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Published in IET Signal Processing

Received on 19th February 2009

Revised on 11th August 2009

doi: 10.1049/iet-spr.2009.0039

Special Issue on Time-Frequency Approach to Radar

Detection, Imaging, and Classification

ISSN 1751-9675

Dual-frequency Doppler radars for indoorrange estimation: CramerRao bound analysisP. Setlur M. Amin F. AhmadRadar Imaging Lab, Center for Advanced Communications, Villanova University, 800 Lancaster Ave, Villanova, PA 19085, USA

E-mail: [email protected]

Abstract: Single frequency Doppler radars are effective in distinguishing moving targets from stationary targets,

but suffer from inherent range ambiguity. With a dual-frequency operation, a second carrier frequency is utilised

to overcome the range ambiguity problem. In urban sensing applications, the dual-frequency approach offers the

benefit of reduced complexity, fast computation time and real-time target tracking. The authors consider a single

moving target with three commonly exhibited indoor motion profiles, namely, constant velocity motion,

accelerating target motion and micro-Doppler (MD) motion. RF signatures of indoor inanimate objects, such

as fans, vibrating machineries and clock pendulums, are characterised by MD motion, whereas animate

translation movements produce linear FM Doppler. CramerRao bounds (CRBs) for the parameters defining

indoor target motions under dual-frequency implementations are derived. Experimental data are used to

estimate MD parameters and to validate the CRBs.

1 Introduction

The emerging area of through-the-wall sensing addresses thedesire to see inside enclosed structures and behind wallsin order to determine building layouts, scene contents andoccupant locations. This capability has merits in a variety ofcivilian and military applications, as it provides vision intootherwise obscured areas, thereby, facilitating information-gathering and intelligent decision-making [14].

Motion detection is a highly desirable feature inmany through-the-wall applications, such as firefighterssearching for survivors or law-enforcement officers involvedin hostage rescue missions. Doppler discrimination ofmovement from stationary background clutter can bereadily used in these applications. Such systems can beclassified into zero- (0D), one-, two- or three-dimensionalsystems [1]. A 0D system simply provides a go/no-gomotion detection capability and, as such, can employ singlefrequency continuous wave (CW) waveforms for sceneinterrogation. A 1D system can provide range to moving

targets by employing modulated or pulsed signals. A 2Dsystem provides both target range and azimuth information, whereas a 3D system adds the dimension of height to the

offerings of a 2D system. However, the higher level ofsituational awareness offered by 2D and 3D systems isobtained at the expense of increased hardware and softwarecomplexity and higher computational load. A 1D systemprovides a good compromise between the level ofsituational awareness and cost against size and weight. Thisis specifically the case when localising moving targets is ofprimary interest.

In this paper, we consider a 1D system for range-to-motion estimation based on Doppler radars. Singlefrequency CW radars suffer from range ambiguity [5].

This problem is more pronounced for through-the- wall applications since the carrier frequencies are in thefew GHz range because of antenna size and frequencyallocation management issues. An additional carrierfrequency can be introduced to resolve the uncertainty inrange. Two carrier frequencies can be chosen such that themaximum unambiguous range is either equal to or greaterthan the spatial extent of the urban structure. In this case,the true solution is the one which corresponds to the target

inside the enclosed structure. It is noted that other radartechniques, such as the swept frequency, pulse compressionetc., can be used to reduce the uncertainty in range [5].

256 IET Signal Process., 2010, Vol. 4, Iss. 3, pp. 256271

& The Institution of Engineering and Technology 2010 doi: 10.1049/iet-spr.2009.0039

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However, the operational logistics and system requirementsfor through-the-wall operations, such as cost, hardwarecomplexity and portability, may prohibit the use of suchtechniques. The dual-frequency approach, although notnew, meets all of the above requirements and is likely toemerge as a leading forerunner in modern urban sensing

systems [69].

We focus on three target motion and range profiles,namely, linear translation, acceleration and the micro-Doppler (MD) motions that are commonly encounteredin urban sensing applications. For example, the swingingof the arms and the legs of a person during walking can bemodelled as MD, whereas the gross motion of the torsocan be represented by translation. We develop Cramer Rao bounds (CRBs) for the parameters correspondingto the aforementioned three indoor motion profiles.

We provide the CRB for the range estimate in addition

to motion speed, harmonic frequency and chirpparameters. Specifically, we show that the dual-frequencyapproach increases the Fisher information compared to asingle frequency operation and, therefore lowers the CRBs[10]. In the analysis, the radar aspect angle u isincorporated into the general CRB derivations. When theaspect angle is assumed unknown, the Fisher informationmatrix (FIM) becomes singular, rendering the parametersnon-identifiable. It is noted that the main contributionof the paper is the derivation of the CRBs, and noestimation technique is advocated here. For MD motion,

we use experimental data to estimate MD parametersand to validate the CRBs. We employ suboptimaltechniques that make use of the inherent impulsivestructure of the spectrum of the returns to estimate some ofthe parameters in the MD model [10]. Maximumlikelihood optimal estimator for MD motion profile ispresented in [10].

2 General signal model

Below, we briefly discuss the range ambiguity problem,and introduce the motivation for adopting carrier frequencydiversity in CW radar for range estimation.

2.1 Range ambiguity in dual-frequencyradar

For a dual-frequency radar employing two known carrierfrequencies f1 and f2, the baseband signal returns because asingle moving target are expressed as

w1(nTs) 4pf1R(nTs)

c, w2(nTs)

4pf2R(nTs)

c(1)

where Ts is the sampling period, R(nTs) is the range of

the target at sample n and c is the speed of light in freespace. Dropping Ts in the phase and range notations forconvenience, and without considering phase wrapping, the

range of the target is given by [6]

R(n) c(w2(n) w1(n))

4p( f2 f1)(2)

In reality, however, phase observations are wrappedwithin the [0, 2p) range. Therefore the true phase can beexpressed as

w(true)(n) w2(n) w1(n) 2mp (3a)

where m is an unknown integer. Accordingly, the rangeestimate is subject to range ambiguity [7], that is

R(n) c[w2(n) w1(n)]

4p( f2 f1)

cm

2( f2 f1)(3b)

The second term in the above equation induces ambiguity inrange. For the same phase difference, the range can assumeinfinite values separated by

Ru c

(2( f2 f1))(3c)

which is referred to as the maximum unambiguous range.From the above equation, Ru is inversely proportional tothe difference of the carrier frequencies. Consider a singlefrequency Doppler radar operating at f1 1 GHz. In thiscase, Ru c/2f1 15 cm. For dual-frequency operation at

f1

1 GHz, and f2

1.01 GHz, Ru

15 m, according to(3a). The increase in Ru from 15 cm to 15 m is sufficientfor target location in rooms and small buildings. The othermultiple range solutions given by (2) and (3b) are simplyrejected since they do not fall within the spatial extentof the urban structure. In the following sections, we,therefore, assume m 0, for simplicity. By choosingclosely separated carrier frequencies, any maximumunambiguous range can be achieved. Coherent phaseestimation may, however, be compromised if the frequencydifference is too small to overcome noise effects andfrequency drifts in down conversions [11]. Theperformance analysis of the dual-frequency radar in

the presence of noise and frequency drifts is provided in[12, 13].

It is noted that in the presence of multiple movingtargets, the phase extracted from the radar returns cannotbe used for range estimation of each target separately. Thisis because the phase terms corresponding to differenttargets are superimposed and cannot be separated in thetime domain. This drawback of the dual-frequency schemecan be overcome, provided that the target Dopplersignatures are separable in the frequency or the timefrequency (TF) domain [14]. With the extraction of the

phase of the individual targets in the Doppler domain, (2)can then be used to obtain the corresponding target range.Likewise for multipath which is similar to multiple targets

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the same procedure can be applied, provided they areseparable in the frequency or TF domain.

2.2 Signal model for CRB derivation

For a dual-frequency radar employing two known carrierfrequencies f1 and f2, the baseband signal returns because ofa single moving target are expressed as

x1(n) x1(nTs) (r1 h1(n; c))exp j4pf1R(n)

c

v1(n)

s1(n) v1(n)

x2(n) x2(nTs) (r2 h2(n; c))exp j4pf2R(n)

c

v2(n)

s2(n) v2(n)

{n j n [ Z, n 0, 1, 2, 3 . . .N 1} (4)

where ri is the amplitude of the return at frequency fi,i 1, 2. The scalar function hi(n; c), dependent on boththe sample index n and a vector c of desired parameters,is added to represent possible cyclic amplitude fluctuationsfor the MD model. It is noted that for the case ofconstant Doppler and accelerating target motion profiles,hi(n; c) 0, since for these targets the radar cross section(RCS) can be assumed constant for a relatively small timeon target. Z is the domain of positive integers. The two

noise sequences, corresponding to the two frequencies ofoperation, over the observation period N are complexadditive white Gaussian noise and uncorrelated, that is,v1(n) v1(nTs)CN(0, s

21), v2(n) v2(nTs)CN(0, s

22),

E{v1(n)v2(n)} 0, where s

21 and s

22 represent the noise

variance for the two frequencies, respectively, and is thecomplex conjugate operator. To aid in the derivation, we

vectorise the observations and describe the signal returnsby a joint probability density function (pdf). The receivedsignals at the two frequencies are appended to form a long

vector

x s v [x1(0), x1(1), x1(2) . . . x1(N 1), x2(0),

x2(1) . . . x2(N 1)]T [ x1 x2 ]

T (5)

where

s[s1(0), s1(1), s1(2).. .s1(N1), s2(0), s2(1).. .s2(N1)]T

[s1

s2

]T

v[v1(0),v1(1), v1(2).. .v1(N1),v2(0),v2(1).. .v2(N1)]T

[v1 v2 ]T

(6)

The mean and covariance matrix of x are given by

E{x}m[s1

s2

]T,

E{(xm)(xm)H}Cs21I 0NN

0NN s22I

" #(7)

The joint pdf of the data, assuming a multivariate complexGaussian distribution, is

p(x;s)1

p2Ndet(C)exp((xm)HC1(xm)) (8)

2.2.1 Constant doppler frequency: For the constantDoppler frequency model, the target range is parameterisedas

R(n)R0vcos(u)n, u[ p

2,

p

2

(9)

where R0 is the initial range of the target at time n 0. Thediscrete velocityvof the target is its actual velocity multipliedby the sampling period. The parameter u is the viewing,or aspect, angle of the radar with respect to the target. Inother words, the component v cos(u) is the identifiableDoppler component registered by the radar system.Inserting (9) into (4), it is clear that the constant

Doppler model is analogous to the sinusoidal parameterestimation [15, 16]. Hence, the identifiability criteria of theparameters are similar to the traditional single frequencyestimation problem. However, some details must be added.

The identifiability criteria are developed by putting u 0in (9) because for any other value of u in the defined range,the criteria are valid. The terms 2f1v/c and 2f2v/c in thephase of the signal returns are the Doppler frequencies,analogous to the frequencies (radian/s) in the sinusoidalparameter estimation problem. Hence, for v to beidentifiable, 20.5 2f1v/c, 0.5 and 20.5 2f2v/c, 0.5[15]. Since we have assumed f2.f1 implicitly,2c/4f2 v, c/4f2 represents the bounds on the positive

and negative Doppler, respectively. For example, iff2 1 GHz, then the discrete velocity is constrained toa maximum of 0.075, assuming a nominal samplingfrequency of 100 Hz and the analog velocity is 7.5 m/s. It isnoted that indoor animate moving targets, such as humansand pets, do not assume such high velocities. Typically, inindoor settings, we expect analog velocities below 1 m/s,

which allows the sampling frequency to be reduced toapproximately 15 Hz. The terms 4pf1R0/c and 4pf2R0/crepresent the phase in the sinusoidal parameter estimationmodel. Clearly, R0 can be uniquely estimated if themaximum unambiguous range is larger than the building

extent, as determined by (3c). For this motion profile, asmentioned earlier, we use hi(n; c) 0, since the RCS isassumed to be constant.



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2.2.2 Micro-Doppler: The range for this motion profileis parameterised as [17]

R(n) R0 dcos(u)cos(v0n w0), (d, v0)= 0,

u[ p

2

,p

2 (10)

In the above equation, the parameter v0 is the discrete version of the rotational or the vibrational frequency. Theparameter d cos(u) is the maximum displacement fromthe initial range R0 along the radars line of sight (LOS).For the specific case of rotation, 2d cos(u) represents thediameter of the circular trajectory in the direction ofthe LOS. The parameter f0 denotes the initial phase ofthe MD motion. The MD signal returns represent thetraditional sinusoidal FM model [10, 11], and are obtainedby substituting (10) into (4). It is noted that the sinusoidalFM model is a good fit for motion profiles of indoor

inanimate objects, such as fans, vibrating machineries andclock pendulums. For the more complicated humanmotion, the sinusoidal FM model is a rough approximationto the swinging of the arms and the legs during walking.

The MD model can be further decomposed into vibrationand rotation models based on the scalar function hi(n; c).For typical indoor targets undergoing pure vibration (i.e.the target moves back and forth), the displacementsare small relative to the target range, especially for longerradar standoff distances from the wall, and the RCSis considered to be constant, that is, hi(n; c) 0. On the

other hand, for a rotating target, a fixed-location radarobserves different elevation aspects of the target, therebyinducing a cyclic amplitude modulation in the radarreturns. These amplitude fluctuations are indeed dependenton the target geometry. In this case, the scalar functionhi(n; c) can be modelled as

hi(n; c) Dricos(v0n fi) (11)

where the parameterDri represents the maximum deviationfrom the nominal amplitude ri, whereas the parameterwi[[0, 2p) represents the phase at time n 0. As anexample, consider a rotating fan with one blade, at one

extreme instant of time the tip of the blade is seenrepresenting a near-zero RCS, whereas at another extremetime instant the entire length of the blade is seen by theradar representing a maximum RCS. The cyclic amplitudemodulations will be demonstrated experimentally in thesimulations section and are in agreement with the modelin (11).

The identifiability criteria for MD signals are developedby substituting u 0 in (10). For the variable R0, this caseis identical to the case of constant Doppler. However, it isnot straightforward to derive the parameter identifiability

criteria for the rest of the parameters using the time-domain model alone, as was the case in the constantDoppler model. We use the Fourier transform to aid in the

derivation and to shed more light on the identifiabilitybounds and the discrete parameterisation in the MDmodel. The noise-free MD returns are given by

si(n) (ri hi(n; c))exp j4pfiR0

cj

4pfidcos(v0nw0)

c ,8i 1, 2

(12)

Let Jk(.) be the kth order Bessel function of the first kind.Accordingly [18, 19]

exp j4pfid

ccos(v0n w0)

X1

k1

jk exp(jk(v0n w0))Jk4pfid

c

(13)

Using (12) and the Fourier representation of (13), andignoring hi(n; c), as it does not affect identifiability, weobtain

Si(ejv) ri exp j

4pfiR0c

X1k1

jkd(v kv0)

exp(jkw0)Jk4pfid

c

, 8i 1, 2 (14)

As in traditional multiple sinusoidal parameter estimation,for the frequencies to be identifiable, they must lie in theinterval [2p, p). This condition, which translates in theunderlying case to 2p kv0 , p, 8k, k [ Z, cannot beimposed, since there is an infinite number of harmonics.Moreover, it is clear from (14) that the signal is notanalytic because of the presence of harmonics in thenegative frequencies. However, it is well known thatthe Bessel functions decrease rapidly with k. Therefore

we impose the following two assumptions to derive theidentifiability constraints: (i) Most of the energy in thesignal can be represented by a finite number of harmonics;say K[ Z. (ii) The Nyquist theorem is satisfied.

Assumption (i) implies that the Fourier transform in (14)

can be replaced by

Si(ejv) ri exp j

4pfiR0c

XKkK

jkd(v kv0)

exp(jkw0)Jk4pfid

c

, 8i 1, 2 (15)

Assumption (ii) then implies that2p kv0 , p, 8k,2K k K in order to avoid aliasing of the K harmonics.Since negative vibrational and rotational frequencies haveno physical meaning, it is straightforward to note that from

the above assumptions v0 [(0, p). Furthermore, since theharmonics are symmetric about DC, and knowing a priorithat an MD target is present, it is sufficient to identify the



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first harmonic corresponding to k +1, that is, jv0j , p. The criterion 0 f0 , 2p is chosen for identifying f0.Surprisingly, the identifiability criteria are similar, althoughnot identical, to the case of sinusoidal parameterestimation. However, the bounds on v0 and f0 are carrierindependent, unlike the constant Doppler case, where the

velocity bound was constrained by the higher carrierfrequency, f2.

The Bessel functions take arguments 4pfid/c. Thereforeany real argument, excluding zero, is valid becauseJk(0) 0, 8k= 0. Since negative values of the maximumdisplacement, d, have no physical meaning, then d. 0.Moreover, in traditional sinusoidal FM, the argument4pfid/c is analogous to the modulation index [11], whichis strictly real and positive. Therefore 0 , d, 1. I t i simportant to note, however, that it is impractical toconstrain the initial range of the target to be within

the maximum unambiguous range, while allowing themaximum displacement to be infinite. Therefore, to beconsistent, we impose the criterion for d as 0 , dc/4( f22f1) Ru/2.

It is noted that typical indoor MD targets, such as rotatingfans, pendulums etc., are essentially narrowband FM(NBFM). For example, consider the motion of a human,the maximum displacement of either the arm or the leg ismuch less than a metre, and for a carrier in the low GHzrange, the bandwidth is typically a few Hz. In the case ofan indoor rotating fan, the received signal bandwidth isproportional to the length of the fans blade, and is clearlya few Hz for such carrier frequencies.

2.2.3 Accelerating target: The range for this motionprofile is given by

R(n) R0 cos(u)(an bn2), u[

p

2,

p

2

(16)

where the parameters a and b represent the initial velocityand half the acceleration (or deceleration) of the target,respectively. As before, R0 is the initial range of thetarget and its identifiability is identical to the constant

Doppler case. Substituting (16) into (4), we obtain thedesired signal returns for the accelerating target. It is clearthat an accelerating target represents a linear chirp, or asecond-order Polynomial-phase signal (PPS) [20]. Theidentifiability criteria are developed by putting u 0 in(16). The model is similar to the chirp parameterestimation model, albeit a few differences in identifiability[21]. The terms 4pfia/c and 4pfib/c8i, i 1, 2 represent

the frequency and chirp rate in the chirp parameterestimation model, respectively. Hence for identifiability,along with the fact that f2 .f1, we must have 2c/4f2 a , c/4f2 and 2c/8f2 b, c/8f2, which are carrierfrequency dependent. It is noted that the bounds are forthe discretised versions of the parameters. For this motion

profile, hi(n; c) 0.

Wenote that in indoor environments, the above three motionprofiles can occur individually or in combination. While arotating fan or a clock pendulum will exhibit purely MDmotion, the motion profile of a human walking can bepredominantly modelled as a combination of translational andMD components. In this paper, however, we focus on threestand-alone motion profiles, namely, translation, MD andacceleration individually. Combinations of these motions arenot considered.

3 CramerRao bounds

In this section, we derive the CRB for the three motion profilesdiscussed previously. The parameter u is assumed known. In

Appendix A, we show that if u is unknown, then the FIMbecomes singular. We begin with the generalised CRBderivation, taking into account the effect of the nuisanceparameters. The parameter vector including the nuisanceparameters is denoted by h [cT, r1, r2, s

21, s

22]

T(p4)1,

where c [c1, c2, c3. . .cp]T. The parameters ck,

1 k p are the p desired parameters, some of which arepresent purely in the phase of the signal returns. In general,the range is a function of both n and c, that is, R(n) R(n;c). For a complex Gaussian vector x, with a covariancematrix C, defined in (7), the Fisher information elements areexpressed as [22]

[F]qr Tr C1 @C

@hqC1

@C

@hr

" # 2Re

@s

@hq

H

C1@s

@hr

( ),

1 (q, r) p 4 (17)

where (see equation at the bottom of the page)

with the exponential taken elementwise, R(c) and ri hi(c)are 1 N row vectors, whose nth elements are R(n; c) andri hi(n; c), i 1, 2 respectively. That is, the dependencyon n is suppressed. The operator W denotes the Hadamardproduct or elementwise multiplication. Following the work

s [s1 s2 ]T (r1 h1(c)) W exp j

4pf1c

R(c)

(r2 h2(c)) W exp j

4pf2c

R(c)

!T



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of [23], and using the vector differential operator, we obtain

@s

@hT

s1

s2

!@

@cT@

@r1

@

@r2

@

@s21

@

@s22

!

@h1(c)

@cT@h2(c)

@cT

2666437775

expj4pf1R(c)

c

expj4pf2R(c)

c

2664 37754pf1

c

@R(c)

@cT4pf2

c

@R(c)

@cT

2666437775

26664

js@s

@h2

@s

@h302N1 02N1

37775

2N(p4)

@s

@h2:

(r1h1(c))TW1

0N1

" #Ws,

@s@h3

: 0N1

(r2h2(c))T81

Ws

"(18)

where denotes the generalised Hadamard product, and()

W1 denotes the Hadamard division of a vector or amatrix, that is, if x [x1,x2 . . . xN], x

W1 [1=x1, 1=x2 . . .1=xN]: (For a matrix A[C

MN a1 a2 . . . aN

andvectorx[ CM1Ax[CMN : a1 x a2 x . . . aN

x :

xA: In (17), the argument of the trace operator is zeroexcept that which corresponds to the Fisher information ofthe noise nuisance parameters namely, s2i, i1, 2:

Partitioning the FIM

FF11 F12

F21 F22

!(19)

The desired CRBs reside on the diagonal of the inverseSchur complement of F, with respect to F22

CRB(c) [(F11F12F122 F21)

1]qq (20)

Using (18)(20), it can be readily shown that

F11 2X2i1

1s2i

@hi(c)

@cT

T@hi(c)

@cT

32p2

c2

X2i1

f2is2i

(rihi(c))@R(c)

@cT

T

(rihi(c))@R(c)

@cT

F12 FT21 [0p 0p 0p 0p]

F22 Diag 2N=s21 2N=s

22 N=s

41 N=s

42

(21)

where Diag[.] is a diagonal matrix with entries as given in thebrackets and the vector1N1 is the row vector of size Nhavingentries all equal to one. Similarly, the vector0p is the column

vector of size p, having all zero entries. The CRB of thedesired parameters is then expressed in a compact form as

CRB([c]q) [F111 ]qq (22)

From (22), the CRBs for the desired parametersare completely decoupled from those of the nuisanceparameters. The decoupling in (22) indicates that the

variance of the range parameters are insensitive tothe knowledge, or the information brought in by theestimation of the nuisance parameters, that is, they areunaffected whether the nuisance parameters are known orunknown [22]. From matrix F11 in (21), we observe thatthe scalar multiplier terms arise as a result of the carrierdiversity, implied by the dual-frequency operation, andtherefore have an additive effect on the net Fisherinformation. It yields lower CRBs, when compared to theno diversity (single carrier frequency) case. In fact,the scalar term 32p2(f21 =s21f22 s22)=c2 indicates that theCRBs are the lowest when f1 f2 0, in which case,Ru 1. However, this is a trivial case, as diversity ceasesto exist. In general, since the FIM elements are afunction of the squared values of f1 and f2, lower CRBsare achieved when higher carrier frequencies are chosen.

The effect of the SNRs on the CRBs is straightforwardto analyse. The addition of Fisher information is exploredin greater detail in the following subsections. In theensuing analysis, we derive the FIM elements for thedesired parameters.

3.1 Constant Doppler CRB

In this case, the parameter vector is c [R0 v]T.

Substituting (9) into (21) for the constant Doppler rangeprofile, we obtain the corresponding FIM. In order tomake the CRB analysis more generic, we introduce theparameters x :f2=f1, and c

0: s21=s

22: It can be readily

shown that for a single frequency radar employing carrierf1, the FIM, Ff1, and its inverse are, respectively, givenby [10]

Ff1

32p2r21 f2

1

c2s21

NN(N 1) cos(u)

2N(N 1) cos(u)

2

N(N 1)(2N 1)cos2(u)

6

2664

3775

(23)

F1f1 c2s21

Kf21 r21

2(2N 1)

N(N 1)

6

cos(u)N(N 1)6

cos(u)N(N 1)

12

cos2(u)N(N2 1)

2664

3775

(24)

where K 32p2 is a constant. However, for the proposed



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dual-frequency scheme, the FIM is given by

F32p2f21

c2s21(r21 r

22x

2c0)

N

N(N 1) cos(u)

2N(N 1) cos(u)

2

cos2(u)N(N 1)(2N 1)

6

2664 3775(25)

F1 c2s21

K(r21 r22x

2c0)

2(2N 1)

N(N 1)

6

cos(u)N(N 1)6

cos(u)N(N 1)

12

cos2(u)N(N2 1)

2664

3775

(26)

It is clear from (23) and (25) that the Fisher information forthe dual-frequency scheme can be written as

F Ff1 Ff2 (27)

The diagonal elements in (26) represent the CRBs for theinitial range and velocity, respectively. It is evident fromthe scalar matrix multiplicative factor in (24) and (26)that the CRBs for both the initial range and velocity forthe dual-frequency case consistently assume smaller valuesas compared to their counterparts in single frequency

operation. Hence, carrier frequency diversity lowers theCRBs for the range parameters. In (26), the matrixelements containing N characterise the temporal diversityin the Fisher information, whereas the scalar multiplier,containing the SNR terms and carriers, represents thecarrier diversity in the Fisher information. Hence, thetemporal diversity factor is identical for both frequencies,

whereas the diversity induced by the carriers is different.If x% 1 and c0 1, with ri% r, i 1, 2, then thecarrier frequencies are closely separated and the SNRcorresponding to the two frequencies are equal. In thiscase, the net Fisher information is simply 2Ff1 and theCRB are then 0.5CRB(c)f1, where CRB(c)f1 is the CRBof the parameter vector c, associated with the singlefrequency f1.

3.2 MD CRB

The derivation of the CRBs for the MD motion profileis different from that presented in [24], which uses ahybrid combination of PPS and MD signals. The resultson the FIM elements presented in [24] are a special case ofthose presented here and do not provide closed-formexpressions of the FIM elements for MD signals, neither

do they include the aspect angle and range parameters inthe employed radar model. Since the frequency diversityappends the individual Fisher information regardless of the

motion profile, as shown in (27), we can simplify notationby suppressing the terms involving the carrier frequenciesand noise variances, that is, 11i 2=s

2i, 8i 1, 2 and

12i 16p2f2i 11i=c

2:

3.2.1 MD Fisher information: In this case, theparameter vector is c [ R0 d v0 w0 Dr f]

T:

The FIM can be derived from the general expression (21)using the range (10). It is straightforward to show that

FR0R0 X2i1

XN1n0

12i(ri Dri cos(v0n fi))2 (28)

FR0d cos(u)X2i1

XN1n0

12i(ri Dri cos(v0n fi))2

cos(v0n w0) (29)

FR0v0 cos(u)X2i1

XN1n0

12i(ri Dri cos(v0n fi))2 dn

sin(v0n w0) (30)

FR0w0 cos(u)X2i1

XN1n0

12i(ri Dricos(v0n fi))2 d

sin(v0n w0) (31)

FR0Dri 0, FR0fi 0, FdDri 0, Fdfi 0 (32)

Fdd cos2(u)

X2i1

XN1n0

12i(ri Dri cos(v0n fi))2

cos2(v0n w0) (33)

Fdv0 cos2(u)

X2i1

XN1n0

12i(ri Dri cos(v0n fi))2 dn

sin(v0n w0)cos(v0n w0) (34)

Fdw0 cos2(u)X

2

i1 XN1

n0

12i(ri Dri cos(v0n fi))2 d

sin(v0n w0) cos(v0n w0) (35)

Fv0v0 X2i1

XN1n0

11iDr2in

2 sin2(v0n fi)

cos2(u)X2i1

XN1n0

12i(ri Dricos(von fi))2d2n2

sin2(v0n w0) (36)

Fv0w0 cos2(u)X

2

i1X

N1

n0

12i(ri Dri cos(v0n fi))2d2n

sin2(v0n w0) (37)



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Fv0Dri XN1n0

11iDrin sin(v0n fi) cos(v0n fi),

Fw0Dri 0, Fw0fi 0 (38)

Fv0fi XN1n0 11iDr

2

in sin

2

(v0n fi),

FDrifi XN1n0

11iDri cos(v0n fi)sin(v0n fi) (39)

Fw0w0 cos2(u)

X2i1

XN1n0

12i(ri Dri cos(v0n fi))2d2

sin2(v0n w0) (40)

FDriDri XN1

n0

11i cos2(v0n fi),

Ffifi XN1n0

11iDr2i sin

2(v0n fi) (41)

Since hi(n; c) depends only on v0 and not on R0, dand f0,the FIM elements in (32) are zero. In fact, the FIM elementsforDri and wi depend on v0 only and not on the rest of theparameters. The effect of hi(n; c) and hence Dri is explainedin the simulations section. Further simplifications of theFIM elements in (28)(41) are shown in Appendix B. Forthe special case of hi(n; c) 0, ) Dri 0, 8i 1, 2corresponding to a vibrating target, the FIM is a function of

the aspect angleu

, and the desired parameters arec

[ R0 d v0 w0 ]T. In order to simplify notation, the terms

with uare separated. We, therefore, obtain the final FIM as aHadamard product of two matrices, the matrix A containingthe terms in uand the FIM, F, corresponding to u 0

F(u) AWFF

1F

3

FT3

F2

!(42)

where

A

1 cos(u) cos(u) cos(u)

cos(u) cos2(u) cos2(u) cos2(u)



26664 37775,

F

FR0R0 FR0d FR0v0 FR0w0FdR0 Fdd Fdv0 Fdw0

Fv0R0 Fv0d Fv0v0 Fv0w0Fw0R0 Fw0d Fw0v0 Fw0w0

26664

37775 (43)

Using numerical inversion it can be verified that F is positivedefinite (PD). However, from (43), A is rank one with onepositive eigenvalue, 1 3 cos2(u). Thus, A is positive semi-

definite (PSD). Indeed, the Hadamard product of a PSD anda PD matrix is a cause for alarm, and prompts us to verify thenon-singularity of F(u), without which the CRBs are

unobtainable. Moreover, since F(u) represents a Fisherinformation matrix, it must at least be PSD, in which case theCRBs are infinite. The positive definiteness of F(u) can beproven using Lemma 1, implying non-singularity of the FIM[25]. In Lemma 1, the matrices are assumed complex;however, the results can be generalised to real matrices. In

[25], the authors have provided the complete definitenesscharacteristics of the Hadamard product of two matrices.However, in our case, it is sufficient to prove that F(u) is PD.

Lemma 1: Consider matrices A, B[ CNN, where A isHermitian PSD and B is Hermitian PD. Then, A W B isPD if all the diagonal elements of A are not equal to zero.

Proof: This lemma is proved in [25, p. 309, Theorem 5.2.1].The proof uses the Sylvesters law of inertia extensively. A

The CRBs are present along the diagonal ofF(u)21. FromLemma 1, F(u) is invertible, we can simplify F(u)21 as

F(u)1

1 1= cos(u) 1= cos(u) 1= cos(u)

1= cos(u) 1= cos2(u) 1= cos2(u) 1= cos2(u)



26664

37775

W

FR0R0 FR0d FR0v0 FR0w0FdR0 Fdd Fdv0 Fdw0

Fv0R0 Fv0d Fv0v0 Fv0w0Fw0R0 Fw0d Fw0v0 Fw0w0

26664

377751

(44)

For deriving (44), the theorem in Appendix C was used. The inverse of the second matrix in (44) is tedious and,therefore, is computed using numerical techniques. It isevident from (44) that the parameter R0 is insensitive to u,

whereas the other parameters are sensitive to u, dependingon cos2(u).

3.3 Accelerating target CRB

In this case, the parameter vector is defined byc R0 a b

T: The FIM can be derived from the

general expression (21) using the range equation (16).Proceeding in the same fashion as before, that is, using theHadamard product approach, we obtain

F(u)

1 cos(u) cos(u)

cos(u) cos2(u) cos2(u)

cos(u) cos2(u) cos2(u)

264

375

W

FR0R0 FR0a FR0b

FaR0 Faa Fab

FbR0 Fba Fbb

264 375 (45)



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where

FR0R0 X2i1

r2i12iN,

FR0a X2i1

XN1n0

r2i12in

N(N 1)

2X2i1

r2i12i (46a)

FR0b X2i1

XN1n0

r2i12in2

N(N 1)(2N 1)

6

X2i1

r2i12i

(46b)

Faa X2i1

XN1n0

r2i12in

2 N(N 1)(2N 1)

6

X2i1

r2i12i

(47a)

Fab X2i1

XN1n0

r2i12in3 N

2(N

1)2

4X2i1

r2i12i (47b)

Fbb X2i1

XN1n0

r2i12in4

N5

5

N4

2

N3

3

N

30

X2i1

r2i12i

(48)

The CRBs, after inverting the FIM, are given by

CRB(R0) 3

(r21121 r22122)

3N2 3N 2

N(N2 3N 2)(49)

CRB(a) 12

cos2(u)(r21121 r22122)

16N2 30N 11

N(N4 5N2 4)

(50)

CRB(b) 1

cos2(u)(r21121 r22122)

180

N(N4 5N2 4)

(51)

where 12i is defined before and includes both carrierfrequency and noise variance terms.

4 Simulation and quasi-experimental results

4.1 Numerical simulations

We begin by examining the effect of u on the CRBs. Theterm 1/cos2(u) in the CRBs will simply increase the CRBsfor increased value of u. The CRBs become infinite whenu+p/2. Moreover, u+p/2 yields a singular FIM.

The minimum CRB is achieved when u 0, that is, whenthe target is moving along the LOS of the radar system.

Without loss of generality, in the simulations we assume,

ri

r and hi(n; c)

h(n; c), 8 i

1, 2. This thenimplies Dri Dr and wi w. The SNR for constantDoppler and accelerating target motion profiles are denoted

by SNRi r2=s2i, i 1, 2 for the two carriers, respectively.

Fig. 1 illustrates the sensitivity of the CRBs for theconstant Doppler and accelerating target motion profiles,the carrier frequencies were chosen to be f1 906 MHzand f2 919 MHz. Fig. 2 shows the CRBs for the MDmotion profile for the vibration model (Dr 0), andthe rotation model. We define SNR for MD asSNRi (r Dr)

2=s2i, i 1, 2. In order to be fair in our

comparison, the SNR for both the vibration and rotationmodels was assumed identical in Fig. 2, and the parametersfor vibration and rotation model are subscripted by v andr respectively. Fig. 2 clearly shows that the CRBs for therotation model are higher than those of the vibration modelfor the same SNRs. It is well known that the CRBs of thephase parameters increase with the bandwidth of the signalamplitude [22, 26]. The bandwidth of the amplitude for

Figure 2 MD CRBs

Figure 1 Constant Doppler and accelerating target CRBs

for(SNR1, SNR2)

(10, 20) dB



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the vibration model is zero (DC), whereas the bandwidth ofthe amplitude for the rotation model is v0. Indeed, the scalarfunction h(n; c) in the rotation model smears the exponentialpart of the returns and makes parameter estimation moredifficult. In Fig. 1 we observe that v has higher sensitivitythan R0, since it has a lower CRB for increasing N [22].

Similarly, from Fig. 1, the chirp, or the accelerating targetparameter b, is the most sensitive, whereas R0 is the leastsensitive. From Fig. 2, for data record length N. 50, theparameter v0 is the most sensitive. It is noteworthy that theCRB for R0 is much lower than the CRB for parameter d,both of which have the same units of measurement andrepresent a distance and displacement, respectively. thesensitivity of the parameters, as evidenced by the CRBs, isan indication of some order of estimation of the parameters.For example, we have seen that v is the most sensitiveparameter for the constant Doppler motion and it representsthe frequency in the sinusoidal parameter estimation

problem, whereas R0 represents the phase in the sinusoidalparameter estimation model. It is well known that themaximum likelihood estimator for the sinusoidal parametermodel estimates the frequency first and then the phase [15].Similarly, for the PPS signal model, the highest order PPScoefficient is estimated first, and the lower orders areestimated subsequently [23, 26 28]. In our case, this agrees

with b being the most sensitive for a second-order PPSsignal, that is, the accelerating target returns. For the case ofMD, we have a suboptimal estimator already in place in (13)and (14). Clearly, taking the Fourier transform of theMD returns, using the DC as a reference and selecting thetwo nearest peaks around 0 Hz is the quickest way toestimate v0.

4.2 Quasi-experimental results

In urban sensing applications, the rotational or the vibrationalfrequency is an important parameter for differentiatinganimate from inanimate targets. An indoor rotating fan

with three plastic blades was used as the target for the real-data collection experiment. In order to obtain strong singlecomponent returns, one of the blades was wrapped in analuminium foil. The carrier frequencies of the dual-

frequency radar were chosen to be f1 906.3 MHz andf2 919.8 MHz, and the sampling frequency was set to1 kHz. The radar returns at the two frequencies weresimultaneously collected for a total duration of 1 s. Thecentre of rotation of the fan was at a distance of 1.22 mfrom the antenna feed point along the LOS. The rotationspeed of the fan was measured to be 780 rpm, which is inagreement with the technical specifications of the fan, andis considered to be the true value of the parameter v0. Theobjective is to estimate the rotational frequency of the fanfor varying SNRs, and compute its variance for comparison

with the CRB. The data are at high SNR and is virtually

noise free. In order to vary the SNR, we add complex white Gaussian noise. Hence the term quasi-experimentalis used. Since no estimation technique has been advocated

in this paper, we use the Fourier domain analysis, as givenby (13) and (14) and extract the frequency of the firstharmonic. This procedure is definitely suboptimal, theoptimal method of maximum likelihood is not pursuedhere because of its associated computational complexity[24]. Extracting the first harmonic frequency is sufficient

to obtain the estimate of the rotational frequency, sincetypically indoor inanimate targets are NBFM, see [24] formore details on estimating harmonics of NBFM. The FIMfor MD is a function of the parameters d and f0, whichare unknown. Furthermore, the amplitudes of the returnsare also unknown. We employ some suboptimal schemesto estimate these and use these in our FIM calculations.

The estimation of the unknown parameters is performedusing the real data without adding artificial noise; theseestimates are then considered as the true parameters. Inparticular, since the amplitudes of the returns are unknown,and, in general, are time varying, the amplitudes can be

efficiently estimated using the second-order cyclic moment,see [28] and the references therein. On the other hand, the

TF distribution is particularly useful in extracting theparameter d. The maximum and minimum instantaneousfrequencies of the MD signal are given by +2fidv0/c,corresponding to the two carrier frequencies, respectively.

The bandwidth of the MD signal can then be defined as4fidv0/c. The bandwidth is estimated from the originalreturns using the spectrogram, and since the rotationalfrequency is known, computation of d is straightforward.

This value is then taken as the true parameter value. Thereturn for the first carrier was used for calculating d. Forestimating the phase f0, we use the least-squares estimatoras in [24], with K 1 using the returns corresponding tothe first carrier frequency. For this procedure, the returns

were zero padded and an 8192 point fast Fourier transform(FFT) was computed. These parameters were then used incomputing the CRBs. Fig. 3a shows the spectrogram ofthe original returns corresponding to the carrier f1. Italso shows the signature using the estimated d and f0, andthe true rotational frequency v0 as a solid black line, whichclosely follows the TF signature of the returns. Themaximum and minimum of instantaneous frequency aredepicted with the dotted black lines. In Fig. 3b, the rawFFT of the returns corresponding to the carrier f1 is

shown, Fig. 3c shows the estimated second-order cyclicmoment for the carrier frequency f1 and Fig. 3d shows thereal part of the returns for the first carrier. From Fig. 3cit is clear that the second-order cyclic moment showsharmonics at the rotational frequency v0, indicating thecyclic RCS changes, and hence validating the cyclicamplitude modulation we have assumed for the MDrotational model. We are now in a position to compute theCRB and the simulated mean square error (MSE) usingMonte Carlo simulations. Fig. 4 shows the CRB for therotational frequency and the MSE of its estimates againstSNRs, 100 Monte Carlo trials was used in the simulations.

For each carrier frequency, the initial estimate was derivedfrom a raw FFT search, and subsequently the chirp-ztransform was used to compute the final estimate of the



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first harmonic around the coarse initial estimate from theFFT, the search span of the chirp-z transform was 8192points. This estimation procedure is identical to that used

in [24] with K

1. The final estimate of the rotationalfrequency was the average of these estimates for the twocarrier frequencies. A strong and consistent bias of the

order 1023 was observed in the raw MSE for SNR!20 db, as shown in Fig. 4, the bias-corrected MSE isalso shown and comes close to the CRB for the rotationalmodel. The strong bias maybe due to the chirp-ztransform, necessitating optimisation methods to beemployed for estimation, for example the non-linear leastsquares. For comparison, the CRB for the vibrationmodel (the single frequency and dual frequency) are alsoshown, the simulated MSE is far away from the vibration

model CRBs.

5 Conclusions

In this paper, we considered dual-frequency radar for rangeestimation with application to urban sensing. Three importantand commonly encountered indoor range profiles wereconsidered, namely, linear translation, MD and acceleratingtarget motion profiles. The CRBs were derived for the initialrange and key target classification parameters, namely, velocity,acceleration and oscillatory frequency of targets, respectively,

encountering linear, accelerating and simple harmonicmotions. A parametric model specific to the dual-frequencyradar technique was developed for the range profiles,

Figure 4 Simulation results with respect to SNR

Figure 3 Experimental results

a Spectrogram of the returnsb FFT of raw returnsc Second-order cyclic momentd Real part of the returns



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incorporating the target aspect angles. It was shown that if theaspect angle is unknown, then the FIM becomes singular,leading to unidentifiability of the parameters. Closed-formexpressions for CRBs were derived for the parametersassociated with translational motion profiles. However, for theMD case, the CRBs were intractable in closed form, and

numerical matrix inversion was pursued. It was also shownthat the dual-frequency approach provides lower bounds ascompared to the single frequency counterpart.

6 Acknowledgments

This work was supported in part by DARPA undercontract HR0011-07-1-0001 and in part by ONR undergrant N00014-07-1-0043. The content of the informationdoes not necessarily reflect the position or policy ofthe government, and no official endorsement should beinferred. Approved for public release.

7 References

[1] A MIN M . (ED .): J. Franklin Inst., Special Issue on

Advances in Indoor Radar Imaging, 2008, 345 , (6),

pp. 556722

[2] BARANASKI E., AHMAD F.: Proc. 2008 IEEE Int. Conf.

Acoustics, Speech, and Signal Processing, Special Session

on Through-the-Wall Radar Imaging, Las Vegas, USA,

April 2008, pp. 51735196

[3] BOREK S.E.: An overview of through the wall surveillance

for homeland security. Proc. 34th Applied Imagery and

Pattern Recognition Workshop, Washington, DC, USA,

October 2005

[4] ENGHETA N., HOORFAR A., FATHY A.: Proc. 2005 IEEE AP-S Int.

Symp., Special Session on Through-Wall Imaging and

Sensing, Washington, DC, USA, July 2005, 3B, pp. 317341

[5] SKOLNIK M.I.: Introduction to radar systems (McGraw-Hill, New York, 2002, 3rd edn.)

[6] RIDENOUR L.N.: Radar system engineering, vol. 1 of MIT

radiation labaratory series, (McGraw-Hill, NY, 1947)

[7] AMIN M.Z., SETLUR P., AHMAD F.: Moving target localization

for indoor imaging using dual frequency CW radars. Proc.

IEEE Workshop on Sensor Array and Multi-channel

Processing, Waltham, MA, USA, 1214 July 2006,

pp. 367371

[8] BOYER W.D.: A diplex, Doppler, phase comparison radar,IEEE Trans. Aerosp. Navig. Electron., 1963, ANE-10, (3),

pp. 2733

[9] DINGLEY G., ALABASTER C.: Radar based automatic target

system. Proc. Fourth Int. Waveform Diversity and Design

Conf., Orlando, FL, USA, February 2009, pp. 2225

[10] SETLUR P., AMIN M., AHMAD F.: CramerRao bounds for

range and motion parameter estimations using dual

frequency radars. Proc. Int. Conf. Acoust., Speech SignalProcess., Honolulu, Hawaii, 1520 April 2007

[11] HAYKIN S.: Communication systems (John Wiley,

New York, 2000, 4th edn.)

[12] AHMAD F., AMIN M.G., ZEMANY P.D.: Performance analysis

of dual-frequency CW radars for motion detection

and ranging in urban sensing applications. Proc. SPIE

Defense and Security Symp., Orlando, FL, USA, April 2007

[13] AHMAD F., AMIN M.G., ZEMANY P.D.: Dual-frequency radars

for target localization in urban sensing, IEEE Trans.Aerosp. Electron. Syst., 2009, 45, (4), pp. 15981609

[14] ZHANG Y., AMIN M.G., AHMAD F.: Time frequency analysis

for the localization of multiple moving targets using dual-

frequency radars, IEEE Signal Process. Lett., 2008, 15,

pp. 777780

[15] KAY S.M.: Modern spectral estimation: theory and

application (Prentice Hall, Englewood Cliffs, NJ, 1988)

[16] RIFE D., BOORSTYN R.: Single tone parameter estimation

from discrete-time observations, IEEE Trans. Inf. Theory,

1974, 20, (5), pp. 581598

[17] CHEN V.C., LI F., HO S. S., WECHSLER H.: Micro-Doppler

effect in radar: phenomenon, model, and simulation

study, IEEE Trans. Aerosp. Electron. Syst., 2006, 42, (1),

pp. 2 21

[18] ABROMOWITZ M., STEGUN I.A.: Handbook of mathematical

functions with formulas, graphs, and mathematical tables

(Dover Publications, New York, 1972)

[19] GREWAL B.S.: Higher engineering mathematics (Khanna

Publishers, New Delhi, India, 2002)

[20] PELEG S., PORAT B.: The Cramer Rao lower bound

for signals with constant amplitude and polynomial

phase, IEEE Trans. Signal Process., 1991, 39, ( 3) ,

pp. 749752

[21] ZOUBIR A.M., TALEB A.: The CramerRao bound for

the estimation of noisy phase signals. Proc. Int. Conf.

Acoust., Speech Signal Process., Salt Lake City, UT, USA,

May 2001

[22]KAY S.M.

: Fundamentals of statistical signal processing,volume: I, estimation theory (Prentice-Hall, Englewood

Cliffs, NJ, 1993)



www.ietdl.org

8/3/2019 Range Came

13/16

[23] CIRILLO L., ZOUBIR A., AMIN M.: Parameter estimation for

locally linear FM signals using a time-frequency Hough

transfrom, IEEE Trans. Signal Process., 2008, 56, (9),

pp. 41624175

[24] GINI F., GIANNAKIS G.B.: Hybrid FM-polynomial phase

signal modeling: parameter estimation and CramerRaobounds, IEEE Trans. Signal Process., 1999, 47, ( 2) ,

pp. 363377

[25] HORN R.A., JOHNSON C.R.: Topics in matrix analysis

(Cambridge University Press, Cambridge, UK, 1991)

[26] GHOGHO M., SWAMI A., NANDI A.K.: Cramer Rao bounds

and maximum likelihood estimation for random

amplitude phase modulated signals, IEEE Trans. Signal

Process., 1999, 47, (11), pp. 29052916

[27] BARBAROSSA S., SCAGLIONE A., GIANNAKIS G.B.: Product high-order ambiguity function for multicomponent polynomial-

phase signal modeling, IEEE Trans. Signal Process., 1998,

46, (3), pp. 691708

[28] GHOGHO M., SWAMI A., GAREL B.: Performance analysis of

cyclic statistics for the estimation of harmonics in

multiplicative and additive noise, IEEE Trans. Signal

Process., 1999, 47, (12), pp. 32353249

[29] STOICA P., MARZETTA T.L.: Parameter estimation problems

with singular information matrices, IEEE Trans. Signal

Process., 2001, 49, (1), pp. 8790

8 Appendix A

We demonstrate by example, when u is unknown and is adesired parameter, the FIM is singular. Specifically, thederivation is for the constant Doppler range profile, thederivation can be generalised to any other motion profile.For convenience, and without loss of generality, we assumeri r. The parameter vector sought after isc [ R0 v u]. Using (17) and (9), and skipping thetrivial details (see (52))

It can be shown that the FIM is singular, implying non-identifiable parameters for the chosen model [29]. Thiscan, however, be overcome if an additional carrier diverseradar system, oriented differently from the original radarsystem, is used. This involves an additional angle diversityterm in the Fisher information. Moreover, the orientation

of the two radars with respect to each other must be knownfor identifiability.

9 Appendix B

In order to bring the FIM elements for MD in closed form, we need the following which can be derived easily usingelementary trigonometry and elementary calculus. Theseries are assumed to uniformly converge so that thedifferential and summation operators can be interchanged

g1(v, c) XN1n0

cos(vn c) ReXN1n0

ej(vnc)( )

cos(v(N 1)=2 c)sin(vN=2)

sin(v=2)(53)

g2(v, c) XN1n0

sin(vn c) ImXN1n0

ej(vnc)( )

sin(v(N 1)=2 c)sin(vN=2)

sin(v=2)(54)

g3(v, c) XN1n0

n sin(vn c)

@g1(4, c)

@v

1

4sin2(v=2)

(2N 1)sinv

2cos v

2N 1

2 c

cosv

2sin v

2N 1

2 c

sin(c)

0BB@

1CCA(55)

F(u)

FR0R0 FR0n FR0uFR0n Fnn FnuFR0u Fnu Fuu

24

35

1N1 cos(u)N(N 1)

2

v1 sin(u)N(N 1)

21 cos(u)N(N 1)

2

1 cos2(u)N(N 1)(2N 1)

6

v1 sin(2u)N(N 1)(2N 1)

12v1 sin(u)N(N 1)

2

v1 sin(2u)N(N 1)(2N 1)

12

1v2 sin2(u)N(N 1)(2N 1)

6

26666664

37777775 (52)



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g4(v, c) XN1n0

n2 sin2(vnc) 1

2

XN1n0

(n2 n2 cos(2vn 2c))

1

2

N(N 1)(2N 1)

6

@g3(v, c)

@vjv2v,c2c

1

2

N(N 1)(2N 1)

6

1

8sin3 (v)

2cos2(v)sin(v(2N 1) 2c)

2cos(v) sin(2c)

(2N 1) sin(2v)

cos(v(2N 1) 2c)

4N(N 1)sin2(v)

sin(v(2N 1) 2c)

0BBBBBBBBBB@

1CCCCCCCCCCA

0BBBBBBBBBB@

1CCCCCCCCCCA

(56)

g5(v, c)

XN1

n0

n2 cos2(vn c) N(N 1)(2N 1)

6g4(v, c) (57)

g6(v, c) XN1n0

n2 cos(vn c) @g3(v, c)

@v

N(N 1)(2N 1)

6 2g4

v

2,

c

2

(58)

g7(v, c) XN1n0

n cos(vn c) @g2(vn c)

@v

1

4sin2(v=2)

(2N 1)sinv

2sin v

2N 1

2 c

cosv

2cos v

2N 1

2 c

cos(c)

0BB@

1CCA (59)

g8(v, c) XN1

n0

n sin2(vn c) 1

2

N(N 1)

2

g7(2v, 2c) (60)

We are now in a position to express the FIM elements in closed form. The following can be readily shown

FR0R0 X2i1

12i r2iN 2riDrig1(v0, fi)

Dr2i(g1(2v0, 2fi) N)

2

(61)

FR0d cos(u)X2

i1

12ir2i

Dr2i2

g1(v0, w0)

Dr2i(g1(3v0, w0 2fi) g1(v0, 2fi w0)

4

riDri(g1(2v0, w0 fi) N cos(w0 fi)

0

B@

1

CA(62)

FR0v0 cos(u)X2i1

12i

r2idDr2id

2

g3(v0, w0) riDridg3(2v0, w0 fi)

riDridsin(fi w0)N(N 1)

2 Dr2idg3

(3v0, w0 2fi)

4 Dr2idg3

(v0, w0 2fi)

4

0BBB@

1CCCA (63)

FR0w0 cos(u)X2i1

12i

r2idDr2id

2

g2(v0, w0) riDridg2(2v0, w0 fi) riDridsin(fi w0)N

Dr2idg2(3v0, w0 2fi)

4 Dr2idg2

(v0, w0 2fi)

4

0BBB@

1CCCA

(64)



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10 Appendix C

Theorem 1: Let A, B be two complex matrices [CNN.The rank ofA is one with non-zero diagonal elements andthe rank of B is N. Then (AWB)1 B1 W (A

W1)T B1 W (AT)

W1, where AW1 [a1ij ].

Proof: Since A is rank one, it can be rewritten as an outerproduct, given by

A xyH, AT yxT, with x, y [CN1 (75)

The Hadamard product can then be expressed as

AWB B W xyH DxBDH

y (76)

where

Dx Diag[x]

x1:

:

xN

2666437775,

Dy Diag[y]

y1

:

:

yN

26664

37775 (77)

It is also clear from (76) that A WB is non-singular. At thisstage it is instructive to note that if any diagonal element of

A is zero, which is tantamount to any element in (77)being zero, then the inverse of AWB does not exist. This isprimarily the reason for imposing the condition, that noneof the diagonal elements of A are zero in the statement ofthe theorem. The proof is straightforward from hereon,from (76)

(AWB)1 (DHy )1B1D1x (78)

Rewriting the diagonal matrices in terms of the Hadamardand the outer products as in (76), we obtain

(AWB)1 B1 W

1=y1:

:

:

1=yN

26666664

37777775

1=x1 : : : 1=xN

B1 W (yW1)(x

W1)T (79)

Expanding the outer product and rewriting the elements interms ofA, we obtain

(AWB)1 B1 W (AW1)T B1 W (AT)

W1 (80)

It must be noted that ifA has a rank greater than one, thenthe inverse of the Hadamard product is quite complicatedto derive. Fortunately, such a situation does not arise inthis paper.


d i 10 1049/i t 2009 0039 & Th I tit ti f E i i d T h l 2010

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