 119875fa when SNRis fixed to minus20 dB The results demonstrate that detectionperformance decreases with the increasing of target height inlow grazing angle scenario This can be explained that as thetarget height increases the grazing angle gets larger and thecomplex reflection coefficient over indirect paths becomessmaller which is shown in Figure 7 Thus the detectionperformance deteriorates

In Figure 8 we analyze the effect of varying the directionsof target velocity vector on the detection performance Herek119879 = [100 0] [717 717] and [0 100]ms respectivelyFigures 8(a) and 8(b) show 119875119889 versus SNR when 119875fa = 10

minus3

and 119875119889 versus 119875fa when SNR is fixed to minus20 dB respectivelyIn this paper we take advantage of multipath propagationby exploiting multiple Doppler shifts corresponding to theprojections of target velocity on each of the multipathcomponents When the angle between the target velocityvector and radar LOS decreases the Doppler frequency of

8 International Journal of Antennas and Propagation

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1 Pfa = 10minus3

P = 3 Pfa = 10minus3P = 1 Pfa = 10minus5

P = 3 Pfa = 10minus5

(a) 119875119889 versus SNR with different 119875fas

02

03

04

05

06

07

08

09

1

Probability of false alarm (Pfa)10minus3 10minus2 10minus1 100

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1

P = 3

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 4 Effect of the path number on the detection performance

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 1

N = 3

N = 5

(a) 119875119889 versus SNR when 119875fa = 10minus3

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 1

N = 3

N = 5

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 5 Effect of the subcarrier number on the detection performance

received signal will increaseThus the detection performanceis improved

In Figure 9 we show the detection performance improve-ment due to the adaptive waveform design Figure 9(a) shows119875119889 versus SNR when 119875fa = 10

minus3 while Figure 9(b) shows 119875119889versus 119875fa when SNR is fixed to minus20 dB We assume that thetransmitted waveform in the first119872 pulses has equal weightsthat is w = 1radic119873I Then based on (49) we compute woptfor the next 119872 pulses As is shown compared with fixed

waveform the detection performance is improved due to theadaptive waveform method proposed in this paper

Finally to validate our method in non-Gaussian scenar-ios we conduct additional experiments using compound K-distribution whose PDF is [29]

119891 (119911) =radic2V120583Γ (V)

(radic2V120583119911)

V

119870Vminus1(radic2V120583119911) (51)

International Journal of Antennas and Propagation 9

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

ht = 200mht = 400mht = 600m

(a) 119875119889 versus SNR when 119875fa = 10minus3

01

02

03

04

05

06

07

08

09

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

ht = 200mht = 400mht = 600m

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 6 Detection performance varies with target height

0 500 1000 1500 20000

02

04

06

08

1

Am

plitu

de o

f com

plex

refle

ctio

n co

effici

ent

Horizontal polarizationVertical polarization

Target height m

Figure 7 Amplitude of complex reflection coefficient varies with target height

and the texture follows a gamma distribution with PDF

119891 (119906) =1

Γ (V)(V120583)

V

119906Vminus1

119890minusV120583119906

120581 (119906) (52)

where Γ(sdot) is the Eulerian gamma function 120581(sdot) is the unitstep function and 119870Vminus1(sdot) is the modified Bessel function ofsecond kind with order V minus 1 120583 and V are the scale and shapeparameter respectively The speckle component is assumed

to be a Gaussian distribution with exponential correlationstructure covariance matrix [30]

[C]119894119895 = 120588|119894minus119895|

for 119894 119895 = 0 1 119873 minus 1 (53)

where 120588 is one-lag correlation coefficient The other parame-ters are the same as in Table 1

Figure 10 shows the detection performance with differentpath numbers (119875 = 1 3) in Gaussian and compound

10 International Journal of Antennas and Propagation

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 8 Effect of different directions of target velocity on detection performance

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 9 Comparison of detection performance with fixed waveform and adaptive waveform

K-distribution clutter and Figure 11 shows the detectionperformance with fixed and adaptive waveform design inGaussian and compound K-distribution clutter We find thatin compound K-distribution the detector derived underGaussian assumption also yields improved performance bytaking advantage of multipath effect and adaptive waveformdesign of OFDM radar However the performance gap existsdue to the model mismatch

7 Conclusions

In this paper we address the problem of target detectionin low grazing angle scenario with OFDM radar We con-sider the realistic physical and statistical effects to makethe propagation model more realistic Based on the char-acteristics of OFDM signal we develop a GLRT detectorand show that detection performance is improved due tothe multipath utilization and frequency diversity of OFDM

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1 Pfa = 10minus3

P = 3 Pfa = 10minus3P = 1 Pfa = 10minus5

P = 3 Pfa = 10minus5

(a) 119875119889 versus SNR with different 119875fas

02

03

04

05

06

07

08

09

1

Probability of false alarm (Pfa)10minus3 10minus2 10minus1 100

Prob

abili

ty o

f det

ectio

n (P

d)

P = 1

P = 3

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 4 Effect of the path number on the detection performance

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 1

N = 3

N = 5

(a) 119875119889 versus SNR when 119875fa = 10minus3

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 1

N = 3

N = 5

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 5 Effect of the subcarrier number on the detection performance

received signal will increaseThus the detection performanceis improved

In Figure 9 we show the detection performance improve-ment due to the adaptive waveform design Figure 9(a) shows119875119889 versus SNR when 119875fa = 10

minus3 while Figure 9(b) shows 119875119889versus 119875fa when SNR is fixed to minus20 dB We assume that thetransmitted waveform in the first119872 pulses has equal weightsthat is w = 1radic119873I Then based on (49) we compute woptfor the next 119872 pulses As is shown compared with fixed

waveform the detection performance is improved due to theadaptive waveform method proposed in this paper

Finally to validate our method in non-Gaussian scenar-ios we conduct additional experiments using compound K-distribution whose PDF is [29]

119891 (119911) =radic2V120583Γ (V)

(radic2V120583119911)

V

119870Vminus1(radic2V120583119911) (51)

International Journal of Antennas and Propagation 9

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

ht = 200mht = 400mht = 600m

(a) 119875119889 versus SNR when 119875fa = 10minus3

01

02

03

04

05

06

07

08

09

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

ht = 200mht = 400mht = 600m

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 6 Detection performance varies with target height

0 500 1000 1500 20000

02

04

06

08

1

Am

plitu

de o

f com

plex

refle

ctio

n co

effici

ent

Horizontal polarizationVertical polarization

Target height m

Figure 7 Amplitude of complex reflection coefficient varies with target height

and the texture follows a gamma distribution with PDF

119891 (119906) =1

Γ (V)(V120583)

V

119906Vminus1

119890minusV120583119906

120581 (119906) (52)

where Γ(sdot) is the Eulerian gamma function 120581(sdot) is the unitstep function and 119870Vminus1(sdot) is the modified Bessel function ofsecond kind with order V minus 1 120583 and V are the scale and shapeparameter respectively The speckle component is assumed

to be a Gaussian distribution with exponential correlationstructure covariance matrix [30]

[C]119894119895 = 120588|119894minus119895|

for 119894 119895 = 0 1 119873 minus 1 (53)

where 120588 is one-lag correlation coefficient The other parame-ters are the same as in Table 1

Figure 10 shows the detection performance with differentpath numbers (119875 = 1 3) in Gaussian and compound

10 International Journal of Antennas and Propagation

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 8 Effect of different directions of target velocity on detection performance

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 9 Comparison of detection performance with fixed waveform and adaptive waveform

K-distribution clutter and Figure 11 shows the detectionperformance with fixed and adaptive waveform design inGaussian and compound K-distribution clutter We find thatin compound K-distribution the detector derived underGaussian assumption also yields improved performance bytaking advantage of multipath effect and adaptive waveformdesign of OFDM radar However the performance gap existsdue to the model mismatch

7 Conclusions

In this paper we address the problem of target detectionin low grazing angle scenario with OFDM radar We con-sider the realistic physical and statistical effects to makethe propagation model more realistic Based on the char-acteristics of OFDM signal we develop a GLRT detectorand show that detection performance is improved due tothe multipath utilization and frequency diversity of OFDM

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 9

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

ht = 200mht = 400mht = 600m

(a) 119875119889 versus SNR when 119875fa = 10minus3

01

02

03

04

05

06

07

08

09

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

ht = 200mht = 400mht = 600m

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 6 Detection performance varies with target height

0 500 1000 1500 20000

02

04

06

08

1

Am

plitu

de o

f com

plex

refle

ctio

n co

effici

ent

Horizontal polarizationVertical polarization

Target height m

Figure 7 Amplitude of complex reflection coefficient varies with target height

and the texture follows a gamma distribution with PDF

119891 (119906) =1

Γ (V)(V120583)

V

119906Vminus1

119890minusV120583119906

120581 (119906) (52)

where Γ(sdot) is the Eulerian gamma function 120581(sdot) is the unitstep function and 119870Vminus1(sdot) is the modified Bessel function ofsecond kind with order V minus 1 120583 and V are the scale and shapeparameter respectively The speckle component is assumed

to be a Gaussian distribution with exponential correlationstructure covariance matrix [30]

[C]119894119895 = 120588|119894minus119895|

for 119894 119895 = 0 1 119873 minus 1 (53)

where 120588 is one-lag correlation coefficient The other parame-ters are the same as in Table 1

Figure 10 shows the detection performance with differentpath numbers (119875 = 1 3) in Gaussian and compound

10 International Journal of Antennas and Propagation

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 8 Effect of different directions of target velocity on detection performance

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 9 Comparison of detection performance with fixed waveform and adaptive waveform

K-distribution clutter and Figure 11 shows the detectionperformance with fixed and adaptive waveform design inGaussian and compound K-distribution clutter We find thatin compound K-distribution the detector derived underGaussian assumption also yields improved performance bytaking advantage of multipath effect and adaptive waveformdesign of OFDM radar However the performance gap existsdue to the model mismatch

7 Conclusions

In this paper we address the problem of target detectionin low grazing angle scenario with OFDM radar We con-sider the realistic physical and statistical effects to makethe propagation model more realistic Based on the char-acteristics of OFDM signal we develop a GLRT detectorand show that detection performance is improved due tothe multipath utilization and frequency diversity of OFDM

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Antennas and Propagation

minus30 minus20 minus10 00

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

x = 100 y = 0msx = y = 717msx = 0 y = 100ms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 8 Effect of different directions of target velocity on detection performance

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 50

02

04

06

08

1

SNR (dB)

Prob

abili

ty o

f det

ectio

n (P

d)

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(a) 119875119889 versus SNR when 119875fa = 10minus3

04

05

06

07

08

09

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

N = 3 fixed waveformsN = 3 adaptive waveforms

N = 5 fixed waveformsN = 5 adaptive waveforms

(b) 119875119889 versus 119875fa when SNR = minus20 dB

Figure 9 Comparison of detection performance with fixed waveform and adaptive waveform

K-distribution clutter and Figure 11 shows the detectionperformance with fixed and adaptive waveform design inGaussian and compound K-distribution clutter We find thatin compound K-distribution the detector derived underGaussian assumption also yields improved performance bytaking advantage of multipath effect and adaptive waveformdesign of OFDM radar However the performance gap existsdue to the model mismatch

7 Conclusions

In this paper we address the problem of target detectionin low grazing angle scenario with OFDM radar We con-sider the realistic physical and statistical effects to makethe propagation model more realistic Based on the char-acteristics of OFDM signal we develop a GLRT detectorand show that detection performance is improved due tothe multipath utilization and frequency diversity of OFDM

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 11

0

02

04

06

08

1

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10010minus110minus210minus3

P = 1 GaussianP = 3 Gaussian

P = 1 compound KP = 3 compound K

Figure 10 Effect of themultipath number ondetection performancein Gaussian and compound K-distribution clutter

02

03

04

05

06

07

08

09

1

Fixed waveform GaussianAdaptive waveform GaussianFixed waveform compound KAdaptive waveform compound K

Probability of false alarm (Pfa)

Prob

abili

ty o

f det

ectio

n (P

d)

10minus3 10minus2 10minus1 100

Figure 11 Comparison of detection performance with fixed andadaptive waveform in Gaussian and compound K-distributionclutter (119875 = 3)

radar Then an adaptive waveform design method basedon Mahalanobis-distance is proposed and the detectionperformance is improved due to the optimized transmittedfrequency weights Finally we show that the GLRT detectorderived under Gaussian assumption yields improved perfor-mance in compound K-distribution clutter due to the multi-path utilization and adaptive waveform design However theperformance gap exists and we will extend our work to moregeneral situations in the future work

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ScienceFoundation of China under Grant no 61401475

References

[1] D K Barton ldquoLow-angle radar trackingrdquo Proceedings of theIEEE vol 62 no 6 pp 687ndash704 1974

[2] S L Silon and B D Carlson ldquoRadar detection in multipathrdquoIEE ProceedingsmdashRadar Sonar and Navigation vol 146 no 1pp 45ndash54 1999

[3] J-H Zhao and J-Y Yang ldquoFrequency diversity to low-angledetecting using a highly deterministic multipath signal modelrdquoin Proceedings of the 6th CIE International Conference on Radar(ICR rsquo06) pp 1ndash5 IEEE Shanghai China October 2006

[4] M Jankiraman B J Wessels and P van Genderen ldquoDesign ofa multifrequency FMCW radarrdquo in Proceedings of the 28thEuropean Microwave Conference (EuMC rsquo98) pp 584ndash589Amsterdam The Netherlands October 1998

[5] N Levanon ldquoMultifrequency complementary phase-codedradar signalrdquo IEE Proceedings Radar Sonar andNavigation vol147 no 6 pp 276ndash284 2000

[6] R Mohseni A Sheikhi and M A Masnadi-Shirazi ldquoMulticar-rier constant envelope OFDM signal design for radar applica-tionsrdquo AEU International Journal of Electronics and Communi-cations vol 64 no 11 pp 999ndash1008 2010

[7] G Lellouch A Mishra and M Inggs ldquoImpact of the Dopplermodulation on the range and Doppler processing in OFDMradarrdquo in Proceedings of the IEEE Radar Conference (RadarConrsquo14) pp 803ndash808 Cincinnati Ohio USA May 2014

[8] S Sen and A Nehorai ldquoAdaptive design of OFDM radarsignal with improved wideband ambiguity functionrdquo IEEETransactions on Signal Processing vol 58 no 2 pp 928ndash9332010

[9] G Lellouch and A K Mishra ldquoOptimization of OFDM radarwaveforms using geneticalgorithmsrdquo httparxivorgabs14054894

[10] K Huo Z K Qiu Y X Liu and W D Jiang ldquoAn adaptivewaveform design method for OFDM cognitive radarrdquo in Pro-ceedings of the International Radar Conference pp 1ndash5 IEEELille France October 2014

[11] S Sen and A Nehorai ldquoAdaptive OFDM radar for targetdetection in multipath scenariosrdquo IEEE Transactions on SignalProcessing vol 59 no 1 pp 78ndash90 2011

[12] S Sen M Hurtado and A Nehorai ldquoAdaptive OFDM radar fordetecting a moving target in urban scenariosrdquo in Proceedingsof the International Waveform Diversity and Design Conference(WDD rsquo09) pp 268ndash272 IEEE Orlando Fla USA February2009

[13] S Sen G Tang andANehorai ldquoMultiobjective optimization ofOFDM radar waveform for target detectionrdquo IEEE Transactionson Signal Processing vol 59 no 2 pp 639ndash652 2011

[14] S Sen G Tang andANehorai ldquoOFDMradarwaveformdesignfor sparsity-based multi-target trackingrdquo in Proceedings of the

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 International Journal of Antennas and Propagation

InternationalWaveformDiversity and Design Conference (WDDrsquo10) pp 18ndash22 IEEE Niagara Falls Canada August 2010

[15] S Kafshgari and R Mohseni ldquoFluctuating target detectionin presence of non Gaussian clutter in OFDM radarsrdquo AEUInternational Journal of Electronics andCommunications vol 67no 10 pp 885ndash893 2013

[16] W D White ldquoLow-angle radar tracking in the presence ofmultipathrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 10 no 6 pp 835ndash852 1974

[17] S Sen and A Nehorai ldquoOFDM MIMO radar with mutual-information waveform design for low-grazing angle trackingrdquoIEEE Transactions on Signal Processing vol 58 no 6 pp 3152ndash3162 2010

[18] H V Hitney J H Richter R A Pappert K D Anderson andG B Baumgartner Jr ldquoTropospheric radio propagation assess-mentrdquo Proceedings of the IEEE vol 73 no 2 pp 265ndash283 1985

[19] H Hitney and R Vieth ldquoStatistical assessment of evaporationduct propagationrdquo IEEE Transactions on Antennas and Propa-gation vol 38 no 6 pp 794ndash799 1990

[20] M W Long Radar Reflectivity of Land Sea Artech House 3rdedition 2001

[21] T Lo and J Litva ldquoUse of a highly deterministicmultipath signalmodel in low-angle trackingrdquo IEE Proceedings FmdashRadar andSignal Processing vol 138 no 2 pp 163ndash171 1991

[22] F Gini M V Greco M Diani and L Verrazzani ldquoPerformanceanalysis of two adaptive radar detectors against non-Gaussianreal sea clutter datardquo IEEE Transactions on Aerospace andElectronic Systems vol 36 no 4 pp 1429ndash1439 2000

[23] K James Sangston F Gini M V Greco and A Farina ldquoStruc-tures for radar detection in compound gaussian clutterrdquo IEEETransactions on Aerospace and Electronic Systems vol 35 no 2pp 445ndash458 1999

[24] S M Kay Fundamentals of Statistical Signal Processing Detec-tion Theory Prentice Hall PTR Upper Saddle River NJ USA1998

[25] T W Anderson An Introduction to Multivariate StatisticalAnalysis Wiley Series in Probability and Statistics Wiley-Inter-science John Wiley amp Sons Hoboken NJ USA 3rd edition2003

[26] L Xu P Stoica and J Li ldquoA block-diagonal growth curvemodelrdquo Digital Signal Processing vol 16 no 6 pp 902ndash9122006

[27] A Dogandzic and A Nehorai ldquoGeneralized multivariate anal-ysis of variance a unified framework for signal processing incorrelated noiserdquo IEEE Signal Processing Magazine vol 20 no5 pp 39ndash54 2003

[28] R DeMaesschalck D Jouan-Rimbaud and D L Massart ldquoTheMahalanobis distancerdquo Chemometrics and Intelligent Labora-tory Systems vol 50 no 1 pp 1ndash18 2000

[29] E Conte M Longo andM Lops ldquoModelling and simulation ofnon-Rayleigh radar clutterrdquo IEE Proceedings F Radar and SignalProcessing vol 138 no 2 pp 121ndash130 1991

[30] G Cui L Kong and X Yang ldquoMultiple-input multiple-outputradar detectors design in non-Gaussian clutterrdquo IET RadarSonar amp Navigation vol 4 no 5 pp 724ndash732 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of