IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 1, JANUARY 2007 187Detection by Time Reversal: Single AntennaJosé M. F. Moura, Fellow, IEEE, and Yuanwei Jin, Member, IEEEAbstract—This paper studies the binary hypothesis test of de-tecting the presence or absence of a target in a highly cluttered envi-ronment by using time reversal. In time reversal, the backscatter ofa signal transmitted into a scattering environment is recorded, de-layed, energy normalized, and retransmitted through the medium.We consider two versions of the test—target channel frequencyresponse assumed known or unknown—and, for each version, con-trast two approaches: conventional detection (where no time re-versal occurs) and time reversal detection. This leads to four al-ternative formulations for which we derive the optimal detectorand the generalized likelihood ratio test, when the target channelfrequency response is known or unknown, respectively. We deriveanalytical expressions for the error probabilities and the thresholdfor all detectors, with the exception of the time reversal generalizedlikelihood ratio test. Experiments with real-world electromagneticdata for two channels (free space with a target immersed in 20scatterers and a duct channel) confirm the analytical results andshow that time reversal detection provides significant gains overconventional detection. This gain is explained by the empirical dis-tribution or type of the target channel frequency response—richerscattering channels induce types with heavier tails and larger timereversal detection gains.Index Terms—Adaptive waveform, detection, empirical distribu-tion, matched filter, time reversal, type, waveform reshape.I. INTRODUCTIONCHANNEL multipath significantly affects the performanceof traditional detectors, e.g., the matched filter. Usually,multipath is thought to be detrimental and a negative whoseeffects should be minimized. Time reversal presents the oppo-site opportunity—multipath as a positive, the more the better.In time reversal signal processing, a signal is first radiatedthrough a rich scattering medium. The backscattered signal isthen recorded, delayed, time reversed, energy normalized, andretransmitted. The technique of time reversal is not new, buta thorough theory of detection for this setting is lacking. Thispaper addresses this gap. We study time reversal detection of atarget immersed in a rich scattering environment. We focus ondetermining the performance gain, if any, provided by the timereversal based detector over conventional detection techniques.We carry out the following plan.1) Formulate a time reversal approach to detection and con-trast it with the conventional approach.2) Derive the detectors for each of these approaches.Manuscript received July 15, 2005; accepted March 21, 2006. This work wassupported by the Mathematical Time Reversal Methods Program, DSO-CMP,Defence Advanced Research Projects Agency through the Army Research Of-fice under Grant W911NF-04-1-0031.The associate editor coordinating the re-view of this manuscript and approving it for publication was Prof. Mats Viberg.The authors are with the Department of Electrical and Computer Engineering,Carnegie-Mellon University, Pittsburgh, PA 15213 USA (e-mail: [email protected]; [email protected]).Color versions of Figs. 1 and 3–9 are available online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2006.8821143) Detail the performance of the detectors analytically andexperimentally.4) Test the detectors with real electromagnetic (EM) data col-lected with two different laboratory experiments.Our results are conclusive.1) Time reversal detection provides significant gains overconventional detection.2) The time reversal detection gain is verified experimentallyfor the first time with electromagnetic real-world experi-ments.3) The time reversal detection gain is directly related to thetype1of the target channel frequency response—the gain islarger for heavy tailed channel types.4) The time reversal detection gain arises because the trans-mitter reshapes the waveform to best match the channel.A. On Time ReversalTime reversal (TR), known in optics as phase conjugation,has been used to increase resolution by exploiting scatteringand multipath in inhomogeneous channels. Fink and collabora-tors have published extensively on time reversal in acoustics andultrasound [2]–[6]. These works demonstrated superresolutionfocusing in the ultrasound domain. In their work, an ultrasoundsource is placed in a water tank with a large number of scatterers.The scattered acoustic signal is recorded by an array of sensorsand retransmitted through the same medium after being time re-versed. Their experiments demonstrate that the acoustic energyrefocus at the source with much higher resolution than predictedby the Rayleigh resolution limit, i.e., they demonstrate super-resolution focusing. More recently, large-scale acoustics exper-iments in the ocean confirmed the resolution ability of time re-versal in real acoustic propagation environments [7], [8]. Thereis a growing literature on time reversal in these acoustic and ul-trasound fields, as well as on studies of time reversal in randomenvironments [9] and in several applications domains, includingimaging [10], [11] or communications [12]–[14]. Focusing inthe electromagnetic domain has recently been demonstrated in[15] and [16]. In [17], we presented a TR-based interferencecanceller to mitigate the effect of clutter in the electromagneticdomain. None of these works has studied the problem of detec-tion using time reversal, derived the detectors, and studied timereversal detection analytically and by experimentation with realelectromagnetic data. This is what this paper pursues and ac-complishes. To stress the focus on the impact of time reversal,we consider the detection of a target in clutter with a single an-tenna. This precludes the use of narrow-band multiple signalclassification and subspace type algorithms where the numberof clutter returns is restricted to be smaller than the number ofarray elements.1The expression type is used in its information theoretic sense of empiricaldistribution [1].1053-587X/$20.00 © 2006 IEEE188 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 1, JANUARY 2007The remainder of this paper is organized as follows. InSection II, we describe the time reversal measurement protocoland present the statistics of the measurements. Section IIIformalizes the