Adaptive Distributed MIMO Radar WaveformOptimization Based on Mutual Information

A novel approach to optimizing the waveforms of an adaptive

distributed multiple-input multiple-output (MIMO) radar is

developed. The research work aims at improving the target

detection and feature extraction performance by maximizing the

mutual information (MI) between the target impulse response

and the received echoes in the first step, and then minimizing

the MI between successive backscatter signals in the second

step. These two stages correspond to the design of the ensemble

of excitations and the selection of a suitable signal out of the

ensemble, respectively. The waveform optimization algorithm

is based upon adaptive learning from the radar scene, which

is achieved through a feedback loop from the receiver to the

transmitter. This feedback includes vital information about the

target features derived from the reflected pulses. In this way the

transmitter adjusts its probing signals to suit the dynamically

changing environment. Simulation results demonstrate better

target response extraction using the proposed two-step algorithm

as compared with each single-step optimization method. This

approach also results in improved target detection probability and

delay-Doppler resolution as the number of iterations increases.

I. INTRODUCTION

In recent years, the research on adaptive radarwaveform design has received great impetus.Some of the noteworthy works in this area include[1][5], where the radar transmission parametersare continuously modified in order to improve thetarget parameter estimation in a time-varying radarenvironment. Another related development is theproposal of cognitive radar, which represents aninnovative paradigm to describe brain-empoweredsystem architectures that constantly employinformation-gathering mechanisms to facilitateintelligent illumination of the dynamic radar scene.For a cognitive radar the perception about the radarenvironment formed at the receiver is relayed to

Manuscript received October 11, 2011; revised February 15 andJune 23, 2012; released for publication November 16, 2012.

IEEE Log No. T-AES/49/2/944545.

Refereeing of this contribution was handled by R. Narayanan.

Y. Chen wishes to acknowledge the support by Chinas RecruitmentProgram of Global Experts and South University of Science andTechnology of China. C. Yuen wishes to acknowledge the supportby the International Design Center (Grants IDG31100102 andIDD11100101).

0018-9251/13/$26.00 c 2013 IEEE

the transmitter through a continuous feedback loop.Subsequently, the updated information about theenvironment can be utilized to allocate crucialresources such as transmit power and spectrum ina more efficient manner [6, 7]. Such a constantlearning approach allows the development ofwaveform optimization techniques offering bettertarget resolution capabilities as shown in [8], [9].Recent results have also shown that a

multiple-input multiple-output (MIMO) radar, whichemploys multiple transmit and receive antennas, canfully exploit waveform and spatial diversity gainsby illuminating the target in different directions[1012]. MIMO radars employ orthogonal signalsat distinct transmit antenna elements, which excitedifferent scattering centers on the extended targets,thus enhancing the information content in thereceived backscatter signal. Since the target returnsare strongly dependent on the cross-sectional areasof scatterers in the line-of-sight directions of radarelements, the spatial diversity provided by thedistributed MIMO radar improves the target parameterextraction as shown in [13][16]. In terms of the pulseoptimization, an important school of thought is toapply information theory to radar signal processing[17]. The pioneering work by Bell [18] appliedinformation-theoretic measures for the design ofwaveforms in order to facilitate improved targetdetection and classification. Yang and Blum [19]extended the work in [18] by using the mutualinformation (MI) between the random target responseand the reflected signal as a waveform optimizationcriterion in the MIMO radar configuration. Otherexisting works [1315, 20, 21] also utilize similardesign criteria.Following from the above discussions, it

is interesting to study the performance of aknowledge-aided MIMO radar that combines thestrength of adaptivity and MIMO. Specifically, welook into the problem of excitation waveform designand present a novel two-stage optimization strategy,which can be summarized as follows.1) Step 1. Waveform Design: This module

involves the design of transmission signals for thedistinct MIMO transmit antenna elements. Themain objective is to maximize the MI between thebackscatter signal and the estimated target response,subject to the transmission power constraint [20]. Thisapproach ensures that the received echoes at each timeinstant become more statistically dependent on thetarget features. Once the optimal waveform ensembleis obtained, the next step is to select the appropriatesignals for transmission.2) Step 2. Waveform Selection: This module is

based on the principle of minimizing the MI betweensuccessive received signals. This selection criterionensures that we always acquire target echoes that aremore statistically independent on each other in time,

1374 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 2 APRIL 2013

dawnchiaTypewritten TextY. Nijusre, Y. Chen, C. Yuen, Y. H. Chew, Z. Ding, S. Boussakta

with an intention of gaining more knowledge aboutthe target features at each time instant of reception.Furthermore, the optimization process is preceded

by channel estimation, wherein an estimate of thetarget response and noise characteristics is formedby the receiver through measurements carried outin the previous time instant. A feedback loop fromthe receiver to the transmitter allows the deliveryof this radar scene information to the transmitter.Consequently, the excitation signal optimizationprocess enables the transmitter to dynamicallyadjust its operational mode to suit the changingradar environment. Such architectures are similar tocognitive radars, where the systems utilize a constantlearning approach by updating the estimates on targetscene parameters through multiple interactions withthe environment.The contributions of the current work can be

summarized as follows:

1) development of a practical framework for anadaptive distributed MIMO radar, which benefits fromthe principle of cognition as elaborated in [6];2) design of a novel two-stage algorithm for

waveform optimization in the proposed adaptivedistributed MIMO radar framework;3) evaluation of the performance of the current

approach in terms of target feature extraction, targetdetection, and delay-Doppler resolution; and4) comparison with other prevalent radar system

configurations through analysis of the target scenegeneration and receiver operating characteristics(ROC) in an interference-prone setup.

The rest of the paper is organized as follows. InSection II we describe the adaptive distributed MIMOradar system and formulate the two-step waveformoptimization process. The probing signals aredesigned at step 1 of the algorithm and selected basedon the criterion presented in step 2. In Section IIIwe analyze the delay-Doppler resolution achievedby the optimized waveform ensemble. In Section IVwe present simulation results for the mean squarederror (MSE) of target response estimation, probabilityof detection, and ambiguity function (AF) by usingthe proposed algorithm, and compare it with otherwidely used techniques for a clutter-corrupted targetscene. Finally, some concluding remarks are drawnin Section V. Throughout this work, we use det() todenote the determinant of a matrix, ()T to denote thetranspose, ()H to denote the Hermitian transpose, tr()to denote the trace, and Efg to denote the expectationoperator.

II. WAVEFORM OPTIMIZATION STRATEGY

A. System Architecture

Figure 1 indicates the general architecture of theadaptive distributed MIMO radar under consideration.

Fig. 1. Adaptive distributed MIMO radar architecture.

As discussed extensively in the existing literature,modern radar applications make use of the pulsecompression techniques such as linear frequencymodulation or phase-coded waveforms employingBarker codes or Costas codes in order to improve thetarget delay-Doppler resolution [22]. In this work weadopt the idea of utilizing phase-coded waveformsfor generation of sequences required for transmissionover various transmit antennas. We consider theultrawideband (UWB) probing signals [7], thoughthe general methodology is also applicable to anyother type of excitation. The waveform comprises asequence of UWB Gaussian pulses in which the phaseof each pulse is modulated in accordance with theorthogonal sequences corresponding to the columnvectors of a Hadamard matrix [22]. Each normalizedGaussian pulse takes the following form

u(t) =1p2T

exp t

2

T2

(1)

where T determines the pulsewidth and is assumed tobe 0.2 ns.As indicated in Fig. 1, a set of orthogonal

sequences of Gaussian waveforms ready to be sentover each of the transmit antenna elements areconstructed at the waveform optimization module,which is discussed in detail in Section II-C. Theoptimized waveforms are then transmitted overthe radar channels. The backscatter signals aregathered by each of the receive antenna elementsand passed on to a matched filter bank, whichcorrelates the received signals with each individualtransmit waveform stored in the receiver. Targetresponse is thus extracted by the target detectionand parameter estimation module, which attemptsto discriminate the target from the surroundingclutter. The estimated channel response and receivedsignal characteristics such as noise covariance areforwarded to the waveform optimization modulethrough a feedback link. In light of the updated radarscene, the optimization module designs and selectssuitable sequences for each of the transmit antennasin order to acquire the best knowledge on the targetin the next time instant. This operation facilitates

CORRESPONDENCE 1375

adaptive illumination of the radar environment andessentially leads to a cognitive system featuring thefollowing two important properties described in [6]:1) intelligent signal processing, which builds onreal-time learning through continuous interaction ofthe radar with the surroundings, and 2) feedback fromthe receiver to the transmitter, which is a facilitator ofintelligence.

B. Signal Model

Without loss of generality, we consider thebistatic radar configuration, where the transmitterand receiver are connected but noncollocated. It isassumed that the two direct paths, one between thetransmitter and the target and the other between thetarget and the receiver, have been extracted throughsome preprocessing steps [23]. For example, theMIMO radar may employ beam-steering [7] anddelay windowing [24] to suppress nontarget impulseresponses. Thus we focus on the direct path in thefollowing.Suppose that the distributed MIMO system has

M transmit and N receive antennas. For simplicity ofdiscussion, it is assumed that M =N. We can expressthe received signal vector at the nth (n= 1,2, : : : ,N)antenna element as [12]

yn =MXm=1

hm,nxm,n+ n: (2)

In the preceding equation yn 2 CK1 where C indicatesthe complex number domain. The parameter K =Ks+Kd, where Ks is the length of the pseudorandomsequence generated and Kd is the maximum excessdelay with respect to the first arrival among all thelinks. The term xm,n = [01Lm,n x

Tm 01(KdLm,n)]

T,where 01l is a null vector of length l, Lm,n is thepropagation delay between the mth transmit and thenth receive antennas via the target, and xm 2 CKs1is the probing signal sent by the mth transmitter.hm,n represents the channel response between themth transmit and the nth receive antennas. Finally,n 2 CK1 denotes the noise at the nth receiver, whichcharacterizes the combination of both additive whiteGaussian noise (AWGN) and joint nuisance of thechannel estimation and measurement errors. In thiswork we consider the scenario that the length ofthe pseudorandom sequence is much larger than theexcess delay, which is generally valid for short-rangeradar applications when the propagation distanceis within tens of meters. As a result, K Ks andxm can be used to approximate xm,n. A similarcondition has also been imposed in other recentworks like [20]. We further assume that the minimumtransmit/receive antenna spacing is sufficientlylarger than half wavelength (distributed MIMOconfiguration). Hence, the correlation introduced byfinite antenna element spacing is low enough that the

fades associated with two different antenna elementscan be considered independent. Subsequently, thereflection coefficient hm,n is assumed to be differentfor various pairs of transmitter and receiver, and itsphase is assumed to be uniformly distributed between0 and 2.Let Y= [y1 y2 yN] 2 CKN be the ensemble

of received signals, X= [x1 x2 xM] 2 CKM bethe set of transmission sequences to be used, H=[hm,n]MN 2 CMN be the target response matrix, and = [1 2 N] 2 CKN be the noise matrix. We canconveniently express (2) as

Y=XH+: (3)

Each hm,n in the scattering matrix H is proportionalto the target radar cross section (RCS), whosescintillation can vary slowly or rapidly dependingon the target size, shape, dynamics, and its relativemotion with respect to the radar. The two randommatrices H and are assumed to be independent ofeach other. A detailed description of the constructionof H is presented in Appendix I.We consider the target as a point scatterer amidst

several clutter sources. Depending upon the motionand clutter characteristics, the radar target has beenclassified into various Swerling models [25]. InSwerling III the RCS samples measured by the radarare correlated throughout an entire scan, but areuncorrelated from scan to scan (slow fluctuation) andthe radar scene comprises a single powerful scatteringcenter and many weak reflectors in its vicinity. Thismodel is applied in the current work, where weassume that the radar scene is dominated by the targetand the amplitude returns from nontarget scatterersare lower than those from the target. The random RCS = jhm,nj2 is exponentially distributed, which takes thefollowing form

f() =1avexp av

(4)

where > 0 represents the variance of RCSfluctuations and av is the average RCS.

C. Two-Stage Waveform Optimization

The waveform design and selection processes canbe formulated as the following two-step algorithm.Note that the subscript t is used to indicate theparameters for a particular round of radar systemadaptation at time t.Step 1 Maximization of MI between the estimated

target response and the received target echoes attime t.Following the definition of MI,

I(Yt;Ht jXt) =H(Yt jXt)H(Yt jHt,Xt)=H(Yt jXt)H(t) (5)

1376 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 2 APRIL 2013

where I(Yt;Ht jXt) is the MI between two randomvariates Yt and Ht given the transmission matrix Xt,and H(Yt jXt) represents the conditional entropy orthe average information that Xt conveys about Yt.Our aim is to maximize I(Yt;Ht jXt) between Yt

and Ht given Xt, i.e., we intend to maximize the MIbetween the received target echoes and the channelresponse given the ensemble of transmit waveforms.This implies that the backscatter signals would bemore statistically dependent upon the actual radarscene. We can simplify (5) by applying the definitionof entropy as follows

H(Yt jXt) =Zp(Yt jXt) ln[p(Yt jXt)]dYt (6)

where p(Yt jXt) denotes the conditional probabilitydensity function (pdf) of Yt given Xt. The aboveexpression for the entropy can be further simplifiedby evaluating p(Yt jXt) to be

p(Yt jXt) =NYn=1

p(yn jX)

=NYn=1

1K det(XHt RHtXt+Rt )

exp[yHn,t(XHt RHtXt+Rt )1yn,t]

=1

NK[det(XHt RHtXt+Rt )]N

expftr[(XHt RHtXt+Rt )1YHt Yt]g(7)

where RHt = EfHHt Htg and Rt = EfHt tg are thecovariance matrices of the target response Ht and thenoise t, respectively. Solving (6) and (7) gives riseto the following result for the entropy [20]

H(Yt jXt) =NK ln() +NK +N ln[det(XHt RHtXt+Rt )]:(8)

Similarly, we can derive the entropy of the noise as

H(t) =NK ln() +NK +N ln[det(Rt )]: (9)Using (5), (8), and (9), we can compute the MI as

I(Yt;Ht jXt) =N ln[det(XHt RHtXt +Rt )]N ln[det(Rt )]:(10)

Hence, the maximization in step 1 can be simplified as

maxXtfN ln[det(XHt RHtXt+Rt )]N ln[det(Rt )]g

subject to tr[XHt Xt] P0(11)

with P0 being the total transmission power.A rigorous solution of (11) has been provided

in [20]. We can then find the optimal waveformensemble SXt out of the entire set of orthogonalsequences from the Hadamard matrix, and the

corresponding power allocation vector over differentantenna elements, Xt , for each Xt 2 SXt . In otherwords, we start our waveform design with theorthogonal sequences from the Hadamard matrix, andmodulate the power of the waveform on an individualpulse level as well as across the transmit antennaelements using the maximization criterion presentedin (11).Step 2 Minimization of MI between the received

target echoes at time t and the estimated target echoesat time t+1.We now proceed to the second module of the

waveform optimization process, in which we intendto ensure that successive target echoes are as differentfrom each other as possible. This would ensure that atevery instant of reception, we learn something moreabout the radar scene.We can express the MI between the received

signals in two consecutive times, t and t+1, as

I(Yt,Yt+1) =H(Yt jXt) +H(Yt+1 jXt+1)H(Yt,Yt+1 jXt,Xt+1): (12)

In the preceding equation H(Yt jXt) (or H(Yt+1 jXt+1))denotes the measure of the uncertainty in the receivedsignal at time t (or t+1) given the knowledge ofthe transmitted signal Xt (or Xt+1). Furthermore,H(Yt,Yt+1 jXt,Xt+1) is the entropy of the receivedsignal pair (Yt,Yt+1) given the transmitted signal pair(Xt,Xt+1). We can simplify (12) in the same way aswe did for (6) to obtain the following results

H(Yt jXt) =Zp(Yt jXt) ln[p(Yt jXt)]dYt

=NK ln() +NK

+N ln[det(XHt RHtXt+Rt )] (13)

H(Yt+1 jXt+1) =Zp(Yt+1 jXt+1) ln[p(Yt+1 jXt+1)]dYt+1

=NK ln() +NK

+N ln[det(XHt+1RHtXt+1 +Rt )] (14)and

H(Yt,Yt+1 j Xt,Xt+1)= 2NK ln() +2NK

+N ln[det(XHt RHtXt+Rt )]

+N ln[det(XHt+1RHtXt+1 +Rt )]

+N lnfdetfIMM [D(t,t+1)]2gg: (15)In the preceding equation IMM is the identity matrixof dimension M M and D(t,t+1) is the diagonalmatrix obtained by singular value decomposition(SVD) of the covariance matrix RYt,Yt+1 , given as thecross-covariance of the whitened expressions for Yt

CORRESPONDENCE 1377

and Yt+1:

RYt,Yt+1 = EfYHt Yt+1g= E

(YtqR1Yt

HYt+1

qR1Yt+1

)

=q

R1Yt

HRYt,Yt+1

qR1Yt+1 (16)

where

RYt = EfYHt Ytg=XHt RHtXt+RtRYt+1 = EfYHt+1Yt+1g=XHt+1RHtXt+1 +Rt

RYt,Yt+1 = EfYHt Yt+1g=XHt RHtXt+1:

(17)

Note that RYt ,Yt+1 in (17) does not include a noise termas noise at two different time instants is assumed to beuncorrelated. Furthermore, the covariance matrices ofH and estimated at time t are used to approximatethe two matrices at time t+1 in (14)(17). Solving(13)(15), we obtain

I(Yt,Yt+1) =N lnfdetfIMM [D(t,t+1)]2gg

=NMXm=1

lnf1 [d(t,t+1)m ]2g (18)

where d(t,t+1)m are the diagonal elements of the matrixD(t,t+1) arranged in the descending order as d(t,t+1)1 d(t,t+1)2 d(t,t+1)3 d(t,t+1)M .Finally, we can form the minimization problem in

step 2 as

minXt+12SXt

(N

MXm=1

lnf1 [d(t,t+1)m ]2g)

subject to tr[XHt+1Xt+1] P0:(19)

Essentially, the minimization criterion presented in(19) is solved by choosing Xt+1 2 SXt such that thecorresponding singular values of RYt ,Yt+1 (cf. (16) and(17)) minimize the expression in (19).The waveform ensemble SXt obtained in step 1

are designed with the purpose of maximizing MI overthe spatial domain, and step 2 selects the transmissionsequence for each transmit antenna element from SXtwith an objective of minimizing MI over the temporaldomain. The proposed waveform optimizationalgorithm can be summarized as follows.

1) At the initial time t= 0, RH0 and R0 can beestimated through successive measurements withuniform power allocation over the transmit antennaelements by solving equations in (17) simultaneously.This can be achieved through estimation of the targetechoes in the next time instant by using (3) andthe prospective transmission waveforms from theensemble SX0 .

2) Solve for the optimum power allocation X0and the set of optimal waveforms SX0 as per themaximization criterion stated in (11).3) Form an estimate of the received signal Y1 at

time t= 1 based on the current estimate for targetimpulse response by using (3). This impulse responseis estimated by de-convolving the received signalwith the transmitted signal. Since it is assumed thatthe target is the most dominant scatterer in the radarenvironment, the result of this de-convolution isassumed to be the actual impulse response.4) Solve for X1 2 SX0 using the minimization

approach as stated in (19).5) Transmit X1 and process the received signal to

obtain the updated RH1 and R1 at time t= 1.6) Repeat steps 25 iteratively.

It is worth emphasizing that cognition is integratedin the above waveform optimization process throughthe feedback operation implemented in step 5 andmultiple interactions with the radar channel instep 6.

III. DELAY-DOPPLER RESOLUTION OF DISTRIBUTEDMIMO RADAR

The radar AF represents the time response of afilter matched to a given finite energy signal when thesignal is received with a delay and a Doppler shift relative to the nominal values expected by the filter asdescribed in [22]. Different from the communicationsystems, the matched filter for a radar receiver isdesigned to match the transmit waveform but notthe channel itself. The radar AF thus explains theability of the radar receiver to boost the backscattersignal from the target (assumed to be at the originof the AF plot) in comparison with the backscattersignal from nontarget scatterers [22]. The closerthe AF response to unity at the origin the better thedelay-Doppler resolution. Ideally the AF plot mustbe a thumbtack response at the origin as suggestedin [22].The radar AF can be mathematically represented

as [25]

0( ,) =Z 11u(t)u(t+ )exp(j2t)

(20)

where u is the complex envelope of the signal. Apositive implies a target moving toward the radar,whereas a positive implies a target being fartherfrom the radar than the reference position with = 0.The radar AF for a single UWB Gaussian pulse asshown in (1) can be represented as

1( ,) =

Z 11exp

t

2

T2

exp

(t+ )

2

T2

exp(j2t)

(21)

1378 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 2 APRIL 2013

where T determines the Gaussian pulsewidth. For atrain of UWB pulses, the radar AF is

2( ,) =1Q

Q1Xq=(Q1)

1( qT,)

sin( jqjT)sin(T)

:

(22)However, the preceding equations are only

applicable to the single-input single-output (SISO)radar architecture. For an MIMO radar the equationneeds to be modified and is derived in [23]. Thereceived signal after matched filtering can beexpressed as

3( ,,f) =Z[y(t, 0,0,f 0)]Hy(t, ,,f)dt

(23)

where , , f represent the delay, Doppler shift, andspatial frequency, respectively, and 0, 0, f 0 are thecorresponding parameters used by the matched filterat the receiver. We can match the spatial frequencyat each of the receive antenna element by adoptingreceiver beam-forming. In terms of the transmittedwaveform, the above equation can be written as

3( ,,f) =

NXn=1

exp[j2(ff 0)n]

MX

m1=1

MXm2=1

jm1,m2 ( ,)jexp[j2(fm1 fm2 )]

(24)

where M and N are the number of transmit andreceive antenna elements. The cross AF is obtained as

m1,m2 ( ,) =

Zum1 (t )u

Hm2(t 0)exp[j2( 0)t]dt:

(25)As the radar AF is a function of the transmit

signal, we can evaluate the performance of theproposed waveform optimization strategy in terms ofthe delay-Doppler resolution by using (24).

IV. SIMULATION RESULTS

A. Simulation Parameters and Target ImpulseResponse Extraction

We employ orthogonal sequences of the Gaussianpulse over the transmit antenna elements. Thebackscatter signals are matched filtered at thereceivers and the transmitted signals are latermodified by the waveform optimization module asshown in Fig. 1. Figure 2(a) indicates the optimizedtransmission sequence at one particular transmitantenna after the two-step optimization process.Figure 2(b) shows the received pulses for thetransmission sequence in Fig. 2(a). Figure 2(c)shows the target response extracted from the receivedtarget echoes after matched filtering at the end of

20 iterations of the algorithm, where an excellentperformance of the target response estimation can beobserved. At each iteration, the scattering coefficientsfor the target and nontarget scatterers in H vary asdescribed by the Swerling III model. This causesthe amplitude returns of the backscatter signals tovary at each instance. However, the amplitudes ofthe echoes from the target are always assumed tobe stronger than those from the clutter sources. Itis difficult to analytically determine the numberof successive measurements required to achieve asufficiently accurate estimate of the actual targetresponse, which depends upon various target sceneparameters such as the distribution of scatterers andtheir relative motion and orientation. Nevertheless,simulation results show that, on average, 20 iterationsof the waveform optimization process are required inorder to achieve convergence of the target responseestimation for a wide range of radar scenes.

B. Probability of Target Detection and MSEPerformance

Figure 3(a) indicates the MSE achieved by thealgorithm with regard to the estimation of targetresponse. This plot demonstrates an improvedMSE performance for the two-step optimization ascompared with the individual maximization (step 1)and minimization (step 2) modules, particularlyfor the first few iterations. Figure 3(b) indicatesthe probability of target detection achieved by theproposed method, which is obtained by averagingover 1000 simulations each at a particular valueof the received signal-to-clutter-plus-noise ratio(SCNR). We apply the hypothesis testing methodbased on the optimal Neyman-Pearson algorithm[22]. Figure 3(b) shows that for a fixed probabilityof detection, the required SCNR value decreases asthe number of iterations increases. An SCNR gain ofmore than 5.5 dB at a detection probability of 0.8 isobserved between iterations 1 and 20. Nevertheless,the system performance does not show any significantimprovement beyond 20 iterations.

C. Delay-Doppler Resolution Performance

Figure 4(a) illustrates the normalized 2 2distributed MIMO radar AF contours after the 1stiteration of the optimization algorithm. As can beseen from Fig. 4(a), the resolution of the delayand Doppler of the target deteriorates due to thepresence of surrounding clutter. This is particularlyevident if the nontarget scatterers are positionedin the vicinity along the line joining the target andthe antenna element, assuming that the target islocated at the origin of the plot. However, as weincrease the iteration number, we observe that thetarget discrimination ability is greatly improved asshown in Fig. 4(b). Specifically, at the end of 20

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Fig. 2. (a) Transmission waveform xt at particular antenna aftertwo-step optimization. (b) Received signal after matched filteringfor transmitted signal shown in (a). (c) Target response extraction.

iterations, the clutter interference is suppressed byapproximately 2 dB. The delay-Doppler resolutionis directly related to the AF of the radar waveforms.It represents the matched filter output of the radarreceiver and should ideally be a thumbtack response

(with unity at the origin corresponding to thedelay-Doppler for the target). The ideal AF responsewould be observed if we use statistically independentwaveforms for transmission with optimum phase shiftsand amplitudes for the pulses. With the two-stepwaveform optimization, we design and select thepulses matched to the approximated target response,which is updated at each iteration. Furthermore, theoptimum power allocation ensures that we suppressthe clutter interference over the distributed MIMOradar channels and thus improve the SCNR of thereceived signals. The probability of target detectionpresented in Fig. 3(b) displays this improvementin SCNR with iteration number in the presence ofstrong clutter. This result demonstrates the improvedcapability of the radar system to discriminate thetarget from its surroundings and resolve it in termsof its range and velocity.

D. Target Discrimination Performance

In Fig. 5 we provide simulation data for the targetscene shown in Fig. 1. In this simulation we comparethe performance of different radar configurations withrespect to their target discrimination capabilities. Theradar scene is simulated by placing the target (whichis the vehicle in Fig. 1) and nontarget scatterers(which are trees and brick walls) at fixed locationson the two-dimensional (2D) map. The map isspatially discretized in the form of range bins inboth the X and Y directions. The boundary cellsof all the scatterers fluctuate over subsequent scansas defined by the Swerling III model described inSection II-B and Appendix I. The target scene isilluminated by UWB Gaussian pulses over eachof the antenna elements, as described in the earliersections. The samples of the signals received at eachantenna are captured and utilized to perform rangeestimation for the scatterer boundaries, which isachieved by calculating the delays that maximize thecross-correlation between the received signal samplesand the delayed version of the transmission waveform.The range cells corresponding to the boundary ofeach scatterer are identified by using these estimateddelays. This approach to target range estimation issimilar to the one adopted in [26]. Subsequently, anapproximated map of the target scene can be created.Figure 5(a) shows the target scene image recreatedusing the conventional synthetic aperture radar (SAR),where the phased antenna arrays are employed atboth the transmitter and the receiver. For the SARwe utilize the same Gaussian UWB sequence overeach of the antenna elements but with different delaysand with the uniform power allocation. As can beseen from Fig. 5(a), the target discrimination offeredby this technique is poor. Although the presence ofthe object can be successfully detected, the SARfails to discriminate the object from the surrounding

1380 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 2 APRIL 2013

Fig. 3. (a) MSE in target response extraction. (b) Probability of target detection.

Fig. 4. AF contours indicating target resolution at (a) iteration 1 and (b) iteration 20 for 2 2 MIMO radar, which demonstratessmaller focal area and improved signal-to-clutter ratio in (b) as compared with (a).

clutter. Furthermore, the target signature extractionis suboptimal as shown in the image. Figure 5(b)represents the approximated target scene by using the4 4 distributed MIMO radar and employing the MImaximization approach for designing waveforms asdiscussed in [20]. The target scene is clearer since thescatterers are better resolved spatially. The enhancedspatial resolution can be attributed to the MIMO radarconfiguration, which exploits the spatial diversityby illuminating the target from different directions.The waveform design solution employed in this caseresults in excitation signals better matched to thetarget response. Hence, the target signature extractionis superior compared with the SAR. Figure 5(c) showsthe recreated radar scene by employing the proposedtwo-step waveform optimization strategy. As can beseen from the image, the target resolution has beensignificantly improved. Finally, Fig. 5(d) depicts theROC for four different radar configurations, which are

1) constant false alarm rate (CFAR) detectionusing the phased antenna arrays at both the transmitterand the receiver as indicated in [27];2) 4 4 distributed MIMO radar employing the

waveform design solution based on maximization ofMI as indicated in [20];3) 4 4 distributed MIMO radar employing the

proposed two-step waveform optimization approach;4) conventional SAR architecture employing the

phased antenna arrays.

For a probability of false alarm at 0.02, theprobability of target detection offered by the proposedscheme is approximately 0.8 as compared with 0.64offered by the MI maximization technique, 0.42by the CFAR, and 0.38 by the SAR. The plots forboth the MI maximization and two-step waveformoptimization algorithms were generated at the endof 20 iterations. No significant improvement wasobserved after 20 iterations.

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Fig. 5. (a) Target scene generated through SAR. (b) Target scene generated through 4 4 MIMO radar (MI maximization algorithm).(c) Target scene generated through 4 4 MIMO radar (two-step MI optimization). (d) ROC for different radar systems.

E. Detection Variation Performance

The detection constraint optimization has beenrecently explored in works like [28], where theauthors address the problem of radar phase-codedwaveform design for extended target recognitionin the presence of colored Gaussian disturbance.The objective function in [28] aims to maximize theweighted average Mahalanobis distance or Euclideandistance between the ideal echoes from differenttarget hypotheses. Furthermore, additional practicalconstraints are considered in [28]. For example,the modulus of the waveform is restricted to be aconstant and the detection constraints require thatthe achievable signal-to-noise ratio for each targethypothesis is larger than a desired threshold. Figure 6represents the detection variation brought aboutby the proposed waveform optimization algorithmsubject to the detection constraint. Specifically, weconsider a radar scene with 7 target scatterers. The

backscatter signal from the radar scene is normalizedand the radar attempts to discriminate the targetsbased on a fixed detection threshold. The detectionof the multiple targets is improved as the numberof iterations of the proposed algorithm increases.This phenomenon also agrees with the probabilityof detection result in Fig. 3(b). As seen in Fig. 6 theradar is able to discriminate 7 targets successfully atthe end of 20 iterations by suppressing the clutter andnoise.This improved detection performance is due to the

waveform design step in the proposed method, whichensures that the Euclidean distance between the idealechoes from different targets is maximized throughthe optimization process in (11). This methodologyis similar to the objective function in [28]. As thenumber of iterations increases, better estimates of thetarget responses are generated resulting in improveddetection of the targets.

1382 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 49, NO. 2 APRIL 2013

Fig. 6. Backscatter signal profiles at (a) iteration 1, (b) iteration10, (c) iteration 15, and (d) iteration 20.

V. CONCLUSION

We have presented a two-stage waveformoptimization algorithm for an adaptive distributedMIMO radar, which unifies the signal design andselection procedures. The proposed algorithm is basedupon constant learning of the radar environmentat the receivers and adaptation of the transmitwaveforms to suit the dynamic radar scene. Thisensures maximum information extraction fromthe target of interest. The proposed approach canprovide high performance gains in terms of targetresponse extraction, probability of target detection,delay-Doppler resolution, and radar scene generationfor real-time practical scenarios as demonstratedthrough the simulation results. Nevertheless, thereis also an increase in the computational load due tothe additional step in the waveform optimization.Future works will look into the tradeoff between theperformance enhancement provided by the proposedmethod and the computational complexity involved.

APPENDIX I. SCATTERING CHANNEL MODEL

The scattering channel model used in this workis as shown in [12]. We consider an extended target,which consists of Q scattering centers located atS,q = fxq,yqg with reflectivity q (q= 1,2, : : : ,Q). Thereflectivity of each scattering center is modeled bya zero-mean, independent and identically distributed(IID) complex random variable with varianceE[jqj2] = 1=Q. We organize the reflectivity valuesin a diagonal QQ matrix, = diag(1, : : : ,Q).Subsequently, the average RCS of the target isE[tr(H)] = 1, independent of the number ofscatterers in the model. If the RCS fluctuations arefixed during an antenna scan but vary independentlyfrom scan to scan, our target model represents a

classical Swerling case III. Now let the target beilluminated by M transmitters arbitrarily locatedat coordinates T,m = fxm,ymg, m= 1, : : : ,M . Thesignals scattered by the target are collected by Nreceivers placed at arbitrary coordinates R,n = fxn,yng,n= 1, : : : ,N. The set of transmitted waveforms in thelow-pass equivalent form is

pE=Mxm(t), m= 1, : : : ,M,

whereRTpjxmj2 = 1, E is the total transmitted energy,

and Tp is the waveform duration. Normalizationby M makes the total energy independent of thenumber of transmitters. The low-pass equivalent ofthe signal observed at the nth receive antenna due tothe probing signal sent from the mth transmit antennaand reflected from the qth scatterer (excluding noise)is given by

y(q)m,n =

rE

Mqxm(t T,m(S,q) R,n(S,q))

expfj2fc[T,m(S,q) + R,n(S,q)]g (26)where T,m(S,q) and R,n(S,q) are the propagationdelays between the mth transmit antenna and the qthscatterer, and between the qth scatterer and the nthreceive antenna, respectively. Continuing with thesignal model in (26), we can interpret the term

h(q)m,n = q expfj2fc[T,m(S,q)+ R,n(S,q)]g(27)

as the equivalent channel for the radar signal pathfrom transmitter m to receiver n via scatterer q.Summing over all the scatterers that make up thetarget, the model in (26) becomes

ym,n(t) =

rE

M

QXq=1

h(q)m,nxm(t T,m(S,q) R,n(S,q)):

(28)

We assume that the target is situated in the farfield and the radar elements at the transmitter andreceiver sites are noncollocated. The target has anRCS center of gravity at S,0 such that xm(t T,m (S,q) R,n(S,q)) xm(t T,m(S,0) R,n(S,0))for all q= 1, : : : ,Q [12]. With these conditions, (28)becomes

ym,n(t) =

rE

Mhm,nxm(t T,m(S,0) R,n(S,0))

(29)

where hm,n =PQq=1h

(q)m,n. The phase of the path gain

hm,n is assumed to be uniformly distributed between0 and 2, and the RCS = jhm,nj2 is exponentiallydistributed as described in Section II-B. In case of aradar scene with clutter sources present, the modelfor hm,n can be extended to accommodate surroundingnontarget scatterers in the form of extended cluttersources in addition to the desired target.

CORRESPONDENCE 1383

YIFAN CHENDepartment of Electronic and ComputerEngineeringSouth University of Science and Technology of China1088 Xueyuan Blvd., Nanshan District, ShenzhenGuangdong, China 518055E-mail: (chen.yf@sustc.edu.cn)

YOGESH NIJSURESchool of Electrical and Electronic EngineeringNanyang Technological University50 Nanyang AvenueSingapore 639798

CHAU YUENSingapore University of Technology and Design20 Dover DriveSingapore 138682

YONG HUAT CHEWInstitute for Infocomm Research1 Fusionopolis Way, 21-01 ConnexisSingapore 138632

ZHIGUO DINGSAID BOUSSAKTASchool of Electrical & Electronic EngineeringNewcastle UniversityNewcastle, NE1 7RUUnited Kingdom

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On the Reversibility of Randomizers andDerandomizers in Aeronautical Telemetry

Data may be recovered when randomization and

derandomization are applied in the reverse order as long as

the initial register conditions for both the randomizer and

derandomizer are identical. In the (very realistic) case where

the initial register contents for the derandomizer are not known,

simple procedures for identifying the correct initial conditions

are described. The procedures leverage the presence of frame

synchronization words in the telemetry data.

I. INTRODUCTION

Long sequences of consecutive ones or zeroscan cause problems in binary data storage andtransmission. Ultimately, these problems can be tracedto the absence of transitions in the waveforms usedto carry the bits. The approaches used to deal withthis problem may be classified in one of two broadcategories. The first category comprises solutionsbased on pulse shapes. Here, a pulse shape thatguarantees waveform transitions every bit interval,even in the presence of consecutive ones or zeros, isused. Examples include the return-to-zero (RZ) orManchester pulse shapes [1]. The popularity of thisapproach can be limited by practical considerations.In high speed applications (hundreds of Mbits/s toseveral Gbits/s over fiber), the complexity generatingthe pulse shapes can be a problem (see [2] and thereferences cited therein). In wireless applications withsevere bandwidth constraints (such as aeronauticaltelemetry), the bandwidth required by the RZ orManchester pulse shapes often renders their useimpractical.The second category is based on the application

of line coding to the bits prior to waveform mapping.Scrambling (or randomizing) is an important specialcase of line coding. The goal of scrambling is torandomize the source bits thereby breaking up longsequences of consecutive ones and zeros. At thedestination the inverse operation, called descrambling,is applied to the bits. This category comprises threebasic approaches [2].

Manuscript received April 17, 2012; revised July 13 and September18, 2012; released for publication November 14, 2012.

IEEE Log No. T-AES/49/2/944546.

Refereeing of this contribution was handled by L. Kaplan.

0018-9251/13/$26.00 c 2013 IEEE

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