Research Article Space-Time Radar Waveforms: …downloads.hindawi.com/journals/jece/2013/809691.pdfSpace-Time Radar Waveforms: Circulating Codes ... the same waveform circulating with

Hindawi Publishing CorporationJournal of Electrical and Computer EngineeringVolume 2013 Article ID 809691 8 pageshttpdxdoiorg1011552013809691

Research ArticleSpace-Time Radar Waveforms Circulating Codes

G Babur P Aubry and F Le Chevalier

Microwave Sensing Systems and Signals (MS3) Delft University of Technology 2628 CD Delft The Netherlands

Correspondence should be addressed to G Babur gbaburtudelftnl

Received 18 May 2013 Accepted 1 August 2013

Academic Editor Sandra Costanzo

Copyright copy 2013 G Babur 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

This paper describes a concept of the circulating codes covering the whole class of the space-time codes The circulating codesdo not narrow the radiated pattern of the antenna array thus providing a wide angular coverage possibly tunable In turn thebeam-forming on transmit is achievable by means of the signal processing in one (or each) receiver channel The modelling resultsdemonstrate the efficiency of the circulating codes based on their multidimensional ambiguity functions

1 Introduction

In many modern radar applications pulse compression is akey element to achieve the required performance The codesto be used must possess appropriate properties of auto- andcross-correlations In turn beam-forming is another well-known signal processing technique used in sensor arrays fordirectional signal transmission or reception [1]

Many engineers have noted that if one were able to formtruly simultaneous transmit beams in independent direc-tions then a flexible radar system could be build that wouldbe capable of true simultaneous multifunction operation [2]Moreover since digital arrayMIMOradars can be operated ineither the MIMO or the more conventional transmit modes(as needed) a great deal of flexibility can be provided

Until recently radar transmitters have not enjoyed thesame degree of software-driven agility as radar receivers[3] Nowadays radar transmitters incorporate highly agilesoftware-driven waveform generators In principle the radardesignerrsquos choice of beam shape and waveform can be usedto spread energy over space and time inmany interesting anddifferent ways [2]

The coloured transmission principles have been pre-sented in [4] They allow the transmission of the soundingsignals in a wide angular angle and at the same time toperform the beam-forming on transmit in one (or each) of thereceiving channels They allow for an effective monitoring ofany number of beams formed on transmit by means of signalprocessing on receive The technique describing application

of the digital beam-forming processing with a single receiverchannel can be found for example in [5] At the sametime the amount of measurements stays the same as for onebeam with some increase of the computational cost in thereceiver So the colouredwaveform transmission offers vastlyincreased transmitter agility to identify ways to exploit agiletransmitter capabilities in practice

In the literature on MIMO radar it is often assumedthat the transmitted waveforms associated with each trans-mit antenna or subarray are orthogonal and have perfectwaveform cross-correlation properties In practice this ishardly achievable when the transmitted signals occupythe same time interval within the same bandwidth Whenwe consider the same waveform circulating through eachantenna element with a very small relative time shift aspresented in this paper it seems that achievement of goodcross-correlation properties is simply impossible Howeverthe received results demonstrate the opposite Consequentlythe circulating codes for coherent MIMO deserve carefulconsideration and analysis

Implicit in the waveform design is that the observed radarscene should be estimated accuratelyThe scattering responseof the clutter and targetsmust be disentangled by the receiverA commonly used measure for assessing waveform selectionin radar systems is the ambiguity function which will be usedhere for analyzing the properties of the proposed code Inconventional radar systems Woodwardrsquos ambiguity functionis used to characterize waveform resolution performance[6 7] For analysis of space-time radar waveforms such as

2 Journal of Electrical and Computer Engineering

antenna array

1205790

12

3

s1(t)

s2(t)

s3(t)

sN(t) N

sT(t 1205790)

TxRx

Figure 1 MIMO transmission

the circulating codes presented in this paper themultidimen-sional ambiguity function should be employed [8 9]

This paper describes a solution for the proposed wave-forms based on the use of a novel and technically simplesolution of transmitting only one waveform circulating fromone antenna element to another (or from one subarray toanother) with a very small relative time shift In this waythe radiated signals have exactly the same coding complexenvelope highly overlapping both in the time and frequencydomains

Thepaper is organized as follows In Section 2 the conceptof the circulating space-time codes which applies to allpossible waveforms having good autocorrelation proper-ties is introduced Section 3 describes the multidimensionalambiguity function used for the space-time signals analysisSection 4 gives the examples of the ambiguity functions fora number of circulating codes examples The capabilities ofthe waveforms for the beam-forming on transmit and pulsecompression are shown Section 5 contains the conclusions

2 Circulating Codes Concept

The transmitted waveforms are assumed to be encoded bythe same waveform 119904(119905) circulating with a relative time shiftthrough 119873 MIMO transmitter channels (see Figure 1) Thewaveform 119904

119899

(119905) circulating through the 119899th channel can bewritten as

119904119899

(119905) = 119886119899sdot 119904 (119905 minus (119899 minus 1) sdot Δ119905) (1)

where the index 119899 represents both the number of thetransmitting channels and the number of the transmittedwaveforms The relative time shift Δ119905 between the circulatingsignals is equal to 1-sample time Δ119905 = 1Δ119865 where Δ119865is the signal bandwidth For waveforms with a large BT-product (compression ratio) the relative time shift is verysmall compared to the pulse duration 119879

119901because in fact

BT = 119879119901Δ119905 For the sake of simplicity the amplitudes 119886

119899are

assumed to be the same and equal to one and will be omittedfurther

The simplicity of implementation of the circulating codesshould be emphasized Because of the use of only one wave-form only one complex signal with a large BT product shouldbe generated and then it circulates in all the transmitterchannels with a small relative offset

s1(t)

s2(t)

s3(t)

sN(t)

0

0

0

0

Tp

t

t

t

t

Δt

s(t)

s(t minus Δt)

s(t minus 2 middot Δt)

s(t minus (N minus 1) middot Δt)

middot middot middot

Figure 2 Circulating signals

Figure 2 shows the circulating signals 119904119899(119905) 119899 = 1 sdot sdot sdot 119873The total radiated signal 119904

119879(119905) has the following properties It

has the same duration119879 as the signal 119904(119905)This is explained bythe fact that the envelope of the sum signal has a trapeziumshape with duration 119879 at minus3 dB level The relative time delayof the signal 119904

119879(119905) compared to 119904(119905) is equal to the half of the

relative time shift between the first and the last circulatingsignals This time delay is constant and it is equal to (119873 minus

1) sdot Δ1199052 It can be compensated at the signal generation stagein the MIMO radar transmitter by the equal shift of all theradiated signals or by compensation of this known time delayat the signal processing stage in the MIMO radar receiver

The time offset used in the circulating signal can belooked at as a time-delay beam-steering (see eg [10]) Thetime-delay steering of array antennas is used to steer thenarrowmain beam of the antenna array physically When thecirculating signals are used for sounding the radiated beamis not steered and the signals are radiated within the wideangle The time shifts within the circulating codes are usednot to steer the beam but to keep it as wide as possible Thebeam-forming on transmit (narrowing of the main beam) isimplemented by the signal processing means on receive

Array theory tends to discuss things in phase ratherthan delay so we can convert the delay experienced by thecirculating signal into a phase shift between adjacent antennaelements at a given frequency

Δ120593 = 2120587 sdot 119891 sdot Δ119905 =

2120587 sdot 119891

Δ119865

(2)

where 119891 is the radiated signalrsquos frequency Since the radiatedsignal bandwidth Δ119865 is not equal to zero the phase shift isnot fixedThe phase shift is a function of the frequencyΔ120593(119891)where the radiated signalrsquos frequency belongs to the followinginterval 119891 isin [(119891

0minus Δ1198652) sdot sdot sdot (119891

0+ Δ1198652)] where 119891

0is the

radiated signalrsquos central frequencyTherefore the range of thephase shift described by (2) can be written as

(Δ120593min sdot sdot sdot Δ120593max) = ((Δ1205930 minus 120587) sdot sdot sdot (Δ1205930 + 120587)) (3)

where Δ1205930is the phase shift provided by the central radiated

frequency Equation (3) shows that the phase shift within theradiated signalrsquos bandwidth changes by 2120587 which character-izes the full angular coverage

Journal of Electrical and Computer Engineering 3

Theuniformly spaced linear array can be steered by apply-ing a phase shift such as Δ120593 = 0 at the angle of interest [11]

Δ1205930= 2120587 sdot

119889

119888

sdot 119891 sdot sin 1205790 (4)

where 119889 is the spacing between the antenna elements and119888 is the speed of light However for the circulating signalsthe phase shift varies by 2120587 along the array for any 120579

0 It

means that regardless of the value of Δ1205930or on the steering

direction the physically radiated circulating signals providethe full angular coverage in practice limited only by theradiated patterns of the antenna elements So in fact thesignals are radiated within a wide angular range and thebeam-forming on transmit (forming of multiple beams) isperformed on receive by the signal processing means whichis explained further in the paper

The group transmitted signal (see Figure 1) is angulardependent For ease of exposition let us make the usualnarrowband assumption this is not a requirement butsimplifies the discussion So in a given direction 120579

0the

aggregate transmitted signal can be written as the sum ofall transmitted signals with appropriate phase shifts for thisdirection defined by the wavenumber

119896(1205790)

119904119879(119905 1205790) =

119873

sum

119899=1

119890119895119896(1205790)sdot119909(119899)

sdot 119904119899

(119905) (5)

where 119896 is a wave-vector 119909(119899) is the position vector of the 119899thradiating element and 119904119899(119905) is a circulating signal describedby (1)

The signal received by one receiver channel 119903 is written as

119904119903

119877(119905 1205790) = 119890minus119895sdot119896(1205790)sdot119909(119903)

sdot sdot 119904119879(119905 minus 1205910 1205790) + 119890 (119905) (6)

where is the complex scattering coefficient of an observedtarget 120591

0is a time delay defined by the travelling distance

of the signal and 119890(119905) is noise at time 119905 In this work weassume that within one pulse the phase variance due to theDoppler shift is negligible and the Doppler shift becomesjust a constant phase term The multidimensional transmitambiguity function presented in the next section is calculatedfor one pulse as a squared modulus of the matched filteroutput (see the end of Section 3) Since we take the moduluswe neglect the phase term defined by the Doppler shiftimposed on the pulse So the use of the bank of Dopplerfilters together with the sounding circulating signals makesthe observation of moving objects possible and at the sametime provides a wide angular coverage with possibility of thebeam-forming on transmit performed in radar receiver

3 Transmit Ambiguity FunctionCirculating Codes

As known the matched filter is the solution for maximizingthe output signal-to-noise ratio (SNR) So the beam-formingon transmit is performed on the received signals in the 119903th

channel as the filtering matched with the transmitted signal119904119879(119905 1205790)

119877119903

1205790

(120591 1205791015840

) = int 119904119879(119905 minus 120591 120579

1015840

) sdot (119904119903

119877(119905 1205790))lowast

119889119905 (7)

where 1205791015840 is a hypothesis about the arrival direction Thesuperscript ldquolowastrdquo means complex conjugation The extendedform of the compressed signal is

119877119903

1205790

(120591 1205791015840

) = lowast

sdot 119890+119895sdot119896(1205790)sdot119909(119903)

sdot int 119904119879(119905 minus 120591 120579

1015840

) sdot (119904119879(119905 minus 1205910 1205790))lowast

119889119905

(8)

Omitting the constant phase term characteristic for the 119903threceiving channel and changing the time variables 1199051015840 = (119905minus120591

0)

1205911015840

= (1205910minus 120591) we receive

1198771205790(1205911015840

1205791015840

) = int 119904119879(1199051015840

+ 1205911015840

1205791015840

) sdot (119904119879(1199051015840

1205790))

lowast

1198891199051015840

(9)

which can be rewritten as

1198771205790(1205911015840

1205791015840

) = int(

119873

sum

119899=1

119890119895sdot119896(1205791015840)sdot119909(119899)

sdot 119904 (1199051015840

+ 1205911015840

minus 119899 sdot Δ119905 + Δ119905))

sdot (

119873

sum

1198991015840=1

119890119895sdot119896(1205790)sdot119909(119899

1015840)

sdot 119904(1199051015840

minus 1198991015840

sdot Δ119905 + Δ119905))

lowast

1198891199051015840

(10)

where as we remember 119899 is the index for the transmittedsignal and 1198991015840 is the index for its replica

1198771205790(1205911015840

1205791015840

) =

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

sdot int 119904 (1199051015840

+ 1205911015840

minus (119899 minus 1) sdot Δ119905)

sdot 119904lowast

(1199051015840

minus (1198991015840

minus 1) sdot Δ119905) 1198891199051015840

(11)

As was defined before Δ119905 is a 1-sample time shift betweenthe replicas of 119904(119905) in the circulating codeAt the same timeΔ119905is the width of the autocorrelation function ACF

119878(1205911015840

) of thesignal 119904(1199051015840) taking into account the time variablesrsquo change

1198771205790(1205911015840

1205791015840

) =

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

sdot ACF119878(1205911015840

minus (119899 minus 1) sdot Δ119905 + (1198991015840

minus 1) sdot Δ119905)

(12)

The normalized value of the autocorrelation functionis equal to 1 within 1-sample time interval Δ119905 = 1Δ119865

around 1205911015840 = 0 For the rest the ACF is described by itsside-lobe level (SLL) The SLL for signals having large BT


products and providing good autocorrelation properties canbe approximated as 0 So (12) can be written as

1198771205790(1205911015840

1205791015840

) =

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

sdot 1 (1205911015840



(13)

The function 1(1205911015840) is equal to 1 within 1-sample time intervalaround 1205911015840 = 0 and zero for the rest Consider1198771205790(1205911015840

1205791015840

)

=

119873

sum

119899=1

rect (1205911015840 minus (119899 minus 1) sdot Δ119905)119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

(14)

where

rect (1205911015840) = 1 for minus (119873 minus 1) sdot Δ119905 le 1205911015840

le 0

0 for elsewhere 1205911015840(15)

We note that the function rect(1205911015840) for the circulatingsignals is119873 times wider than the achievable range resolutionprovided by one waveform which is equal to 1 bit or Δ119905 Thisgives the proper width of its autocorrelation function (ACF)introduced below and therefore the proper widening of theambiguity function along ranges

Equation (14) can be simplified as follows

1198771205790(1205911015840

1205791015840

)

=

ACFrect (1205911015840

) sdot

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

for 10038161003816100381610038161003816120591101584010038161003816100381610038161003816le

119873 sdot Δ119905

2

0 for elsewhere 1205911015840(16)

Equation (16) is an adequate approximation in case of anideal autocorrelation function of the signal 119904(119905)with one peakand zero side-lobe level (SLL)

The mathematical expression of the multiparameter sig-nal1198771205790(1205911015840

1205791015840

) shows its dependency on the transmittedwave-forms 119904119899(119905) (see (1)) The ambiguity function |119877

1205790(1205911015840

1205791015840

)|

2

is thus a 3-dimensional function (or 4-dimensional if theamplitude is considered as a dimension as well) givingfor each aiming direction 120579

0the delay-angle ambiguity

1198771205790(1205911015840

1205791015840

) being a 3-parameters function cannot be visual-ized easily a preferred way is to look at the angle-angle cutfor 120591 = 0 and at the range-angle ambiguity for a specific 120579

0

(i) |1198771205790(0 sin 120579)|2 = 119863(120579 120579

0) is the angular transmit

diagram (at the exact range of the target 1205911015840 = 1205910) as

a function of the angular aiming position 1205790

(ii) |1198771205790((1198881205911015840

)2 1205791015840

) |

2 where 1205790

is constant is therange-angle ambiguity function for boresight aimingdirectionmdashideally this range angle ambiguity func-tion should also be analyzed for each possible aimingdirection 120579

0

4 Modeling Results

Four very different complex waveforms have been chosen foroperation in the circulating space-time mode as follows

(1) Quadratic Alltop code as a representative of the dis-tinct complex-valued sequences constructed for theminimization of the magnitudes of autocorrelationside lobes and cross-correlations [12]

(2) M-sequence or a maximum length sequence as atype of widely used pseudorandom binary sequenceswhich are inexpensive to implement in hardware orsoftware The autocorrelation function has a uniformside-lobe level

(3) LFM signal which is a very common radar wave-form to realize pulse compression for its fairly readygeneration and easy processing However withoutweighting the autocorrelation function experiences ahigh side-lobe level

(4) Nonlinear LFM signal (NLFM) Nonlinear frequencymodulation can advantageously shape the ldquorectan-gularrdquo energy spectrum of an LFM signals suchthat the autocorrelation function exhibits substan-tially reduced side lobes The NLFM signal with thetangent-based frequency variation law [13] has beenchosen for modeling

It is known that the number of orthogonal LFM signalswithin the same bandwidth and time interval is limited by anumber of 2 In case an LFM signal is used according to theprinciple presented in this paper all the signals circulatingthrough the transmitting channels have the same shape ascan be seen in (1) The important conclusion for all wave-forms not necessarily having LFM modulation is that thenumber of circulated signals (transmittersantenna elements)is not really limited and it can be definitelymore than 2Whatis really needed is the good autocorrelation properties of thecirculating waveform to make the approximation (13) viable

The pulse duration is equal to 100 120583s The carrier fre-quency is 10GHzThe BT product of all the modelled signalswas chosen relatively small 255 for better visibility of theresults The antenna array is made of 119873 = 8 elementaryomnidirectional antennas spaced 1205822 from each other Fordemonstrating the possibilities of the beam-forming ontransmit for the analysed waveforms only one element of thearray was used on receive

In Figure 3 four angle-angle cuts of the ambiguityfunction calculated for four considered types of circulatingwaveforms are shown They are angle versus aiming (beam-formed on transmit) angle taken at the contingently zerorange 1205911015840 = 0 (120591 = 120591

0)These cuts demonstrate the possibilities

of the considered signals to implement beam-forming ontransmit implemented in one (or each) of the receiving radarchannels Every single vertical slice for a definite possibleangular direction 120579

0describes the array patterns formed

on transmit by the signal processing means in the receiverwhile physically the radiation pattern on transmit stays wide(omnidirectional for the presented modelling results)This isillustrated by eight vertical slices of the ambiguity function


Circ quad Alltop 120591 = 12059101

05

0

minus05

minus1minus1 10

1

05

0

minus05

minus1minus1 10

0

minus40

minus30

minus20

minus10

(dB)

0 05 1minus1 minus05

Circ LFM 120591 = 1205910

sin(120579

998400 )sin

(120579998400 )

Cut along 1205790 = 0∘

Cut along 1205790 = 0∘

Cut along 1205790 = 30∘

0 05 1

0

minus1 minus05minus40

minus30minus35

minus20minus25

minus15

minus5minus10

(dB)

0

minus40

minus30

minus20

minus10

(dB)


Cut along 1205790 = 30∘

sin(1205790)

sin(1205790)

0

minus10

minus20

minus30

minus40

0

minus10

minus20

minus30

minus40


0

minus40

minus30

minus20

minus10

(dB)

(a)

Circ M sequence 120591 = 12059101

05

0

minus05

minus1minus1 10

1

05

0

minus05

minus1

minus1 10

0

minus10

minus20

minus30

minus40

0

minus10

minus20

minus30

minus40

Circ NLFM 120591 = 1205910

sin(120579

998400 )0 05 1minus1 minus05

Cut along 1205790 = 0∘

Cut along 1205790 = 0∘

Cut along 1205790 = 30∘

Cut along 1205790 = 30∘

sin(1205790)

sin(1205790)

sin(120579

998400 )

0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)

(b)

Figure 3 Angle-angle cuts of the ambiguity function |11987701205790(1205911015840

1205791015840

)|

2 at 1205911015840 = 0 in dB


minus15 minus10 minus5 0 5 10 15minus50

minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)Ra

nge (

km)

(dB)

minus50

minus40

minus30

minus20

minus10

0

(dB)

Circ quad Alltop 1205790 = 0∘

Circ LFM 1205790 = 0∘

10

15

5

0

minus5

minus10

minus15minus1 minus05 0 05 1

sin(120579998400)

minus15 minus10 minus5 0 5 10 15Range (km)

10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(a)

Circ M sequence 1205790 = 0∘

Circ NLFM 1205790 = 0∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40


minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10

minus15minus05minus1 0 05 1

sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(b)

Figure 4 Range-angle cuts of the ambiguity function |11987701205790(1205911015840

1205791015840

)|

2 at 1205790= 0∘ in dB

cuts namely by eight antenna patterns presented for theconsidered signals in two possible aiming angular directions1205790= 0∘

30∘ in Figure 3The angle-angle cuts demonstrate the

uniform radiation for all the considered angular directions1205791015840 for the circulating quadratic Alltop signal M-sequenceand LFM-signalThere are no fluctuations along the diagonalwhich means that the antenna array gain is constant withrespect to the aiming angle This is a clear demonstrationof the fact that by circulating one waveform through thetransmitting radar channels the orthogonality of the radiatedsignals can be achieved

As for the circulating NLFM signal the beam-formedpatterns have maximal amplitude the main beam-widthincreases in the zero angular direction and the gradualamplitude degrades for farther angles The amplitude degra-dation is in order of 14 dB which is certainly not negligibleThe amplitude degradation is explained by the reducedequivalent bandwidth of the NLFM signal compared to thetotal occupied bandwidth and therefore by the inappropri-ate definition of the bandwidth dependent offset Δ119905 Thisproblem can be solved by recalculation of the relative timeshift Δ119905 between the circulating signals 119904119899(119905) according tothe reduced equivalent bandwidth of the NLFM signal Thereduced angular coverage for the circulating NLFM signalmeans also that the angular coverage of the antenna canbe adapted by reducing the time shift between adjacenttransmitters this is also a valuable property of circulatingcodes for use in different operational uses requiring different

angular coverages In this work for fair comparison therelative time shifts are preserved equal

We note that the results presented in Figure 3 indicatethe inherent properties of the considered signals for beam-forming because they have been obtained with the assump-tion of omnidirectional radiation patterns of the antennaarray elements

One of the important properties of transmitted circulat-ing signals is the ability to detect weak (with small radarcross-sections) targets against strong targets or clutter Thisproperty is normally illustrated by the range side lobes ofthe ambiguity function Figures 4 and 5 demonstrate as anexample this property for two aiming directions while notchanging the transmitted waveforms or array geometry

Figure 4 presents four range-angle cuts of the ambiguityfunction They are angle versus range taken at the aiming(beam-formed) angle 120579

0= 0∘ The zoomed figures display

the main lobe of the ambiguity functions for four analysedsignals In turn the vertical slices demonstrate the obtainedmain lobe and range side lobes for one chosen observed angu-lar direction 1205791015840 = 0∘ In this way these vertical slices receivedfor the space-time radar waveforms can be considered as ananalogue of the traditional autocorrelation function obtainedhowever for each observed angular direction

The range side lobes are maximal for the observed direc-tionwhich is equal to the aiming directionThe side-lobe levelbetween the considered signals is maximal about minus15 dB forthe circulating M-sequence and it is rather homogeneous



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)


Circ LFM 1205790 = 30∘

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(a)



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Circ NLFM 1205790 = 30∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(b)


1205791015840

)|

2 at 1205790= 30∘ in dB

in each chosen observation direction Such side-lobe levelswould certainly be detrimental in case of multiple targetssituations The minimal side-lobe level about minus42 dB hasbeen obtained for the circulating NLFM signal Both thecirculating LFM and NLFM signals are characterized by avery low side-lobe level for all the observed directions 1205791015840 Weremind that the BT product of all the signals has been chosenpretty small 255 The main lobe is widened in range for theNLFM signal which is visible in the corresponding zoomedfigure in Figure 4 This is explained by the widened mainbeam of the antenna array for the circulating NLFM signalcase in the aiming direction 120579

0= 0∘ see the corresponding

angle-angle cut of the ambiguity function in Figure 3Figure 5 shows another cut of the ambiguity function

|1198770

1205790

(1205911015840

1205791015840

)|

2 obtained by means of the signal processing forthe aiming angle 120579

0= 30∘ in one of the receiving channels

Now the main lobe of the ambiguity function is shifted inthe proper angular direction The peak of the ambiguityfunction along the observed angular directions is located atthe aiming angle The zoomed figures display the main lobeof the ambiguity functions for four analysed signals as in theprevious figure The vertical slices demonstrate the obtainedmain lobe and range side lobes for one observed angulardirection 1205791015840 = 30

∘ which is equal to the aiming angle Therange side-lobes are maximal for the aiming directions for allthe analysed waveforms This time the range side-lobes aremaximal for the circulating quadratic Alltop code minus156 dBand lower for the M-sequence minus17 dB The circulating LFM

and NLFM signals are still providing very low side-lobe levelin range for all the observed angles In the aimed directionthe maximal values are minus34 dB and minus44 dB respectivelyIt is quite impressive taking into account the small BT-product of the waveforms For the circulating NLFM signalthe normalized amplitude of the main lobe drops to minus14 dBThis is caused by the amplitude degradation with the angleincrease appropriate for the circulating NLFM signal (see theproper angle-angle function in Figure 3)

5 Conclusions

A concept of the circulating space-time code has beendescribed Basically the presented concept describes thewhole class of space-time codes employing a selected type ofa waveform having a large BT product The circulating codesdo not narrow the radiated pattern of the antenna array thusproviding a wide angular coverage possibly tunable In turnthe beam-forming on transmit is achievable by means of thesignal processing in one (or each) receiver channel

Four types of circulating codes have been analysed in thepaper using their ambiguity functions The circulating LFMand NLFM signals demonstrated a very clean range-angleambiguity function with some angular-dependent amplitudedegradation of the latter

This paper describes a novel and technically simplesolution of transmitting only one waveform circulating witha very small offset from one antenna element to another


(or from one subarray to another) The signals can beuniformly radiated in free space providing the beam-formingon transmit implemented in radar receiver

References

[1] B D Van Veen and K M Buckley ldquoBeamforming a versatileapproach to spatial filteringrdquo IEEE ASSPMagazine vol 5 no 2pp 4ndash24 1988

[2] D J Rabideau andPA ParkerUbiquitousMIMOMultifunctionDigital Array Radar and the Role of Time-Energy Managementin Radar Massachusetts Institute of Technology Lincoln Labo-ratory 2004

[3] D Cochran S Suvorova S D Howard and B MoranldquoWaveform libraries and measures of effectiveness for radarschedulingrdquo IEEE Signal Processing Magazine vol 26 no 1 pp12ndash21 2009

[4] F Le Chevalier ldquoSpace-time transmission and coding forairborne radarsrdquo Radar Science and Technology vol 6 no 6 pp411ndash421 2008

[5] P Calvary andD Janer ldquoSpatio-temporal coding for radar arrayprocessingrdquo in Proceedings of the IEEE International Conferenceon Acoustics Speech and Signal Processing (ICASSP rsquo98) vol 4pp 2509ndash2512 Seattle Wash USA May 1998

[6] P Woodward ldquoRadar ambiguity analysisrdquo RRE Technical Note731 1967

[7] H L Van Trees Detection Estimation and Modulation TheoryPart III John Wiley amp Sons 1971

[8] G S Antonio D R Fuhrmann and F C Robey ldquoMIMOradar ambiguity functionsrdquo in Proceeding of the 40th AsilomarConference on Signals Systems and Computers (ACSSC rsquo06) pp36ndash40 IEEE Pacific Grove Calif USA November 2006

[9] J Li and P Stoica MIMO Radar Signal Processing Wiley-IEEEPress 2008

[10] M I Skolnik Introduction To Radar Systems McGraw-HillNew York NY USA 3rd edition 2001

[11] M Longbrake ldquoTrue time-delay beamsteering for radarrdquo inProceedings of the IEEE National Aerospace and ElectronicsConference (NAECON rsquo12) pp 246ndash249 2012

[12] W O Alltop ldquoComplex sequences with low periodic correla-tionsrdquo IEEE Transactions on Information Theory vol 26 no 3pp 350ndash354 1980

[13] S Boukeffa Y Jiang and T Jiang ldquoSidelobe reduction withnonlinear frequency modulated waveformsrdquo in Proceedings ofthe IEEE 7th International Colloquium on Signal Processingand Its Applications (CSPA rsquo11) pp 399ndash403 Penang MalaysiaMarch 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



antenna array

1205790

12

3

s1(t)

s2(t)

s3(t)

sN(t) N

sT(t 1205790)

TxRx

Figure 1 MIMO transmission

the circulating codes presented in this paper themultidimen-sional ambiguity function should be employed [8 9]

This paper describes a solution for the proposed wave-forms based on the use of a novel and technically simplesolution of transmitting only one waveform circulating fromone antenna element to another (or from one subarray toanother) with a very small relative time shift In this waythe radiated signals have exactly the same coding complexenvelope highly overlapping both in the time and frequencydomains

Thepaper is organized as follows In Section 2 the conceptof the circulating space-time codes which applies to allpossible waveforms having good autocorrelation proper-ties is introduced Section 3 describes the multidimensionalambiguity function used for the space-time signals analysisSection 4 gives the examples of the ambiguity functions fora number of circulating codes examples The capabilities ofthe waveforms for the beam-forming on transmit and pulsecompression are shown Section 5 contains the conclusions

2 Circulating Codes Concept

The transmitted waveforms are assumed to be encoded bythe same waveform 119904(119905) circulating with a relative time shiftthrough 119873 MIMO transmitter channels (see Figure 1) Thewaveform 119904

119899

(119905) circulating through the 119899th channel can bewritten as

119904119899

(119905) = 119886119899sdot 119904 (119905 minus (119899 minus 1) sdot Δ119905) (1)

where the index 119899 represents both the number of thetransmitting channels and the number of the transmittedwaveforms The relative time shift Δ119905 between the circulatingsignals is equal to 1-sample time Δ119905 = 1Δ119865 where Δ119865is the signal bandwidth For waveforms with a large BT-product (compression ratio) the relative time shift is verysmall compared to the pulse duration 119879

119901because in fact

BT = 119879119901Δ119905 For the sake of simplicity the amplitudes 119886

119899are

assumed to be the same and equal to one and will be omittedfurther

The simplicity of implementation of the circulating codesshould be emphasized Because of the use of only one wave-form only one complex signal with a large BT product shouldbe generated and then it circulates in all the transmitterchannels with a small relative offset

s1(t)

s2(t)

s3(t)

sN(t)

0

0

0

0

Tp

t

t

t

t

Δt

s(t)

s(t minus Δt)

s(t minus 2 middot Δt)

s(t minus (N minus 1) middot Δt)

middot middot middot

Figure 2 Circulating signals

Figure 2 shows the circulating signals 119904119899(119905) 119899 = 1 sdot sdot sdot 119873The total radiated signal 119904

119879(119905) has the following properties It

has the same duration119879 as the signal 119904(119905)This is explained bythe fact that the envelope of the sum signal has a trapeziumshape with duration 119879 at minus3 dB level The relative time delayof the signal 119904

119879(119905) compared to 119904(119905) is equal to the half of the

relative time shift between the first and the last circulatingsignals This time delay is constant and it is equal to (119873 minus

1) sdot Δ1199052 It can be compensated at the signal generation stagein the MIMO radar transmitter by the equal shift of all theradiated signals or by compensation of this known time delayat the signal processing stage in the MIMO radar receiver

The time offset used in the circulating signal can belooked at as a time-delay beam-steering (see eg [10]) Thetime-delay steering of array antennas is used to steer thenarrowmain beam of the antenna array physically When thecirculating signals are used for sounding the radiated beamis not steered and the signals are radiated within the wideangle The time shifts within the circulating codes are usednot to steer the beam but to keep it as wide as possible Thebeam-forming on transmit (narrowing of the main beam) isimplemented by the signal processing means on receive

Array theory tends to discuss things in phase ratherthan delay so we can convert the delay experienced by thecirculating signal into a phase shift between adjacent antennaelements at a given frequency

Δ120593 = 2120587 sdot 119891 sdot Δ119905 =

2120587 sdot 119891

Δ119865

(2)

where 119891 is the radiated signalrsquos frequency Since the radiatedsignal bandwidth Δ119865 is not equal to zero the phase shift isnot fixedThe phase shift is a function of the frequencyΔ120593(119891)where the radiated signalrsquos frequency belongs to the followinginterval 119891 isin [(119891

0minus Δ1198652) sdot sdot sdot (119891

0+ Δ1198652)] where 119891

0is the

radiated signalrsquos central frequencyTherefore the range of thephase shift described by (2) can be written as

(Δ120593min sdot sdot sdot Δ120593max) = ((Δ1205930 minus 120587) sdot sdot sdot (Δ1205930 + 120587)) (3)

where Δ1205930is the phase shift provided by the central radiated

frequency Equation (3) shows that the phase shift within theradiated signalrsquos bandwidth changes by 2120587 which character-izes the full angular coverage



Δ1205930= 2120587 sdot

119889

119888

sdot 119891 sdot sin 1205790 (4)


0 It




0the


119896(1205790)

119904119879(119905 1205790) =

119873

sum

119899=1

119890119895119896(1205790)sdot119909(119899)

sdot 119904119899

(119905) (5)



119904119903

119877(119905 1205790) = 119890minus119895sdot119896(1205790)sdot119909(119903)

sdot sdot 119904119879(119905 minus 1205910 1205790) + 119890 (119905) (6)







119877119903

1205790

(120591 1205791015840

) = int 119904119879(119905 minus 120591 120579

1015840

) sdot (119904119903

119877(119905 1205790))lowast

119889119905 (7)


119877119903

1205790

(120591 1205791015840

) = lowast

sdot 119890+119895sdot119896(1205790)sdot119909(119903)

sdot int 119904119879(119905 minus 120591 120579

1015840


119889119905

(8)


0)

1205911015840


1198771205790(1205911015840

1205791015840

) = int 119904119879(1199051015840

+ 1205911015840

1205791015840

) sdot (119904119879(1199051015840

1205790))

lowast

1198891199051015840

(9)


1198771205790(1205911015840

1205791015840

) = int(

119873

sum

119899=1

119890119895sdot119896(1205791015840)sdot119909(119899)

sdot 119904 (1199051015840

+ 1205911015840

minus 119899 sdot Δ119905 + Δ119905))

sdot (

119873

sum

1198991015840=1

119890119895sdot119896(1205790)sdot119909(119899

1015840)

sdot 119904(1199051015840

minus 1198991015840

sdot Δ119905 + Δ119905))

lowast

1198891199051015840

(10)


1198771205790(1205911015840

1205791015840

) =

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

sdot int 119904 (1199051015840

+ 1205911015840


sdot 119904lowast

(1199051015840

minus (1198991015840

minus 1) sdot Δ119905) 1198891199051015840

(11)


119878(1205911015840


1198771205790(1205911015840

1205791015840

) =

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

sdot ACF119878(1205911015840



(12)





1198771205790(1205911015840

1205791015840

) =

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

sdot 1 (1205911015840



(13)


1205791015840

)

=

119873

sum

119899=1


sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

(14)

where


le 0

0 for elsewhere 1205911015840(15)



1198771205790(1205911015840

1205791015840

)

=

ACFrect (1205911015840

) sdot

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

for 10038161003816100381610038161003816120591101584010038161003816100381610038161003816le

119873 sdot Δ119905

2

0 for elsewhere 1205911015840(16)



1205791015840


1205790(1205911015840

1205791015840

)|

2



1198771205790(1205911015840

1205791015840


0

(i) |1198771205790(0 sin 120579)|2 = 119863(120579 120579




(ii) |1198771205790((1198881205911015840

)2 1205791015840

) |

2 where 1205790


0

4 Modeling Results















05

0

minus05

minus1minus1 10

1

05

0

minus05

minus1minus1 10

0

minus40

minus30

minus20

minus10

(dB)


Circ LFM 120591 = 1205910

sin(120579

998400 )sin

(120579998400 )

Cut along 1205790 = 0∘

Cut along 1205790 = 0∘

Cut along 1205790 = 30∘

0 05 1

0


minus30minus35

minus20minus25

minus15

minus5minus10

(dB)

0

minus40

minus30

minus20

minus10

(dB)


Cut along 1205790 = 30∘

sin(1205790)

sin(1205790)

0

minus10

minus20

minus30

minus40

0

minus10

minus20

minus30

minus40


0

minus40

minus30

minus20

minus10

(dB)

(a)

Circ M sequence 120591 = 12059101

05

0

minus05

minus1minus1 10

1

05

0

minus05

minus1

minus1 10

0

minus10

minus20

minus30

minus40

0

minus10

minus20

minus30

minus40

Circ NLFM 120591 = 1205910

sin(120579

998400 )0 05 1minus1 minus05

Cut along 1205790 = 0∘

Cut along 1205790 = 0∘

Cut along 1205790 = 30∘

Cut along 1205790 = 30∘

sin(1205790)

sin(1205790)

sin(120579

998400 )

0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)

(b)


1205791015840

)|

2 at 1205911015840 = 0 in dB



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)Ra

nge (

km)

(dB)

minus50

minus40

minus30

minus20

minus10

0

(dB)


Circ LFM 1205790 = 0∘

10

15

5

0

minus5

minus10


sin(120579998400)


10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(a)


Circ NLFM 1205790 = 0∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40


minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(b)


1205791015840

)|

2 at 1205790= 0∘ in dB














minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)


Circ LFM 1205790 = 30∘

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(a)



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Circ NLFM 1205790 = 30∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(b)


1205791015840

)|

2 at 1205790= 30∘ in dB




|1198770

1205790

(1205911015840

1205791015840

)|






5 Conclusions






References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Δ1205930= 2120587 sdot

119889

119888

sdot 119891 sdot sin 1205790 (4)


0 It




0the


119896(1205790)

119904119879(119905 1205790) =

119873

sum

119899=1

119890119895119896(1205790)sdot119909(119899)

sdot 119904119899

(119905) (5)



119904119903

119877(119905 1205790) = 119890minus119895sdot119896(1205790)sdot119909(119903)

sdot sdot 119904119879(119905 minus 1205910 1205790) + 119890 (119905) (6)







119877119903

1205790

(120591 1205791015840

) = int 119904119879(119905 minus 120591 120579

1015840

) sdot (119904119903

119877(119905 1205790))lowast

119889119905 (7)


119877119903

1205790

(120591 1205791015840

) = lowast

sdot 119890+119895sdot119896(1205790)sdot119909(119903)

sdot int 119904119879(119905 minus 120591 120579

1015840


119889119905

(8)


0)

1205911015840


1198771205790(1205911015840

1205791015840

) = int 119904119879(1199051015840

+ 1205911015840

1205791015840

) sdot (119904119879(1199051015840

1205790))

lowast

1198891199051015840

(9)


1198771205790(1205911015840

1205791015840

) = int(

119873

sum

119899=1

119890119895sdot119896(1205791015840)sdot119909(119899)

sdot 119904 (1199051015840

+ 1205911015840

minus 119899 sdot Δ119905 + Δ119905))

sdot (

119873

sum

1198991015840=1

119890119895sdot119896(1205790)sdot119909(119899

1015840)

sdot 119904(1199051015840

minus 1198991015840

sdot Δ119905 + Δ119905))

lowast

1198891199051015840

(10)


1198771205790(1205911015840

1205791015840

) =

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

sdot int 119904 (1199051015840

+ 1205911015840


sdot 119904lowast

(1199051015840

minus (1198991015840

minus 1) sdot Δ119905) 1198891199051015840

(11)


119878(1205911015840


1198771205790(1205911015840

1205791015840

) =

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

sdot ACF119878(1205911015840



(12)





1198771205790(1205911015840

1205791015840

) =

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

sdot 1 (1205911015840



(13)


1205791015840

)

=

119873

sum

119899=1


sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

(14)

where


le 0

0 for elsewhere 1205911015840(15)



1198771205790(1205911015840

1205791015840

)

=

ACFrect (1205911015840

) sdot

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

for 10038161003816100381610038161003816120591101584010038161003816100381610038161003816le

119873 sdot Δ119905

2

0 for elsewhere 1205911015840(16)



1205791015840


1205790(1205911015840

1205791015840

)|

2



1198771205790(1205911015840

1205791015840


0

(i) |1198771205790(0 sin 120579)|2 = 119863(120579 120579




(ii) |1198771205790((1198881205911015840

)2 1205791015840

) |

2 where 1205790


0

4 Modeling Results















05

0

minus05

minus1minus1 10

1

05

0

minus05

minus1minus1 10

0

minus40

minus30

minus20

minus10

(dB)


Circ LFM 120591 = 1205910

sin(120579

998400 )sin

(120579998400 )

Cut along 1205790 = 0∘

Cut along 1205790 = 0∘

Cut along 1205790 = 30∘

0 05 1

0


minus30minus35

minus20minus25

minus15

minus5minus10

(dB)

0

minus40

minus30

minus20

minus10

(dB)


Cut along 1205790 = 30∘

sin(1205790)

sin(1205790)

0

minus10

minus20

minus30

minus40

0

minus10

minus20

minus30

minus40


0

minus40

minus30

minus20

minus10

(dB)

(a)

Circ M sequence 120591 = 12059101

05

0

minus05

minus1minus1 10

1

05

0

minus05

minus1

minus1 10

0

minus10

minus20

minus30

minus40

0

minus10

minus20

minus30

minus40

Circ NLFM 120591 = 1205910

sin(120579

998400 )0 05 1minus1 minus05

Cut along 1205790 = 0∘

Cut along 1205790 = 0∘

Cut along 1205790 = 30∘

Cut along 1205790 = 30∘

sin(1205790)

sin(1205790)

sin(120579

998400 )

0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)

(b)


1205791015840

)|

2 at 1205911015840 = 0 in dB



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)Ra

nge (

km)

(dB)

minus50

minus40

minus30

minus20

minus10

0

(dB)


Circ LFM 1205790 = 0∘

10

15

5

0

minus5

minus10


sin(120579998400)


10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(a)


Circ NLFM 1205790 = 0∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40


minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(b)


1205791015840

)|

2 at 1205790= 0∘ in dB














minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)


Circ LFM 1205790 = 30∘

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(a)



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Circ NLFM 1205790 = 30∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(b)


1205791015840

)|

2 at 1205790= 30∘ in dB




|1198770

1205790

(1205911015840

1205791015840

)|






5 Conclusions






References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198771205790(1205911015840

1205791015840

) =

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

sdot 1 (1205911015840



(13)


1205791015840

)

=

119873

sum

119899=1


sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

(14)

where


le 0

0 for elsewhere 1205911015840(15)



1198771205790(1205911015840

1205791015840

)

=

ACFrect (1205911015840

) sdot

119873

sum

119899=1

119873

sum

1198991015840=1

119890119895sdot(119896(120579

1015840)sdot119909(119899)minus119896(1205790)sdot119909(119899

1015840))

for 10038161003816100381610038161003816120591101584010038161003816100381610038161003816le

119873 sdot Δ119905

2

0 for elsewhere 1205911015840(16)



1205791015840


1205790(1205911015840

1205791015840

)|

2



1198771205790(1205911015840

1205791015840


0

(i) |1198771205790(0 sin 120579)|2 = 119863(120579 120579




(ii) |1198771205790((1198881205911015840

)2 1205791015840

) |

2 where 1205790


0

4 Modeling Results















05

0

minus05

minus1minus1 10

1

05

0

minus05

minus1minus1 10

0

minus40

minus30

minus20

minus10

(dB)


Circ LFM 120591 = 1205910

sin(120579

998400 )sin

(120579998400 )

Cut along 1205790 = 0∘

Cut along 1205790 = 0∘

Cut along 1205790 = 30∘

0 05 1

0


minus30minus35

minus20minus25

minus15

minus5minus10

(dB)

0

minus40

minus30

minus20

minus10

(dB)


Cut along 1205790 = 30∘

sin(1205790)

sin(1205790)

0

minus10

minus20

minus30

minus40

0

minus10

minus20

minus30

minus40


0

minus40

minus30

minus20

minus10

(dB)

(a)

Circ M sequence 120591 = 12059101

05

0

minus05

minus1minus1 10

1

05

0

minus05

minus1

minus1 10

0

minus10

minus20

minus30

minus40

0

minus10

minus20

minus30

minus40

Circ NLFM 120591 = 1205910

sin(120579

998400 )0 05 1minus1 minus05

Cut along 1205790 = 0∘

Cut along 1205790 = 0∘

Cut along 1205790 = 30∘

Cut along 1205790 = 30∘

sin(1205790)

sin(1205790)

sin(120579

998400 )

0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)

(b)


1205791015840

)|

2 at 1205911015840 = 0 in dB



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)Ra

nge (

km)

(dB)

minus50

minus40

minus30

minus20

minus10

0

(dB)


Circ LFM 1205790 = 0∘

10

15

5

0

minus5

minus10


sin(120579998400)


10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(a)


Circ NLFM 1205790 = 0∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40


minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(b)


1205791015840

)|

2 at 1205790= 0∘ in dB














minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)


Circ LFM 1205790 = 30∘

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(a)



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Circ NLFM 1205790 = 30∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(b)


1205791015840

)|

2 at 1205790= 30∘ in dB




|1198770

1205790

(1205911015840

1205791015840

)|






5 Conclusions






References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











05

0

minus05

minus1minus1 10

1

05

0

minus05

minus1minus1 10

0

minus40

minus30

minus20

minus10

(dB)


Circ LFM 120591 = 1205910

sin(120579

998400 )sin

(120579998400 )

Cut along 1205790 = 0∘

Cut along 1205790 = 0∘

Cut along 1205790 = 30∘

0 05 1

0


minus30minus35

minus20minus25

minus15

minus5minus10

(dB)

0

minus40

minus30

minus20

minus10

(dB)


Cut along 1205790 = 30∘

sin(1205790)

sin(1205790)

0

minus10

minus20

minus30

minus40

0

minus10

minus20

minus30

minus40


0

minus40

minus30

minus20

minus10

(dB)

(a)

Circ M sequence 120591 = 12059101

05

0

minus05

minus1minus1 10

1

05

0

minus05

minus1

minus1 10

0

minus10

minus20

minus30

minus40

0

minus10

minus20

minus30

minus40

Circ NLFM 120591 = 1205910

sin(120579

998400 )0 05 1minus1 minus05

Cut along 1205790 = 0∘

Cut along 1205790 = 0∘

Cut along 1205790 = 30∘

Cut along 1205790 = 30∘

sin(1205790)

sin(1205790)

sin(120579

998400 )

0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)


0

minus40

minus30

minus20

minus10

(dB)

(b)


1205791015840

)|

2 at 1205911015840 = 0 in dB



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)Ra

nge (

km)

(dB)

minus50

minus40

minus30

minus20

minus10

0

(dB)


Circ LFM 1205790 = 0∘

10

15

5

0

minus5

minus10


sin(120579998400)


10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(a)


Circ NLFM 1205790 = 0∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40


minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(b)


1205791015840

)|

2 at 1205790= 0∘ in dB














minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)


Circ LFM 1205790 = 30∘

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(a)



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Circ NLFM 1205790 = 30∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(b)


1205791015840

)|

2 at 1205790= 30∘ in dB




|1198770

1205790

(1205911015840

1205791015840

)|






5 Conclusions






References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)Ra

nge (

km)

(dB)

minus50

minus40

minus30

minus20

minus10

0

(dB)


Circ LFM 1205790 = 0∘

10

15

5

0

minus5

minus10


sin(120579998400)


10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(a)


Circ NLFM 1205790 = 0∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40


minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Cut along 120579998400 = 0∘

Cut along 120579998400 = 0∘

(b)


1205791015840

)|

2 at 1205790= 0∘ in dB














minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)


Circ LFM 1205790 = 30∘

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(a)



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Circ NLFM 1205790 = 30∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(b)


1205791015840

)|

2 at 1205790= 30∘ in dB




|1198770

1205790

(1205911015840

1205791015840

)|






5 Conclusions






References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)


Circ LFM 1205790 = 30∘

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(a)



minus40

minus30

minus20

minus10

0

Range (km)

Rang

e (km

)

(dB)


minus40

minus30

minus20

minus10

0

Range (km)

(dB)

10

15

5

0

minus5

minus10


sin(120579998400)

Rang

e (km

)

10

15

5

0

minus5

minus10


sin(120579998400)

Circ NLFM 1205790 = 30∘

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

0minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

Cut along 120579998400 = 30∘

Cut along 120579998400 = 30∘

(b)


1205791015840

)|

2 at 1205790= 30∘ in dB




|1198770

1205790

(1205911015840

1205791015840

)|






5 Conclusions






References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Space-Time Radar Waveforms: …downloads.hindawi.com/journals/jece/2013/809691.pdfSpace-Time Radar Waveforms: Circulating Codes ... the same waveform circulating with