IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 4, APRIL 2013 327

An Information Theoretic Take on TimeReversal for Nonstationary Channels

Pawan Setlur and Natasha Devroye

Abstract—It has been shown that time-reversal (TR) techniquesfocus energy back to the dominant scatters, lead to super-reso-lution focusing, and gains in detection. Time reversal has so farmainly been studied when the channel remains invariant betweenthe initial and time-reversed signal transmission times. In thisletter, we relax this assumption and study the benefits of TR overtime-varying channels. To do so, we compare a time-reversed anda non time-reversed system by comparing the mutual informationbetween the channel impulse response and channel outputs giventhe transmitted signals. We present analytical results for a simplescalar problem which illustrates the impact of nonstationary chan-nels on TR, and for general channels, numerically evaluate thedifference in mutual informations, which demonstrate that, if thechannels are nonstationary yet correlated, TR may still providemutual information gains over non time-reversed systems.

Index Terms—Mutual information, nonstationary channel,radar, stochastic time-varying channel, time reversal.

I. INTRODUCTION

I N this letter, we analyze radar-based time reversal (TR) overtime-varying channels usingmutual information as compar-

ison metric. In TR, a signal is initially radiated; the backscat-tered signal is then recorded, time-reversed, energy scaled andre-transmitted. TRmay lead to super-resolution spatio-temporalfocusing using multiple antennas, and detection gains for singleand multiple antennas [1]–[6]. Most of these important contri-butions in TRwere derived assuming the channel to be invariantfrom the initial signal transmission to the time-reversed re-trans-mission [1]–[6]. The question of whether time-reversal is bene-ficial in realistic time-varying channels remains. We make ana-lytical progress by introducing and analyzing the mutual infor-mation in TR systems as compared to conventional systems fortime-varying channels.Our TR model is well suited to monostatic radar, where the

receiver and transmitter are co-located but could be extendedto other applications such as communications. In our model,a single antenna transceiver first probes the channel. It sub-sequently transmits the time-reversed signal it received from

Manuscript received November 29, 2012; revised January 16, 2013; acceptedJanuary 24, 2013. Date of publication February 06, 2013; date of current versionFebruary 21, 2013. This work was supported in part by the Air Force Office ofScientific Research under Award FA9550-10-1-0239. The contents of this articleare solely the responsibility of the authors and do not necessarily represent theofficial views of the AFOSR. The associate editor coordinating the review ofthis manuscript and approving it for publication was Prof. Joseph Tabrikian.The authors are with the Department of Electrical and Computer Engineering,

University of Illinois at Chicago, Chicago, IL 60607 USA (e-mail: setlurp@uic.edu; devroye@uic.edu).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/LSP.2013.2245646

the initial probe. We will compare performance with a “con-ventional” model where the channel is probed twice with thesame signal (as opposed to recent waveform scheduling models[7]–[9]). The channels are assumed to be linear, stochastic, sub-ject to additive Gaussian noise, and time-varying.Related Work: For invariant channels time-reversal in op-

tics, ultrasound and acoustics, radar and communications, maybe seen in for example [1]–[6] and references therein. Work onTR in time-varying channels is much more limited. It was ac-knowledged in [4, pg. 36–37] via experimental insights that TRfocusing degrades in nonstationary time-varying environmentssuch as the time-varying ocean surface and its volume. In com-munication applications (rather than our radar-focused applica-tion), it was experimentally shown that time varying channelsaffect the TR performance in [10], [11], and interestingly theconclusions drawn are similar to those drawn here—that TRmay still be beneficial in channels which are correlated but notnecessarily identical over time.Contributions and Organization: We first introduce the TR

and conventional channel models in Section II. We then intro-duce the relevant mutual information quantity for these modelsin Section III, before analytically comparing the differencein mutual information for TR and conventional channels inSection IV, where we analytically work out an intuitive specialscalar case which illustrates the effect of channel correlationbetween the two stages on the mutual information. Finally,in Section V we provide numerical evaluations. Our centralcontributions are 1) analyzing TR using information theoreticmetrics for the first time, and 2) using this framework to quan-titatively analyze the impact of time-varying channels on TRas compared to conventional systems.

II. CHANNEL MODEL

We now outline the two transmission stages of the TRchannel and the conventional channel models. Other “time-re-versal” protocols consisting of more than two stages may bedevised, but for simplicity, and to understand existing TRchannel models, we limit ourselves to two stages.During the first stage, let the baseband transmitted signal

during the first scheduling instant or stage be given by thebaseband samples , and letthe matrix be the Toeplitz convolution matrixcomprised of the samples along with zeros padded appro-priately—i.e. the -th column consists of zeros, thesamples of , and zeros again. Note that

. The received signal at is then given bythe complex samples

(1)

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328 IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 4, APRIL 2013

where , is the additivenoise and is the sampledchannel impulse response.During the second stage, the TR system transmitter then

transmits the scaled, by , time-reversed channeloutput . The con-volution matrix of the time-reversed output is denoted by

. In the meantime the channel has changed and thenew impulse response is given by and the new noiseby , which need not necessarily follow the same statisticsas in the first stage. The “conventional” channel model, whichwe will use as comparison point in evaluating the utility of TR,transmits the same waveform during the second stage. Thus,the second stage outputs of the TR and conventional system,are given, respectively, by

(2)

(3)

For analytical tractability, we assume that isjointly Gaussian with mean and covariance, respectively,

(4)

In (4), , , 2 and ,where is the covariance (matrix) between vectorsand . The concatenated noise vector, is inde-pendent of and is assumed to be zero-mean, jointly Gaussianwith corresponding covariance matrix, given by,

(5)

where , , 2 and .

III. MUTUAL INFORMATION AS COMPARISON METRIC

In this work, we quantify the utility of TR channels by usingthe mutual information between the appropriate channel input/output quantities as our comparison metric. In radar channels,information theoretic metrics such as mutual information andconditional entropy have been used for a variety of purposes in-cluding waveform design [12], [13], waveform scheduling [8],[9], and sensor management [14], [15]. It is often motivated as a“surrogate” metric [7] because of its generic ability to quantifythe information gained by certain measurements, which is notlinked to a specific task such as detection or estimation (thoughlinks between mutual information and SNR or Fisher informa-tion may be made).Over the two stages of transmission, the radar system

wishes to learn about the channel from the received sig-

nals (TR) or (con-ventional), given knowledge of the transmitted waveforms. The amount of information we can gain about from

given knowledge of the transmitted waveformis given by the mutual information (MI) between and, where we note that is a known parameter, denoted by

, or equivalently

where is defined as the conditional differential en-tropy, and is the conditional pdf, and denotesthe expectation operator. We recall the definitions of dif-ferential entropy [16]: and

. For our TR channel underGaussian assumptions, we will be evaluating mutual informa-tion terms for which the following is useful [16]:Fact 1: If is a multivariate Gaussian random vector with

an arbitrary mean vector and a covariance matrix, , then thedifferential entropy (nats), , where isa constant and is related to the dimensions of .Remark 1: Time-Reversal: A channel with feedback or not?

In a mono-static radar system which employs TR, the nextwaveform transmitted depends on the previously receiveddata. Intuitively at least, TR appears analogous to an informa-tion theoretic channel where the encoder employs feedback(i.e. encoders at time have access to previous channel out-puts and may let their subsequent channel inputsbe functions of these outputs, i.e.where is the message). In our scenario, the time-reversedtransmitted waveform appear to be just that—a specificfunction of previously received outputs and hence mayappear to be a feedback channel. For feedback channels, di-rected information (DI) between inputs and outputs, ratherthan mutual information between them, is a more relevantmetric (in terms of channel capacity) [17], [18]. In the con-text of the TR channel the relevant DI would be defined as

. Interestingly,for our Gaussian channel model, the DI and MI may be shownto be equal (see [19]), which at first is somewhat surprising asoften in channels where feedback is employed, DI is strictlyless than MI. This may be explained by the fact that we areinterested in the MI between the channel impulse responseand the output , and the channel impulse response does NOTemploy feedback. Hence, because of the problem’s applicationin learning about the channel rather than the inputs, what mightappear to be a feedback channel does not result in different DIand MI.

IV. COMPARING TR AND CONVENTIONAL CHANNELS

We compare the TR and conventional channels fortime-varying channels using the difference between eitherthe DI or MI (as they are equal) as a metric. These are generallyanalytically intractable and we will use Monte Carlo simula-tions in Section V to evaluate the metric. However, to buildintuition, we do present analytical results for a simple scalarproblem which illustrates the impact of nonstationary channelson TR.We consider the difference between the mutual information

achieved by the TR and the conventional channel:

(6)

(7)

where (7) follows by the chain rule for mutual information [16],and the fact that is the same regardless of whether TR isused or not. If , we may conclude that TR yields more

SETLUR AND DEVROYE: INFORMATION THEORETIC TAKE ON TIME REVERSAL 329

Fig. 1. For ch- metric vs for (a) BPSK, , (b) chirp waveform, .

information about the channel than the conventional channel,and vice-versa if . We note that our derivation thus far hasbeen for Gaussian, nonstationary environments, i.e. involvingcolored noise and channels which are correlated over the twotime-instances.The two terms in (7) may be evaluated as in (8)–(9).

(8)

(9)

Although not explicitly shown, it is noted that is a functionof and stays inside the expectation operation. From (2), notethat the pdf of given is normally distributed with ameangiven by and covariancematrix ,which yields (8). Unfortunately a closed form solution to (8)is not immediate, and hence Monte Carlo simulations are em-ployed here. Similar Monte Carlo analysis was employed in [2],[3] but for comparing the detection performance of TR systemswith its non TR counterparts. We now consider a simple scalarchannel where we are able to evaluate the metric analytically.

A. Scalar Special Case

Consider the following scalar channel model:

Assume for simplicity that the noise covariance matrix,and that the channel has covariance matrix,

. For the scalar case .

Then,

(10)

(11)

where is the square of the cor-relation coefficient between and for the conventionalchannel. Now from (10) is always positive for . Inother words, as long as the channel is correlated, , whichimplies that TR is preferable to using the channel conventionallyin terms of mutual information. Similar to , it is envisioned thatthe matrix

plays an analogous and dominant part in for vectorchannels.

V. SIMULATIONS

Two channels, termed are considered in the simulationswhich differ in the channel covariances . The first is the sim-

plest and is given by . This implies

that the taps in , , 2 areuncorrelated within each stage but are correlated across stages,depending on the values assumed by . In channel ch- thecovariance matrix is that is Hermitian Toeplitz, andhas a covariance function

, correlating channel coeffi-cients within and between stages, depending on . To purelyanalyze the effects of the channel, we assume white noise, i.e.

, , , 2. Two waveforms for the trans-mitted are analyzed, the first is a BPSK symbol waveformcomprising random 1, the other is a radar chirp waveform. Theanalysis is carried out in baseband. We will assume forboth ch- . The SNR is defined as . The number ofMonte Carlo trials were set at 10,000 to evaluate the expectationoperation.In Fig. 1, the value of versus are shown for the BPSK

and chirp waveforms for ch- at , 10, 20 dBs. In

330 IEEE SIGNAL PROCESSING LETTERS, VOL. 20, NO. 4, APRIL 2013

Fig. 2. For ch- metric vs for (a) BPSK, , (b) chirp waveform, .

Fig. 1(a), for , and , we see that theis positive for high correlation ,

indicating superior performance of the TR when compared tothe conventional channel. For medium to low correlations, andnot surprisingly, the opposite is true, i.e. becomes negativeindicating that TR is not preferable when compared to using thechannel conventionally. In particular, we see that for

, and for medium and low correlation, the metric assumeslow values. Similar results are seen for the chirp waveform inFig. 1(b). The break even points for Fig. 1(a) and Fig. 1(b), i.e.

are different for the same SNR, hence a waveform de-pendency is also noted. The processing for the chirp was per-formed in the baseband bandwidth [2] which contains 99% ofthe signal energy. For the implementation, spectral content out-side the band was notched, and an inverse FFT (IFFT) was em-ployed to return to the time domain. Such frequency domainprocessing is not required for the BPSK, as it is wideband.In Fig. 2, the results are shown for the BPSK waveform and

the chirp for ch- . The BPSK waveforms were different inFig. 1(a) and Fig. 2(a). In Fig. 2 the parameter now controlsthe correlation. As increases, the channel coefficients startbecoming uncorrelated. Identical conclusions to Fig. 1 maybe drawn by observing Fig. 2. For example and as before,for reasonable SNR and low correlation scenarios, the metricassumes low values, implying a harsh penalty for using the TRrather than using the channel conventionally.It is stressed that the metric evaluates the TR and con-

ventional channel on the “average”. In other words, for low tomedium correlation, we have seen instances for both ch- ,where the difference between the DI between the TR and theconventional channels are actually positive, whereas on an av-erage it is negative, i.e. .

REFERENCES[1] M. Fink, “Time reversal of ultrasonic fields I. Basic principles,” IEEE

Trans. Ultrason., Ferroelec. Freq. Contr., vol. 39, no. 5, pp. 555–566,Sep. 1992.

[2] J.M. F.Moura andY. Jin, “Detection by time reversal: Single antenna,”IEEE Trans. Signal Process., vol. 55, no. 1, pp. 187–201, Jan. 2007.

[3] Y. Jin and J. M. F. Moura, “Time-reversal detection using antenna ar-rays,” IEEE Trans. Signal Process. vol. 57, no. 4, pp. 1396–1414, Apr.2009 [Online]. Available: http://dx.doi.org/10.1109/TS.2008.2010425

[4] W. A. Kuperman, W. S. Hodgkiss, H. C. Song, T. Akal, C. Ferla, andD. R. Jackson, “Phase conjugation in the ocean: Experimental demon-stration of an acoustic time-reversal mirror,” J. Acoust. Soc. Amer., vol.103, 1998.

[5] B. I. Zeldovich, N. F. Pilipetskii, and V. V. Shkunov, Principles ofPhase Conjugation. Berlin, Germany: Springer-Verlag, 1985.

[6] S. K. Lehman and A. J. Devaney, “Transmission mode time-reversalsuper-resolution imaging,” J. Acoust. Soc. Amer., vol. 113, no. 5, pp.2742–2753, 2003.

[7] D. Cochran, S. Suvorova, S. Howard, and W. Moran, “Waveform li-braries: Measures of effectiveness for radar scheduling,” IEEE SignalProcess. Mag., vol. 26, no. 1, pp. 12–21, 2009.

[8] W. Moran, S. Suvorova, and S. Howard, “Application of sensorscheduling concepts to radar,” Found. Applicat. Sensor Manag., pp.221–256, 2008.

[9] P. Setlur and N. Devroye, “Adaptive waveform scheduling in radar:An information theoretic approach,” in Proc SPIE 8361, Radar SensorTechnology XVI, May 1, 2012, pp. 836 103–836 111 [Online]. Avail-able: doi.org/10.1117/12.919213

[10] H. C. Song, “Time reversal communication in a time-varying sparsechannel,” J. Acoust. Soc. Amer. vol. 130, no. 4, p. EL161, 2011[Online]. Available: http://www.biomedsearch.com/nih/Time-re-versal-communication-in-time/21974486.html

[11] I. Naqvi, P. Besnier, and G. Zein, “Robustness of a time-reversal ultra-wideband system in non-stationary channel environments,” IET Mi-crow. Antennas Propagat. vol. 5, no. 4, pp. 468–475, 2011 [Online].Available: http://link.aip.org/link/?MAP/5/468/1

[12] M. Bell, “Information theory and radar waveform design,” IEEE Trans.Inf. Theory, vol. 39, no. 5, pp. 1578–1597, 1993.

[13] M. Bell, “Information Theory and Radar: Mutual Information and theDesign and Analysis of Radar Waveforms and Systems,” Ph.D. disser-tation, Calif. Inst. Technol., Pasadena, 1988.

[14] A. Hero, C. Kreucher, and D. Blatt, “Information theoretic approachesto sensor management,” Found. Applicat. Sensor Manag., pp. 33–57,2008.

[15] J. Williams, “Information Theoretic Sensor Management,” Ph.D. dis-sertation, Mass. Inst. Technol., Cambridge, 2007.

[16] T. Cover and J. Thomas, Elements of Information Theory. New York,NY, USA: Wiley, 1991.

[17] J. Massey, “Causality, feedback and directed information,” in Proc.IEEE Int. Symp. Inform. Theory and Its Applications, Honolulu, HI,USA, Nov. 1990.

[18] G. Kramer, “Directed Information for Channels with Feedback,” Ph.D.dissertation, Swiss Fed. Inst. Technol., Zurich, 1998.

[19] P. Setlur and N. Devroye, “On the mutual information of time reversalfor non-stationary channels,” in ICASSP 2013, submitted for publica-tion.