210 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4

210 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 1, FEBRUARY 2010

Time Reversal in Multiple-InputMultiple-Output Radar

Yuanwei Jin, Senior Member, IEEE, José M. F. Moura, Fellow, IEEE, andNicholas O’Donoughue, Student Member, IEEE

Abstract—Time reversal explores the rich scattering in amultipath environment to achieve high target detectability. Mul-tiple-input multiple-output (MIMO) radar is an emerging activesensing technology that uses diverse waveforms transmitted fromwidely spaced antennas to achieve increased target sensitivitywhen compared to standard phased arrays. In this paper, wecombine MIMO radar with time reversal to automatically matchwaveforms to a scattering channel and further improve the per-formance of radar detection. We establish a radar target model inmultipath rich environments and develop likelihood ratio tests forthe proposed time-reversal MIMO radar (TR-MIMO). Numericalsimulations demonstrate improved target detectability comparedwith the commonly used statistical MIMO strategy.

Index Terms—Detection, multiple-input multiple-output(MIMO) radar, time reversal, waveform design.

I. INTRODUCTION

M ULTIPATH is a common phenomenon in radar, sonar,and wireless communication applications [1]–[3]. Most

radar systems are designed under line-of-sight (LOS), not inmultipath rich environments. Multipath has noticeable impacton radar performance. For example, the range of radar sensorsis limited by LOS blockage due to buildings, forests, and manyother scatterers. Hence, radar sensors usually attempt to main-tain a very high aspect angle to avoid shadows in urban environ-ments or forest areas, which severely reduces the coverage area.Multipath may also reduce the radar resolution and sensitivityto detecting targets. The multipath limitation has motivated ex-tensive research on radar algorithm and architecture design toovercome the LOS restriction. In this paper, rather than treatingmultipath as an adverse effect, we use time reversal to exploremultipath to enhance radar detection.

Manuscript received February 03, 2009; revised July 06, 2009. Current ver-sion published January 20, 2010. This work is supported in part by the Depart-ment of Energy under Award DE-NT-0004654, the National Science Founda-tion under Award CNS-093-868, and the Defence Advanced Research ProjectsAgency through the Army Research Office under Grant W911NF-04-1-0031.The work of N. O’Donoughue was supported by a National Defense Scienceand Engineering Graduate (NDSEG) Fellowship. The associate editor coordi-nating the review of this manuscript and approving it for publication was Dr.Michael Wicks.

Y. Jin is with Department of Engineering and Aviation Sciences, University ofMaryland Eastern Shore, Princess Anne, MD 21853 USA (e-mail: [email protected]).

J. M. F. Moura and N. O’Donoughue are with Department of Electrical andComputer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 USA(e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTSP.2009.2038983

In time reversal (phase conjugation in the frequency domain),a short pulse, for example, transmitted by a source through a dis-persive medium, is received by an array, then time reversed, en-ergy normalized, and retransmitted through the same medium.If the scattering channel is reciprocal and sufficiently rich, theretransmitted waveform refocuses on the original source. Bytime reversal, the transmission waveforms are tailored to thepropagation medium and the target scattering characteristics.Hence, time reversal is a radar waveform adaptive transmissionscheme. Other radar waveform design strategies are given in,e.g., [4]–[6]. Our recent work, [7]–[9] considers signal detectionusing time reversal with a single antenna pair as well as with an-tenna arrays, and demonstrates that the time reversal generalizedlikelihood ratio detector (TR-GLRT) significantly improves de-tection performance when compared with conventional detec-tors.

Recently, there has been considerable interest in a novelclass of radar systems known as “MIMO radar,” where the termmultiple-input multiple-output (MIMO) refers to the use of mul-tiple-transmit as well as multiple-receive antennas [10]–[15].Most authors assume that the key aspect of a MIMO radarsystem is the use of a set of orthogonal transmit waveforms.The radar return from a given scatterer has various degrees ofcorrelation across the array, depending on the element spacingand array configuration. In particular, the term “statisticalMIMO radar,” [10], refers to the signal model where the signalsmeasured at different antennas are uncorrelated. If the antennasare separated far enough, the target radar cross sections (RCS)for different transmitting paths become independent randomvariables. Thus, each orthogonal waveform carries independentinformation about the target; spatial diversity about the targetis thus created. Exploiting the independence between signals atthe array elements, MIMO radar achieves improved detectionperformance and increased radar sensitivity. This is in contrastwith a conventional phased array that presupposes a highcorrelation between signals either transmitted or received by anarray.

In this paper, we develop a MIMO setup to operate in a richscattering environment and to exploit the multipath propagation.There are many mechanisms that cause multipath in radar detec-tion, for example, the presence of a large number of scatterers inthe vicinity of the target of interest, or tracking and detection oflow-angle targets over a flat surface [16], [17]. Multipath affectsthe level of the energy return from the target due to coherentcombining of the return signals. As a result, we will observefades and enhancements relative to the level that is expected ina free-space environment. In general, the overall target responseis characterized by the target’s radar cross section, the multipath

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JIN et al.: TIME REVERSAL IN MIMO RADAR 211

propagation due to the surrounding scatterers, and the antenna’saspect angle. The unknown nature of the complex target reflec-tion makes the overall target response appear to be random evenfor a point target. Therefore, we adopt a statistical model forthe target in this paper. Although our MIMO model is somehowsimilar to what is used in [10], the difference is clear. In [10],the randomness of the radar target return is caused by many lookangles from extended targets; in our case, the randomness of a(point) target response is the result of multipath.

In this paper, we combine time reversal with MIMO (TR-MIMO) radar technology to improve the signal-to-noise ratio bytailoring the transmitted waveforms to the propagation mediumand the target scattering characteristics. The benefits of the pro-posed TR-MIMO detection include: 1) to exploit the spatial di-versity arising from the multipath propagation; 2) to use timereversal to adaptively adjust the radar waveforms to scatteringcharacteristics of the channel; 3) to employ simple orthogonalwideband waveforms without seeking complicated waveformcoding design methods; and 4) to incorporate the de-correlationbetween the forward channel and the backward channel whenthe reciprocity condition may not strictly hold. We develop abinary hypothesis detector for the TR-MIMO and provide ana-lytical expressions for the test statistic. In previous work [7]–[9],[18], [19], we showed that time reversal offered higher resolu-tion and improved detectability over conventional methods. Inthis paper, we demonstrate that a MIMO radar combined withtime reversal (TR-MIMO) improves target detectability whencompared with statistical MIMO (S-MIMO) [20].

The remainder of the paper is organized as follows. Section IIdiscusses the MIMO modeling in multipath. Section III dis-cusses the TR-MIMO signal modeling. Section IV derives theTR-MIMO detectors. Section V provides performance analysisfor the proposed MIMO detector. Section VI presents theTR-MIMO detection results. Finally, conclusions are drawn inSection VII.

II. MIMO MODELING IN MULTIPATH

We consider the problem of detecting a stationary or slowlymoving target immersed in a multipath rich scattering environ-ment. Such scenarios occur in many radar applications, for ex-ample, detection through tree canopy or low-angle detection andtracking. In this section, we derive the MIMO radar model.

A. Overview of Statistical MIMO Radar Modeling

In statistical MIMO radar [20], antennas at the transmitter andthe receiver of the radar are well separated such that they expe-rience an angular spread caused by variation of the radar crosssection (RCS). Statistical MIMO radar utilizes the spatial di-versity introduced by the radar target fluctuation. For extendedtargets, due to the large inter-element spacing between the an-tennas, each transmitter–receiver pair sees a different aspect ofthe target. This observation yields the following basic MIMOradar modeling for extended targets [20]:

(1)

where is an matrix, referred to as the target channelmatrix, and

(2)

The are independent complex normal random vari-ables. The vector is the collec-tion of received signals at the receive antennas. The vector

is the collection of the transmitsignals at the transmit antennas. The vector is the additivenoise. Given the statistical MIMO model (1), the optimaldetector is a non-coherent detector given by [20]

(3)

Remarks:1) The radar model (2) is justified in that the received echoes

from an extended target between each pair of transmitantenna and receive antenna become independent randomvariables when the antennas are placed sparsely. We willshow that multipath causes a similar effect even when thetarget is pointwise. The superposition of the multipathadding constructively and destructively makes the receivedradar returns appear to be random.

2) This observation can be further justified by examining theradar operating spectrum for a point target. A point targetin a LOS condition yields a flat spectrum for the returnedsignal, but results in a fluctuating spectrum in a non-LOScondition.

3) The radar model (2) and the processing (3) are designedfor narrowband systems. By exploring the orthogonalityof the waveforms, each receiver may match to a specifiedtransmit waveform [20]. To extend this design scheme to awideband system is difficult and may significantly increasethe radar system complexity. This is because of the dif-ficulty in designing orthogonal radar waveforms that canmeet various radar operational conditions (e.g., [21]). Onthe other hand, (3) indicates that the signal cross-correla-tion can not be disregarded; significant correlation reducesthe mainlobe width, which can result into higher ambiguitysidelobe levels. This paper proposes a simple quasi-orthog-onal waveform design for wideband MIMO radar using thetime reversal method.

Next, we introduce a point target MIMO model in multipathenvironments.

B. Multipath MIMO Radar Model

In the absence of multipath, the reflected radar waveformfrom a point target is an amplitude scaled and time delayedreplica of the transmitted waveform. In theory, the targetfrequency response is flat. However, multipath propagation in-duces a rapid fluctuation in the frequency response of the pointtarget. Fig. 1 illustrates a two-way radar propagation modelin multipath. For simplicity, this model considers a two-pathpropagation with only a single reflected ray emanating froma virtual target image. The two-path propagation is causedby scatterers between the receiving array B and the target.This model can be extended to the more general scenario with



Fig. 1. Multipath propagation model. Only the forward propagation (fromarray A to array B) is illustrated.

multiple path propagation due to scatterers in the fields of viewof both the transmitting array A and receiving array B.

Suppose we have a transmitting signal

(4)

where is the baseband radar pulse at frequency fromthe th transmit antenna, is the carrier frequency. We assumethat the signal propagates in the multipath medium and reflectsfrom a target with a possible phase change . The noise-freereceived signal at the th element of array B due to the thtransmit antenna at array A is given by

(5)

where is the phase change due to the multipath; the directpath delay and the multipath delay are given as fol-lows:

(6)

(7)

The symbols , , and are the azimuth coordinates of theth antenna at A, the th antenna at B, and the target, respec-

tively; is the target range; and is the complex amplitudedue to the target characteristics. The complex amplitudes of thedirect and reflected rays are simply related by a complex multi-path reflection coefficient .

By mixing the received signal with the transmitted signal, weobtain the baseband signal as follows:

(8)

With a large number of scatterers and ,, multipath reflection coefficients, we obtain the overall target

reflectivity

(9)

where the target channel response between the th antenna inarray and the th antenna in array is

(10)

We hence assume that, with a large number of scatterers andmultipath reflection coefficients , the overall target reflec-tivity (10) is a random variable. This analysis implies that, evenfor a point target, the multipath effect induces fades and en-hancements in the returned signals relative to the free space re-turned signals. The transmit and receive antenna pairs provideindependent information about the target due to their differentviewing angles.

Hence, from (10), we introduce the forward propagationchannel. We let denote the forward channel response ma-trix between the transmit array A and the receive array B at fre-quency . The th entry of , i.e., the forward channelresponse from antenna , , to antenna ,

, is

(11)

where the symbol stands for “distributed as.” The targetchannel model (11) implies that, at a fixed frequency and foreach transmitter antenna and receiver antenna pair, the targetchannel response is a complex Gaussian random variable withzero mean and variance . This variance (i.e., the powerspectrum density) is frequency dependent, which is caused bythe multipath scattering. Further, we assume that the channel re-sponse from different transmit and receive pairs are statisticallyindependent, identically distributed (i.i.d.) random variables.Note that the frequency-domain representation isrelated to the time-domain channel impulse response ,

(12)

where the RCS of the target is modeled as a zero mean,finite covariance, wide sense stationary uncorrelated scattering(WSSUS) Gaussian process. This assumption is valid for manypractical situations in wave propagation in the radar literature[22], [23]. The variance of is frequency depen-dent. By the WSSUS assumption and the Wiener–Khintchinetheorem [24]

(13)



where is the power spectral density (PSD) of , i.e.,the Fourier transform of the covariance function

(14)

(15)

Hence, by (13)

(16)

In general, pulse wave propagation and scattering in a randommedium can be characterized by the correlation function (seechapter §5 in [22])

(17)

The function is the correlation function between the outputfields due to the time-harmonic inputs at two different frequen-cies and . We should note that the time-varying fre-quency response of the random channel in (17) showsexplicitly the time index that is omitted in (12). If we let

and send two waves at different frequencies and, and observe the fluctuation fields at the same time, as

we separate the frequencies, the correlation of these two fluctu-ation fields decreases. For uncorrelated scattering channel, thefunction is a function of the frequency sep-aration .

In a multipath channel, it is often convenient to consider thecoherence bandwidth measured by the reciprocal of the mul-tipath spread. Two sinusoids with frequency separation greaterthan are affected quite differently by the channel. Hence,the frequency samples taken apart are consideredto be, approximately, independent. Therefore, we use discretefrequency samples , , in developing theTR-MIMO detector. The number is chosen by

(18)

where is the coherence bandwidth of the multipath channel,and is the system bandwidth. The richer the multipath scat-tering, the smaller the coherence bandwidth, and so the largerthe quantity .

C. Wideband Orthogonal Waveform Signaling

MIMO radar typically transmits a set of orthogonal wave-forms from different antennas. In our problem, the simultane-ously transmitted waveforms occupy the same frequency range.We let the transmitting signal from the th antenna be

(19)

To achieve the orthogonality among the transmitted waveforms,we assume that

(20)This assumption is accurate for linear frequency modulated(LFM) signals and provides a good approximation for otherwideband signals such hyperbolic frequency modulation(HFM) signals, as well as signals generated from pseudorandom sequences [23]. Using the orthogonality condition in(20), and converting it into the frequency domain by the Fouriertransform, we get

(21)

(22)

(23)

where , and we assume that .Using discrete frequency samples, for , ,and , we obtain a phase coding scheme, [25],by setting

(24)

for . For transmit antennas and , it isstraightforward to show that

(25)

III. TIME REVERSAL MIMO SIGNAL MODEL

In this section, we describe the time reversal MIMO radarsignal model. In time reversal, the received signal is phaseconjugated, energy normalized, and retransmitted to the samemedium. If the medium is reciprocal, the forward propagationchannel is the same as the backward propagation channel.However, in many radar applications, the reciprocity conditionmay not hold, for example, due to small random perturbationsbetween the forward and the backward channel realizations,or due to slow target motion. Hence, we model the backwardpropagation channel as follows with respect to the forward



channel given in (11): let denote the backwardchannel frequency response matrix between the array and

. The symbol denotes the transpose. The propagationchannel from antenna , , to antenna ,

, is given by

(26)

where (26) models the backward channel as a noisy version ofthe forward channel: the symbol is the correlation coefficientbetween the forward channel and the backward channel

. The term describes the channel disturbancethat occurs between the forward and backward channel for eachpair of transmit and receive antennas in and . We assumethat the disturbance is independent of ; it is distributedas a complex Gaussian with zero mean and variance , i.e.,

. We further impose the constraintthat has the same variance as , i.e.,

(27)

which implies that

(28)

When , the quantity captures the de-correlation betweenthe forward channel and the backward channel. As discussedabove, the de-correlation can happen due to slow changes onthe media [26], [27]; its net effect is to degrade the reciprocitycondition. The characterization of the degradation is importantto analyze the performance of time reversal MIMO radar. Let

(29)

Equation (26) can then be rewritten in matrix form as

(30)

A. TR-MIMO Data Collection and Processing

The time reversal radar data collection and processing aredescribed in three steps as follows.

Step 1 Target Probing: The signal vector received at array Bfor the th data snapshot is

(31)

where

(32)

is a vector. The signal vector transmittedfrom array is

(33)

The transmitted signal from the th antenna is a widebandsignal with Fourier representation at frequency . Weassume that the average transmission power at each antenna isthe same

(34)

In this step, the total transmission energy is . The noisevector is characterized statistically as

(35)

We assume here that the clutter is homogeneous; the noisevector (35) describes both the homogeneous clutter and theadditive noise in the received radar returns. We should note thatthis is a very simplified assumption on the clutter. Statisticalmodels for MIMO radar clutter should incorporate a numberof effects including geometry, coherence, transmit waveform,multipath scattering, etc. Development of statistical modelsfor MIMO radar clutter is beyond the scope of this paper.In general, inhomogeneous clutter can be suppressed usingwhitening filters in the spectral and spatial subspaces (see, e.g.,[28] and [29]). Next, suppose we can collectsnapshots, for a slow varying target channel. The minimummean squared estimate of the returned target signal is

(36)

where . We assume that we canobtain a reasonably accurate estimate of the target channel re-sponse for sufficiently large . In the subsequent derivation, weassume that we know precisely. This assumption yieldsthe ideal scenario for the TR-MIMO detector. In reality, wewould use as the signal to be retransmitted. The problemof obtaining a sufficient number of snapshots for this pur-pose is governed by two important factors: 1) the scale overwhich the response changes with respect to space and time; and2) systems considerations such as the bandwidth that limit thesampling rate. Thus, in a radar setting it is fairly common tohave a snapshot starved scenario. In this case, the noise vari-ance will increase. Our early attempts to analyzing the effect ofnoise variance increase have been reported in [8]. The detectionperformance under this scenario will be studied below by sim-ulations. We note that a larger means longer estimation timethat yields target channel partial de-correlation in time reversal.

Step 2: Time Reversal Probing: Conventional detection pro-cesses the data received at array B. With time reversal, the re-ceived radar return at array B is transmitted back to array A.Prior to retransmission, the data vector is time reversedand energy normalized. The received signal vector atarray A is

(37)

(38)

(39)

(40)

where the scalar is the energy normalization factor

(41)

This factor normalizes the energy of the time reversed retrans-mitted signal to equal the energy of the original transmittedsignal . The target return is defined as

(42)



The noise vector, defined by

(43)

is distributed as . We use in (26)rather than to account for the backward channel de-cor-relation due to the changes on the propagation media.

Step 3: Signal Matched Filtering: The received signal ,in (38) is a vector. The th entry

of , defined in (37), is the received radar return atthe th antenna of array . This radar signal will be matchedwith the originally transmitted signal at the th antenna, i.e.,

defined in (33).1 We repeat this process forantennas at the array , which yields the following datavector

(44)

(45)

...(46)

...

(47)

where in (47) and , , are defined in (42) and(43), respectively. Hence, we can rewrite (45) in vector form as

(48)

where

(49)

(50)

IV. MIMO DETECTORS

In this section, we formulate the MIMO radar detectionproblem. The binary hypothesis test for TR-MIMO is

(51)

1One may argue that the received radar signal at the �th antenna, � , canbe matched with the individual waveforms � �� . We show inSection IV that the output of the matched filter of � with � �� , � �� isrelatively small and then can be ignored.

The optimal detector, in the Neyman–Pearson sense, is the like-lihood ratio test (LRT), i.e.,

(52)

where is the decision threshold. The function ,are the probability density functions of the received signal

under and , respectively. The detection problem (51) is acommon problem: detecting a nonwhite Gaussian process im-mersed in additive nonwhite (or white) Gaussian noise in sonaror radar (see, e.g., [28]–[31]). A closed form for the probabilitydensity function for the binary hypothesis (51) is often difficultto obtain. Thus, we will rely on approximations and on the cen-tral limit theorem to study the data statistics and to derive thetest statistics.

A. Data Statistics

By (49), the th entry of is

(53)

Note that, the sum of the terms in (53) can be ap-proximated by

(54)

where and are independent Gaussian randomvariables for . Equation (54) greatly simplifies (53), whichyields

(55)

(56)

where by (24). However, by eliminating the un-matched phase terms in the analysis, (54) may introduce a small



prediction error compared with the results from numerical sim-ulation. We will show by Monte Carol simulation in Section VIthat this prediction error is within a fraction of a dB.

Next, we use to represent the central Chi-squared distri-bution with degrees of freedom. Therefore, [32]

(57)

which leads to , a frequency-depen-dent real constant. Hence, we can further approximate (56) as

(58)

Equations (56) and (58) show that the diagonal componentsare the focused target response; in a statistical sense, by av-eraging out the perturbation term in (56), they are zero-phasequantities. This observation leads to the following approxima-tion:

(59)

By ignoring the perturbation term, we should note that, whenthe forward channel and backward channel are fully correlated,i.e., , the above approximation is exact. We will developthe TR-MIMO detector under this assumption. We will test forcases when using numerical simulations. Let’s furtherdefine

(60)

where, the symbol stands for “distributed as” andstands for “distributed approximately as.” The symbol

. For the sake of simplicity, herewe treat and as two independent randomvariables. We further approximate by the constant .We show in (129) and (133) in Appendix I that this is a validapproximation: the random variable is a low variance distri-bution with a constant mean. Next, we rewrite (59) as

(61)

The sequence are independent random variableswith finite variance. By the Lindeberg–Lyapunov central limittheorem on sums of independent random variables [32],2 thesequence is asymptotically Gaussian for large . Hence,(61) yields

(62)

2The Lindeberg–Lyapunov central limit theorem [33] generalizes the clas-sical central limit theorem [34] by removing the identically distributed condi-tion.

where

(63)

(64)

B. TR-MIMO Detector

Hence, under in the binary hypothesis (51), the th entryof signal becomes

(65)

(66)

The symbols and denote the real and imaginary partof the complex number . Both quantities andare real numbers. Further notice that

(67)

We obtain

(68)

(69)

We stack the real and the imaginary parts of the in (66) for, to create the vector

(70)

From the results in Appendix II, the test statistic for theTR-MIMO detector is

(71)The detector uses (71) to calculate the decision threshold underthe null hypothesis .

C. About the Output of Matched Filters

In Step 3 of Section III-A, the th entry of definedin (37), is matched with the originally transmitted signal at theth antenna, i.e., defined in (33). One can certainly match

with other signals , to generate a total ofoutputs. Excluding the additive noise terms, these

outputs are

(72)



Since and , are random variables, we takethe expectation of

(73)

(74)

(75)

(76)

Hence

(77)We argue now that (77) is a complex number with small magni-tude compared with (53). This is the result of the orthogonalitybetween and , , chosen in Section II-C. Inparticular, if the term , a constant valueindependent of frequency, (77) becomes zero. Therefore, we donot consider these outputs from the matched filters.

D. Statistical MIMO Detector

The statistical MIMO (S-MIMO) radar discussed in the lit-erature is designed for narrowband radar [10], [20]. We extendthis narrowband signal model, (1) and (2), to wideband. Oneof the key processing steps in S-MIMO is to extract the com-plex gains of a total of channels. When using wide-band orthogonal waveform signaling, we implement the orthog-onal phase coding given in (24), which corresponds to the samewaveforms that we used for TR-MIMO. Similar to the devel-opment of the TR-MIMO detector, we assume a homogeneous

clutter model. In this case, compared with (38), the receivedsignal for S-MIMO is

(78)

where

(79)

(80)

(81)

It is straightforward to derive that

(82)

By the statistical MIMO processing scheme in [20], matched-filtering the received signals with the orthogonal waveforms

given in (24) yields

(83)

where

(84)

(85)

We group the quantities , , and intovectors

(86)

(87)

(88)

Thus, the binary hypothesis test for S-MIMO is given by

(89)

We use the following test statistic for MIMO radar [20]:

(90)

This detector computes the threshold under the null hypothesis.



V. PERFORMANCE ANALYSIS

In this section, we derive the test statistic for time reversalMIMO (TR-MIMO) in (71) and statistical MIMO (S-MIMO)in (90), respectively. To derive the detectors and to analyzetheir performance, we derive the probability density functionsof the test statistics under and , respectively. The per-formance analysis under the homogeneous clutter assumptionprovides insight on the tradeoffs between the number of an-tennas, the number of frequency samples of the proposed wide-band MIMO radar detectors. We rely on numerical simulationsto study channel decorrelation.

A. TR-MIMO

Under and based on (68), we know that are in-dependent and identically distributed normal random variablesfor . Therefore, from (71), we obtain

(91)

(92)

where denotes the non-central Chi-squared distributedrandom variable with degrees of freedom and the non-cen-trality parameter

(93)

Under

(94)

(95)

where is a central Chi-squared random variable withdegrees of freedom. Both (92) and (95) provide a simple

description of the test statistic for the TR-MIMO detectorunder each hypothesis. To obtain the detection probabilityand the decision threshold from (92) and (95), we derive inAppendix III the probability density functions under

and under , respectively. Given the densityfunction and a chosen false alarm rate , we cannumerically calculate the threshold by solving

(96)

Readers can refer to Appendix III for a detailed discussion onhow the threshold is computed. Next, we can compute thedetection probability

(97)

Under simplifying conditions, for example, the radar signalmodel given in [35] and [36], one can develop closed formexpressions for and . In our case, the time reversalradar signal model in (66) becomes, approximately, a nonzeromean, real Gaussian signal immersed in zero-mean complexGaussian noise. The mathematical tractability of closed formexpressions for and becomes difficult. One can resortto importance sampling to efficiently calculate these quantities(see, e.g., [37], [38]).

B. Statistical MIMO Detector

From (90), under the null hypothesis ,. For a fixed

(98)

(99)

where

(100)

(101)

Furthermore, we define

(102)

where the th entry of is given by

(103)



(104)

(105)

We further define the following two matrices and

(106)

(107)

We immediately recognize that

(108)

Hence, (99) can be rewritten as

(109)Note that the matrix has identical singular values

(110)

Each entry of the vector is a complex Gaussian randomvariable with zero mean and variance . Hence, (109) can bewritten as a quadratic form

(111)

where . Since areindependent, complex random variables, we have

(112)Under the alternative hypothesis , for a fixed

(113)Hence,

(114)

The binary hypothesis test (89) for S-MIMO is given by

(115)The detection probability is given by

(116)

C. Discussion—Nominal Performance

Although the probability of detection is the most usefulmetric for comparing performance between different test statis-tics, other performance measures can also provide insight andmathematically tractable approaches; for example, the nominalperformance given in [39, (p. 329)] . Here, we use the followingmetric [20]

(117)

where denotes a test statistic, and represents the normalized-divergence between and hypothesis. Equation (117) is

a simple measure to illustrate the performance of the detector.To simplify the calculation, we assume that

(118)

is frequency independent. We further define .Thus, from (162) in Appendix IV, we obtain (119) shown atthe bottom of the page. Similarly, we obtain the value for theS-MIMO

(120)

To compare (119) and (120), we assume that , which is inhigh signal-to-noise ratio (SNR). In this case, ,while . This implies that, with a few sparsely placedantennas, and can be small. If the channel multipath scat-tering is rich, i.e., , the TR-MIMO shows a higher per-formance than the S-MIMO. Fig. 2 depicts the nominal perfor-mance versus SNR for TR-MIMO and S-MIMO using .

(119)



Fig. 2. Nominal performance versus SNR �� for TR-MIMO and S-MIMOusing � � ��.� � � transmit antennas and � � � receive antennas.

VI. NUMERICAL SIMULATIONS

In this section, we carry out numerical simulations to eval-uate the performance of the proposed detectors. The simulationis carried out as follows: 1) generate random realizations ofindependent frequency samples for the forward channelmatrix and the backward channel matrix ;2) generate orthogonal waveforms based on (24); 3) generatetarget signal independent additive noise and add the noise to thereceived radar returns; 4) calculate the test statistics and deter-mine the decision threshold given the false alarm rate ; and5) calculate the detection probability. The signal-to-noise ratiois defined as

SNR (121)

We choose frequencies, transmit antennas,and receive antennas for simulation purposes. Weshow in Fig. 3 the detection probability versus SNR forTR-MIMO versus S-MIMO under the false alarm rates of

, respectively. The correlationfactor represents the ideal scenario. The analytical resultsare plotted using (97) for the TR-MIMO detector and (116)for the S-MIMO detector, respectively. The lines representthe analytical plots while the markers denote the Monte Carlosimulation results. Fig. 3(a)–(d) shows that the analytical re-sults match the Monte Carlo simulation results quite well. Theproposed TR-MIMO has about 14-dB gain over S-MIMO forthe simulation setup . There appears a small predictionbias (within a fraction of a dB) for the analytical ROC resultcompared with the Monte Carol simulation. This small predic-tion bias can be explained by the approximation we made in ouranalysis, for example, see (54). This equation argues that theunmatched phase terms are zero, which of course is not true inpractice. This approximation yields slightly better performanceand ROC curve than the Monte Carlo simulation shows, seeFig. 3. This prediction bias can be reduced by increasing the

Fig. 3. Detection probability versus SNR for TR-MIMO and S-MIMO using� � ��. The analytical results in the legend are marked with subscripts “ana”for the TR-MIMO detector and the S-MIMO detector. The simulation uses� �� transmit antennas and � � � receive antennas. The correlation between theforward channel and the backward channel is � � �. The false alarm rates are:(a) � � �� , (b) � � �� , (c) � � �� , and (d) � � �� .



Fig. 4. (a) Average coherence bandwidth of scattering channels estimated fromexperimental electromagnetic data collected in lab, [9]. The average coherencebandwidth decreases from 912 to 120 MHz as the propagation medium be-comes increasingly multipath rich. (b) Detection probability of TR-MIMO andS-MIMO by simulations assuming� � �� , respectively. � �

�� , � � �, � � �.

size of the receiving antenna or the number of independentfrequency bands .

The number of independent frequencies depends on thescattering properties of the channel and can be determinedby experiments. In our case, the choice of is based on ourexperimental electromagnetic data collected in an increasinglyrich scattering lab environment. The scattering environmentis created using dielectric solid rods of 3.2-cm diameter, [9].The details of the experiments are reported in [9]. The ex-periment synthesizes a wideband signal of 2 GHz (4–6 GHz)using stepped frequency signals generated by a vector networkanalyzer (VNA). The scattering environment is created bygradually increasing the number of dielectric rods in a 4 4square feet wood platform. The wood platform has a total of46 holes that can hold the dielectric rods vertically. First, thetarget response is measured for a copper target; then, we addone dielectric rod into the scene to create multipath. We keepadding dielectric rods until all the rods are filled in the holes.We then measure the coherence bandwidth of the channelversus the number of dielectric rods. Fig. 4(a) shows thatthe coherence bandwidth decreases from MHz to

MHz. To further enhance the channel scattering, weplace a metal shield behind these 46 rods, which brings down

Fig. 5. Detection probability for TR-MIMO versus decorrelation factor �. Thenumber of frequencies � � ��.� � �, � � �. � � �� .

the coherence bandwidth to 120 MHz. The- GHz band-width corresponds to , respectively.Using this sequence of values, we generate, by Monte Carlosimulations, the detection of probability versus in Fig. 4(b)for TR-MIMO and S-MIMO. We fix the average channel gainto noise ratio at 2 dB and change the value.Fig. 4(b) shows that the performance gain of TR-MIMO versusS-MIMO increases as the scattering becomes dense. This isbecause the TR-MIMO detector coherently process signals byadaptive time reversal transmission while the S-MIMO detectornon-coherently add up signals from different frequencies.

To study the de-correlation between the forward channel andthe backward channel in time reversal, we vary the correlationfactor in the range of 0 and 1. Fig. 5 depicts the detectionprobability of TR-MIMO and S-MIMO when the decorrelationfactor decreases from 1 to 0. The performance of TR-MIMOis robust against channel decorrelation.

VII. CONCLUSION

This paper develops the time reversal MIMO radar detectorand provides an analytical expression for the probability distri-bution of TR-MIMO. The TR-MIMO exploits the spatial diver-sity arising from the multipath and adjusts the waveforms to thescattering properties of the medium by using time reversal. Thepaper presents a model for TR-MIMO that accounts for possibledecorrelation between the forward channel and the backwardchannel and derives the test statistic, threshold, and probabilityof detection for the TR-MIMO Neyman–Pearson detector. Thealgorithm we develop is robust in rich multipath environmentsand shows a significant gain over the statistical MIMO detector.

APPENDIX IENERGY NORMALIZATION SCALAR

The scalar in (41) is a random variable because it dependson the channel matrix whose elements are random. In thedevelopment of the statistics for TR-MIMO, we wish to charac-terize the mean of this random variable as , where

(122)



where . Let the vector

, where

(123)

where , ,and we assume for simplicity . It is straightforwardto see that

(124)

Hence, using[40, Theorem 3.1], we obtain that the followingweighted sum of independent Chi-squared random variables is

(125)

where

(126)

(127)

Hence, we obtain

(128)

Hence, is distributed as a scaled inverse-chi-squared randomvariable. It is well known that, for ,has the mean , , and

, . Hence, we obtain the mean ofas follows:

(129)

In probability theory and statistics, the coefficient of variation(CV) is a normalized measure of the dispersion of a probabilitydistribution. It is defined as the ratio of the standard deviation tothe mean as

(130)

(131)

(132)

By the Cauchy–Schwartz inequality

Hence, for , we obtain

(133)

which implies that has a low variance. This result impliesthat the variance of will be a very small number when thenumber of frequencies and the number of antennas are large.Hence, it suffices to say that is approximately a constant.

APPENDIX IIDERIVATION OF THE TR-MIMO DETECTOR

By (70), the probability density function of under is

(134)where

(135)

(136)

The symbols , , stand for an vector that con-tains all 1s, an vector that contains all 0s, and anidentity matrix, respectively. Similarly, the probability densityfunction of under is

(137)

where

(138)

The likelihood ratio test becomes

(139)

Using (70), and discarding the constant terms 1/2 in (139), werewrite (139) as

(140)



(141)

(142)

Discarding the common denominator in (142), we obtain (71).APPENDIX III

DERIVATION OF DETECTION PROBABILITY AND THRESHOLD

FOR TR-MIMO DETECTOR

We first rewrite (92) and (95) in the form of

(143)

where we have the equation shown at the bottom of the page.Note that for , where and are independent

random variables with probability density function and, respectively, the probability density function of is the

convolution of and , [34], i.e.,. For , where is a constant, the probability

density function of is given by .Hence, under

(145)where the random variable , and .Note that the probability density function of a non-central Chi-squared random variable is

(146)

where is a modified Bessel function of the first kind

(147)

Further, notice that the Gaussian random variable has theprobability density function

(148)

which yields

(149)

Similarly, under , the probability density function

(150)where the random variable , and .Note that the probability density function of a centralChi-squared random variable is

(151)

The Gaussian random variable has the probability densityfunction

(152)

which yields

(153)

(144)



Both (149) and (153) can be evaluated numerically.To calculate the decision threshold and the detection

probability in (97), we start from the probability den-sity function (pdf) expressions in (145) and (150). We take thecumulative density function (cdf) as the integral along from

to ,

(154)Shifting the order of integration allows us the expression

(155)We note that the integral produces the cdf of the Gaussianrandom variable . Thus,

(156)

This was solved with numerical integration. The inverse wasaccomplished via a simple search algorithm. The cdf is a non-decreasing function; therefore, comparison of the received cdffor some test point against the desired cdf determinesin which direction to increment the search term. For these tests,the stopping criteria was

(157)

We use this inverse function to compute the threshold underthe null hypothesis by calling

(158)

where computes the numerical inverse describedabove. is then used to compute the detection probability

(159)

APPENDIX IVCALCULATION OF THE NOMINAL PERFORMANCE FOR

TR-MIMO AND S-MIMO

Using the statistical properties of the Chi-squared distributionand normal distribution, from (143) and (144), we obtain

(160)

(161)

A straightforward algebraic calculation yields the following:

(162)

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Yuanwei Jin (S’99–M’04–SM’08) received the B.S.and M.S. degrees from East China Normal Univer-sity, Shanghai, China, in 1993 and 1996, respectively,and the Ph.D. degree in electrical and computer en-gineering from the University of California at Davisin 2003.

From 2003 to 2004, he was a Visiting Re-searcher with the University of California at SantaCruz. From 2004 to 2008, he was a Post-DoctoralResearch Fellow, then Project Scientist, with theDepartment of Electrical and Computer Engineering,

Carnegie Mellon University, Pittsburgh, PA. Since August 2008, he has beenan Assistant Professor with Department of Engineering and Aviation Sciencesat the University of Maryland Eastern Shore, Princess Anne. His researchinterests are in the general area of statistical signal and image processing, withapplications in radar/sonar, biomedical imaging, structure health monitoring,and wireless communications. He has published over 40 technical Journal andConference papers.

Dr. Jin was a recipient of an Earle C. Anthony Fellowship from the Univer-sity of California at Davis. He is affiliated with several IEEE societies, SigmaXi, he American Society for Engineering Education (ASEE), and the AmericanAssociation for Cancer Research (AACR).

José M. F. Moura (S’71–M’75–SM’90–F’94)received the engenheiro electrotécnico degree fromthe Instituto Superior Técnico (IST), Lisbon, Por-tugal, and the M.Sc., E.E., and D.Sc. degrees inelectrical engineering and computer science fromthe Massachusetts Institute of Technology (MIT),Cambridge.

He is a Professor of Electrical and ComputerEngineering and, by courtesy, of BioMedical En-gineering, at Carnegie Mellon University (CMU),Pittsburgh, PA. He was on the faculty at IST, has held

visiting faculty appointments at MIT, and was a Visiting Research Scholar atthe University of Southern California, Los Angeles. He is a founding codirectorof the Center for Sensed Critical Infrastructures Research (CenSCIR) andmanages a large education and research program between Carnegie Mellonand Portugal (www.icti.cmu.edu). His research interests include statisticaland algebraic signal processing, image, bioimaging, and video processing,and digital communications. He has published over 380 technical journal andconference papers, is the coeditor of two books, holds six patents, and hasgiven numerous invited seminars at international conferences, U.S., European,and Japanese Universities, and industrial and government Laboratories.

Dr. Moura was the President (2008–09) of the IEEE Signal ProcessingSociety (SPS) and vice-chair of the IEEE Publications Services and ProductsBoard (2008). He was Editor in Chief for the IEEE TRANSACTIONS ON

SIGNAL PROCESSING, interim Editor in Chief for the IEEE SIGNAL PROCESSING

LETTERS, and was on the Editorial Board of several journals, including the IEEEPROCEEDINGS, the IEEE Signal Processing Magazine, and the ACM Trans-actions on Sensor Networks. He chaired the IEEE Transactions Committeethat joins the Editors in Chief of the over 80 IEEE transactions and journals.He was on the steering and technical committees of numerous conferences.He is a Fellow of the American Association for the Advancement of Science(AAAS), and a corresponding member of the Academy of Sciences of Portugal(Section of Sciences). He was awarded the 2003 IEEE Signal ProcessingSociety Meritorious Service Award and in 2000 the IEEE Millennium Medal.In 2007 he received the CMU’s College of Engineering Outstanding ResearchAward and in 2008 the Philip L. Dowd Fellowship Award for contributions toengineering education. He is affiliated with several IEEE societies, Sigma Xi,AMS, IMS, and SIAM.

Nicholas O’Donoughue (S’03) received the B.S.degree in computer engineering from VillanovaUniversity, Villanova, PA, in 2006, and the M.S.degree in electrical and computer engineering fromCarnegie Mellon University, Pittsburgh, PA, in2008. He is currently pursuing the Ph.D. degreein electrical and computer engineering at CarnegieMellon University.

His general research interests lie in the area of sta-tistical signal processing, with applications in radar/sonar and communications, with a special interest in

the technique of Time Reversal signal processing.Mr. O’Donoughue is a recipient of the 2006 National Defense Science and

Engineering Graduate (NDSEG) Fellowship, the 2006 Dean Robert D. LynchAward from the Villanova University Engineering Alumni Society, and the 2006Computer Engineering Outstanding Student Medallion from Villanova Univer-sity. He has published more than a dozen technical journal and conference pa-pers, including two that were chosen as Best Student Paper. He is a member ofseveral IEEE societies, the Acoustical Society of America, Tau Beta Pi, and EtaKappa Nu.


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