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210 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 1, FEBRUARY 2010
Time Reversal in Multiple-Input
Multiple-Output Radar
Yuanwei Jin, Senior Member, IEEE, José M. F. Moura, Fellow, IEEE, and
Nicholas O’Donoughue, Student Member, IEEE
Abstract—Time reversal explores the rich scattering in a
multipath environment to achieve high target detectability. Mul-
tiple-input multiple-output (MIMO) radar is an emerging active
sensing technology that uses diverse waveforms transmitted from
widely spaced antennas to achieve increased target sensitivity
when compared to standard phased arrays. In this paper, we
combine MIMO radar with time reversal to automatically match
waveforms to a scattering channel and further improve the per-
formance of radar detection. We establish a radar target model in
multipath rich environments and develop likelihood ratio tests for
the proposed time-reversal MIMO radar (TR-MIMO). Numerical
simulations demonstrate improved target detectability compared
with the commonly used statistical MIMO strategy.
Index Terms—Detection, multiple-input multiple-output
(MIMO) radar, time reversal, waveform design.
I. INTRODUCTION
MULTIPATH is a common phenomenon in radar, sonar,
and wireless communication applications [1]–[3]. Most
radar systems are designed under line-of-sight (LOS), not in
multipath rich environments. Multipath has noticeable impact
on radar performance. For example, the range of radar sensors
is limited by LOS blockage due to buildings, forests, and many
other 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 sensitivity
to detecting targets. The multipath limitation has motivated ex-
tensive research on radar algorithm and architecture design to
overcome the LOS restriction. In this paper, rather than treating
multipath as an adverse effect, we use time reversal to explore
multipath 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 Projects
Agency through the Army Research Office under Grant W911NF-04-1-0031.
The work of N. O’Donoughue was supported by a National Defense Science
and 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 of
Maryland Eastern Shore, Princess Anne, MD 21853 USA (e-mail: yjin@umes.
edu).
J. M. F. Moura and N. O’Donoughue are with Department of Electrical and
Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 USA
(e-mail: moura@ece.cmu.edu; nodonoug@ece.cmu.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/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, the
retransmitted waveform refocuses on the original source. By
time reversal, the transmission waveforms are tailored to the
propagation medium and the target scattering characteristics.
Hence, time reversal is a radar waveform adaptive transmission
scheme. Other radar waveform design strategies are given in,
e.g., [4]–[6]. Our recent work, [7]–[9] considers signal detection
using time reversal with a single antenna pair as well as with an-
tenna arrays, and demonstrates that the time reversal generalized
likelihood ratio detector (TR-GLRT) significantly improves de-
tection performance when compared with conventional detec-
tors.
Recently, there has been considerable interest in a novel
class of radar systems known as “MIMO radar,” where the term
multiple-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 radar
system is the use of a set of orthogonal transmit waveforms.
The radar return from a given scatterer has various degrees of
correlation across the array, depending on the element spacing
and array configuration. In particular, the term “statistical
MIMO radar,” [10], refers to the signal model where the signals
measured at different antennas are uncorrelated. If the antennas
are separated far enough, the target radar cross sections (RCS)
for different transmitting paths become independent random
variables. Thus, each orthogonal waveform carries independent
information about the target; spatial diversity about the target
is thus created. Exploiting the independence between signals at
the array elements, MIMO radar achieves improved detection
performance and increased radar sensitivity. This is in contrast
with a conventional phased array that presupposes a high
correlation between signals either transmitted or received by an
array.
In this paper, we develop a MIMO setup to operate in a rich
scattering 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 in
the vicinity of the target of interest, or tracking and detection of
low-angle targets over a flat surface [16], [17]. Multipath affects
the level of the energy return from the target due to coherent
combining of the return signals. As a result, we will observe
fades and enhancements relative to the level that is expected in
a free-space environment. In general, the overall target response
is characterized by the target’s radar cross section, the multipath
1932-4553/$26.00 © 2010 IEEE
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JIN et al.: TIME REVERSAL IN MIMO RADAR 211
propagation due to the surrounding scatterers, and the antenna’s
aspect angle. The unknown nature of the complex target reflec-
tion makes the overall target response appear to be random even
for a point target. Therefore, we adopt a statistical model for
the target in this paper. Although our MIMO model is somehow
similar to what is used in [10], the difference is clear. In [10],
the randomness of the radar target return is caused by many look
angles 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 by
tailoring the transmitted waveforms to the propagation medium
and 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 time
reversal to adaptively adjust the radar waveforms to scattering
characteristics of the channel; 3) to employ simple orthogonal
wideband waveforms without seeking complicated waveform
coding design methods; and 4) to incorporate the de-correlation
between the forward channel and the backward channel when
the reciprocity condition may not strictly hold. We develop a
binary 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. In
this paper, we demonstrate that a MIMO radar combined with
time reversal (TR-MIMO) improves target detectability when
compared with statistical MIMO (S-MIMO) [20].
The remainder of the paper is organized as follows. Section II
discusses the MIMO modeling in multipath. Section III dis-
cusses the TR-MIMO signal modeling. Section IV derives the
TR-MIMO detectors. Section V provides performance analysis
for the proposed MIMO detector. Section VI presents the
TR-MIMO detection results. Finally, conclusions are drawn in
Section VII.
II. MIMO MODELING IN MULTIPATH
We consider the problem of detecting a stationary or slowly
moving 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 and
tracking. 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 and
the receiver of the radar are well separated such that they expe-
rience an angular spread caused by variation of the radar cross
section (RCS). Statistical MIMO radar utilizes the spatial di-
versity introduced by the radar target fluctuation. For extended
targets, due to the large inter-element spacing between the an-
tennas, each transmitter–receiver pair sees a different aspect of
the target. This observation yields the following basic MIMO
radar modeling for extended targets [20]:
(1)
where is an matrix, referred to as the target channel
matrix, 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 transmit
signals at the transmit antennas. The vector is the additive
noise. Given the statistical MIMO model (1), the optimal
detector 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 transmit
antenna and receive antenna become independent random
variables when the antennas are placed sparsely. We will
show that multipath causes a similar effect even when the
target is pointwise. The superposition of the multipath
adding constructively and destructively makes the received
radar returns appear to be random.
2) This observation can be further justified by examining the
radar operating spectrum for a point target. A point target
in a LOS condition yields a flat spectrum for the returned
signal, but results in a fluctuating spectrum in a non-LOS
condition.
3) The radar model (2) and the processing (3) are designed
for narrowband systems. By exploring the orthogonality
of the waveforms, each receiver may match to a specified
transmit waveform [20]. To extend this design scheme to a
wideband system is difficult and may significantly increase
the radar system complexity. This is because of the dif-
ficulty in designing orthogonal radar waveforms that can
meet various radar operational conditions (e.g., [21]). On
the other hand, (3) indicates that the signal cross-correla-
tion can not be disregarded; significant correlation reduces
the mainlobe width, which can result into higher ambiguity
sidelobe levels. This paper proposes a simple quasi-orthog-
onal waveform design for wideband MIMO radar using the
time reversal method.
Next, we introduce a point target MIMO model in multipath
environments.
B. Multipath MIMO Radar Model
In the absence of multipath, the reflected radar waveform
from a point target is an amplitude scaled and time delayed
replica of the transmitted waveform. In theory, the target
frequency response is flat. However, multipath propagation in-
duces a rapid fluctuation in the frequency response of the point
target. Fig. 1 illustrates a two-way radar propagation model
in multipath. For simplicity, this model considers a two-path
propagation with only a single reflected ray emanating from
a virtual target image. The two-path propagation is caused
by scatterers between the receiving array B and the target.
This model can be extended to the more general scenario with
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212 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 1, FEBRUARY 2010
Fig. 1. Multipath propagation model. Only the forward propagation (from
array A to array B) is illustrated.
multiple path propagation due to scatterers in the fields of view
of both the transmitting array A and receiving array B.
Suppose we have a transmitting signal
(4)
where is the baseband radar pulse at frequency from
the th transmit antenna, is the carrier frequency. We assume
that the signal propagates in the multipath medium and reflects
from a target with a possible phase change . The noise-free
received signal at the th element of array B due to the th
transmit antenna at array A is given by
(5)
where is the phase change due to the multipath; the direct
path delay and the multipath delay are given as fol-
lows:
(6)
(7)
The symbols , , and are the azimuth coordinates of the
th antenna at A, the th antenna at B, and the target, respec-
tively; is the target range; and is the complex amplitude
due to the target characteristics. The complex amplitudes of the
direct and reflected rays are simply related by a complex multi-
path reflection coefficient .
By mixing the received signal with the transmitted signal, we
obtain 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 in
array and the th antenna in array is
(10)
We hence assume that, with a large number of scatterers and
multipath reflection coefficients , the overall target reflec-
tivity (10) is a random variable. This analysis implies that, even
for 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 provide
independent information about the target due to their different
viewing angles.
Hence, from (10), we introduce the forward propagation
channel. 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 channel
response from antenna , , to antenna ,
,is
(11)
where the symbol stands for “distributed as.” The target
channel model (11) implies that, at a fixed frequency and for
each transmitter antenna and receiver antenna pair, the target
channel response is a complex Gaussian random variable with
zero mean and variance . This variance (i.e., the power
spectrum density) is frequency dependent, which is caused by
the multipath scattering. Further, we assume that the channel re-
sponse from different transmit and receive pairs are statistically
independent, identically distributed (i.i.d.) random variables.
Note that the frequency-domain representation is
related 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 many
practical situations in wave propagation in the radar literature
[22], [23]. The variance of is frequency depen-
dent. By the WSSUS assumption and the Wiener–Khintchine
theorem [24]
(13)
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JIN et al.: TIME REVERSAL IN MIMO RADAR 213
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 random
medium can be characterized by the correlation function (see
chapter §5 in [22])
(17)
The function is the correlation function between the output
fields 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) shows
explicitly 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, the
function is a function of the frequency sep-
aration .
In a multipath channel, it is often convenient to consider the
coherence bandwidth measured by the reciprocal of the mul-
tipath spread. Two sinusoids with frequency separation greater
than are affected quite differently by the channel. Hence,
the frequency samples taken apart are considered
to be, approximately, independent. Therefore, we use discrete
frequency samples , , in developing the
TR-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 larger
the 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 other
wideband signals such hyperbolic frequency modulation
(HFM) signals, as well as signals generated from pseudo
random sequences [23]. Using the orthogonality condition in
(20), and converting it into the frequency domain by the Fourier
transform, 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 ,itis
straightforward to show that
(25)
III. TIME REVERSAL MIMO SIGNAL MODEL
In this section, we describe the time reversal MIMO radar
signal model. In time reversal, the received signal is phase
conjugated, energy normalized, and retransmitted to the same
medium. If the medium is reciprocal, the forward propagation
channel is the same as the backward propagation channel.
However, in many radar applications, the reciprocity condition
may not hold, for example, due to small random perturbations
between the forward and the backward channel realizations,
or due to slow target motion. Hence, we model the backward
propagation channel as follows with respect to the forward
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214 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 1, FEBRUARY 2010
channel given in (11): let denote the backward
channel frequency response matrix between the array and
. The symbol denotes the transpose. The propagation
channel from antenna , , to antenna ,
, is given by
(26)
where (26) models the backward channel as a noisy version of
the forward channel: the symbol is the correlation coefficient
between the forward channel and the backward channel
. The term describes the channel disturbance
that occurs between the forward and backward channel for each
pair of transmit and receive antennas in and . We assume
that the disturbance is independent of ; it is distributed
as a complex Gaussian with zero mean and variance , i.e.,
. We further impose the constraint
that has the same variance as , i.e.,
(27)
which implies that
(28)
When , the quantity captures the de-correlation between
the forward channel and the backward channel. As discussed
above, the de-correlation can happen due to slow changes on
the media [26], [27]; its net effect is to degrade the reciprocity
condition. The characterization of the degradation is important
to 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 are
described in three steps as follows.
Step 1 Target Probing: The signal vector received at array B
for the th data snapshot is
(31)
where
(32)
is a vector. The signal vector transmitted
from array is
(33)
The transmitted signal from the th antenna is a wideband
signal with Fourier representation at frequency .We
assume that the average transmission power at each antenna is
the same
(34)
In this step, the total transmission energy is . The noise
vector is characterized statistically as
(35)
We assume here that the clutter is homogeneous; the noise
vector (35) describes both the homogeneous clutter and the
additive noise in the received radar returns. We should note that
this is a very simplified assumption on the clutter. Statistical
models for MIMO radar clutter should incorporate a number
of effects including geometry, coherence, transmit waveform,
multipath scattering, etc. Development of statistical models
for MIMO radar clutter is beyond the scope of this paper.
In general, inhomogeneous clutter can be suppressed using
whitening filters in the spectral and spatial subspaces (see, e.g.,
[28] and [29]). Next, suppose we can collect
snapshots, for a slow varying target channel. The minimum
mean squared estimate of the returned target signal is
(36)
where . We assume that we can
obtain a reasonably accurate estimate of the target channel re-
sponse for sufficiently large . In the subsequent derivation, we
assume that we know precisely. This assumption yields
the ideal scenario for the TR-MIMO detector. In reality, we
would use as the signal to be retransmitted. The problem
of obtaining a sufficient number of snapshots for this pur-
pose is governed by two important factors: 1) the scale over
which the response changes with respect to space and time; and
2) systems considerations such as the bandwidth that limit the
sampling rate. Thus, in a radar setting it is fairly common to
have a snapshot starved scenario. In this case, the noise vari-
ance will increase. Our early attempts to analyzing the effect of
noise variance increase have been reported in [8]. The detection
performance under this scenario will be studied below by sim-
ulations. We note that a larger means longer estimation time
that 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 reversed
and energy normalized. The received signal vector at
array 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 transmitted
signal . The target return is defined as
(42)
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JIN et al.: TIME REVERSAL IN MIMO RADAR 215
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 at
the th antenna of array . This radar signal will be matched
with the originally transmitted signal at the th antenna, i.e.,
defined in (33).1We repeat this process for
antennas at the array , which yields the following data
vector
(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 detection
problem. The binary hypothesis test for TR-MIMO is
(51)
1One may argue that the received radar signal at the th antenna, , can
be matched with the individual waveforms . We show in
Section IV that the output of the matched filter of with , is
relatively 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 a
common problem: detecting a nonwhite Gaussian process im-
mersed in additive nonwhite (or white) Gaussian noise in sonar
or radar (see, e.g., [28]–[31]). A closed form for the probability
density function for the binary hypothesis (51) is often difficult
to obtain. Thus, we will rely on approximations and on the cen-
tral limit theorem to study the data statistics and to derive the
test 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 random
variables for . Equation (54) greatly simplifies (53), which
yields
(55)
(56)
where by (24). However, by eliminating the un-
matched phase terms in the analysis, (54) may introduce a small
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216 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 1, FEBRUARY 2010
prediction error compared with the results from numerical sim-
ulation. We will show by Monte Carol simulation in Section VI
that 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 components
are the focused target response; in a statistical sense, by av-
eraging out the perturbation term in (56), they are zero-phase
quantities. This observation leads to the following approxima-
tion:
(59)
By ignoring the perturbation term, we should note that, when
the forward channel and backward channel are fully correlated,
i.e., , the above approximation is exact. We will develop
the TR-MIMO detector under this assumption. We will test for
cases when using numerical simulations. Let’s further
define
(60)
where, the symbol stands for “distributed as” and
stands for “distributed approximately as.” The symbol
. For the sake of simplicity, here
we treat and as two independent random
variables. We further approximate by the constant .
We show in (129) and (133) in Appendix I that this is a valid
approximation: the random variable is a low variance distri-
bution with a constant mean. Next, we rewrite (59) as
(61)
The sequence are independent random variables
with finite variance. By the Lindeberg–Lyapunov central limit
theorem on sums of independent random variables [32],2the
sequence 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 entry
of signal becomes
(65)
(66)
The symbols and denote the real and imaginary part
of the complex number . Both quantities and
are 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 the
TR-MIMO detector is
(71)
The detector uses (71) to calculate the decision threshold under
the null hypothesis .
C. About the Output of Matched Filters
In Step 3 of Section III-A, the th entry of defined
in (37), is matched with the originally transmitted signal at the
th antenna, i.e., defined in (33). One can certainly match
with other signals , to generate a total of
outputs. Excluding the additive noise terms, these
outputs are
(72)
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JIN et al.: TIME REVERSAL IN MIMO RADAR 217
Since and , are random variables, we take
the 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 orthogonality
between and , , chosen in Section II-C. In
particular, if the term , a constant value
independent of frequency, (77) becomes zero. Therefore, we do
not 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 extend
this narrowband signal model, (1) and (2), to wideband. One
of 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 same
waveforms 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 received
signal 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 into
vectors
(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
.
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218 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 1, FEBRUARY 2010
V. P ERFORMANCE ANALYSIS
In this section, we derive the test statistic for time reversal
MIMO (TR-MIMO) in (71) and statistical MIMO (S-MIMO)
in (90), respectively. To derive the detectors and to analyze
their performance, we derive the probability density functions
of the test statistics under and , respectively. The per-
formance analysis under the homogeneous clutter assumption
provides 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 simulations
to study channel decorrelation.
A. TR-MIMO
Under and based on (68), we know that are in-
dependent and identically distributed normal random variables
for . Therefore, from (71), we obtain
(91)
(92)
where denotes the non-central Chi-squared distributed
random variable with degrees of freedom and the non-cen-
trality parameter
(93)
Under
(94)
(95)
where is a central Chi-squared random variable with
degrees of freedom. Both (92) and (95) provide a simple
description of the test statistic for the TR-MIMO detector
under each hypothesis. To obtain the detection probability
and the decision threshold from (92) and (95), we derive in
Appendix III the probability density functions under
and under , respectively. Given the density
function and a chosen false alarm rate , we can
numerically calculate the threshold by solving
(96)
Readers can refer to Appendix III for a detailed discussion on
how the threshold is computed. Next, we can compute the
detection probability
(97)
Under simplifying conditions, for example, the radar signal
model given in [35] and [36], one can develop closed form
expressions for and . In our case, the time reversal
radar signal model in (66) becomes, approximately, a nonzero
mean, real Gaussian signal immersed in zero-mean complex
Gaussian noise. The mathematical tractability of closed form
expressions for and becomes difficult. One can resort
to 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)
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JIN et al.: TIME REVERSAL IN MIMO RADAR 219
(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 random
variable with zero mean and variance . Hence, (109) can be
written as a quadratic form
(111)
where . Since are
independent, 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 useful
metric for comparing performance between different test statis-
tics, other performance measures can also provide insight and
mathematically tractable approaches; for example, the nominal
performance given in [39, (p. 329)] . Here, we use the following
metric [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 at
the bottom of the page. Similarly, we obtain the value for the
S-MIMO
(120)
To compare (119) and (120), we assume that , which is in
high signal-to-noise ratio (SNR). In this case, ,
while . This implies that, with a few sparsely placed
antennas, 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)
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220 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 1, FEBRUARY 2010
Fig. 2. Nominal performance versus SNR for TR-MIMO and S-MIMO
using . 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 simulation
is carried out as follows: 1) generate random realizations of
independent frequency samples for the forward channel
matrix and the backward channel matrix ;
2) generate orthogonal waveforms based on (24); 3) generate
target signal independent additive noise and add the noise to the
received radar returns; 4) calculate the test statistics and deter-
mine the decision threshold given the false alarm rate ; and
5) calculate the detection probability. The signal-to-noise ratio
is defined as
SNR (121)
We choose frequencies, transmit antennas,
and receive antennas for simulation purposes. We
show in Fig. 3 the detection probability versus SNR for
TR-MIMO versus S-MIMO under the false alarm rates of
, respectively. The correlation
factor represents the ideal scenario. The analytical results
are plotted using (97) for the TR-MIMO detector and (116)
for the S-MIMO detector, respectively. The lines represent
the analytical plots while the markers denote the Monte Carlo
simulation results. Fig. 3(a)–(d) shows that the analytical re-
sults match the Monte Carlo simulation results quite well. The
proposed TR-MIMO has about 14-dB gain over S-MIMO for
the simulation setup . There appears a small prediction
bias (within a fraction of a dB) for the analytical ROC result
compared with the Monte Carol simulation. This small predic-
tion bias can be explained by the approximation we made in our
analysis, for example, see (54). This equation argues that the
unmatched phase terms are zero, which of course is not true in
practice. This approximation yields slightly better performance
and ROC curve than the Monte Carlo simulation shows, see
Fig. 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 the
forward channel and the backward channel is . The false alarm rates are:
(a) , (b) , (c) , and (d) .
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JIN et al.: TIME REVERSAL IN MIMO RADAR 221
Fig. 4. (a) Average coherence bandwidth of scattering channels estimated from
experimental electromagnetic data collected in lab, [9]. The average coherence
bandwidth decreases from 912 to 120 MHz as the propagation medium be-
comes increasingly multipath rich. (b) Detection probability of TR-MIMO and
S-MIMO by simulations assuming , respectively.
, , .
size of the receiving antenna or the number of independent
frequency bands .
The number of independent frequencies depends on the
scattering properties of the channel and can be determined
by experiments. In our case, the choice of is based on our
experimental electromagnetic data collected in an increasingly
rich scattering lab environment. The scattering environment
is 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 network
analyzer (VNA). The scattering environment is created by
gradually increasing the number of dielectric rods in a 4 4
square feet wood platform. The wood platform has a total of
46 holes that can hold the dielectric rods vertically. First, the
target response is measured for a copper target; then, we add
one dielectric rod into the scene to create multipath. We keep
adding dielectric rods until all the rods are filled in the holes.
We then measure the coherence bandwidth of the channel
versus the number of dielectric rods. Fig. 4(a) shows that
the coherence bandwidth decreases from MHz to
MHz. To further enhance the channel scattering, we
place a metal shield behind these 46 rods, which brings down
Fig. 5. Detection probability for TR-MIMO versus decorrelation factor . The
number of frequencies .,..
the coherence bandwidth to 120 MHz. The- GHz band-
width corresponds to , respectively.
Using this sequence of values, we generate, by Monte Carlo
simulations, the detection of probability versus in Fig. 4(b)
for TR-MIMO and S-MIMO. We fix the average channel gain
to noise ratio at 2 dB and change the value.
Fig. 4(b) shows that the performance gain of TR-MIMO versus
S-MIMO increases as the scattering becomes dense. This is
because the TR-MIMO detector coherently process signals by
adaptive time reversal transmission while the S-MIMO detector
non-coherently add up signals from different frequencies.
To study the de-correlation between the forward channel and
the backward channel in time reversal, we vary the correlation
factor in the range of 0 and 1. Fig. 5 depicts the detection
probability of TR-MIMO and S-MIMO when the decorrelation
factor decreases from 1 to 0. The performance of TR-MIMO
is robust against channel decorrelation.
VII. CONCLUSION
This paper develops the time reversal MIMO radar detector
and 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 the
scattering properties of the medium by using time reversal. The
paper presents a model for TR-MIMO that accounts for possible
decorrelation between the forward channel and the backward
channel and derives the test statistic, threshold, and probability
of detection for the TR-MIMO Neyman–Pearson detector. The
algorithm we develop is robust in rich multipath environments
and shows a significant gain over the statistical MIMO detector.
APPENDIX I
ENERGY NORMALIZATION SCALAR
The scalar in (41) is a random variable because it depends
on the channel matrix whose elements are random. In the
development of the statistics for TR-MIMO, we wish to charac-
terize the mean of this random variable as , where
(122)
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222 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 1, FEBRUARY 2010
where . Let the vector
, where
(123)
where , ,
and we assume for simplicity . It is straightforward
to see that
(124)
Hence, using[40, Theorem 3.1], we obtain that the following
weighted 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 random
variable. It is well known that, for ,
has the mean , , and
, . Hence, we obtain the mean of
as follows:
(129)
In probability theory and statistics, the coefficient of variation
(CV) is a normalized measure of the dispersion of a probability
distribution. It is defined as the ratio of the standard deviation to
the mean as
(130)
(131)
(132)
By the Cauchy–Schwartz inequality
Hence, for , we obtain
(133)
which implies that has a low variance. This result implies
that the variance of will be a very small number when the
number of frequencies and the number of antennas are large.
Hence, it suffices to say that is approximately a constant.
APPENDIX II
DERIVATION 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 an
identity matrix, respectively. Similarly, the probability density
function of under is
(137)
where
(138)
The likelihood ratio test becomes
(139)
Using (70), and discarding the constant terms 1/2 in (139), we
rewrite (139) as
(140)
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JIN et al.: TIME REVERSAL IN MIMO RADAR 223
(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 the
probability 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 central
Chi-squared random variable is
(151)
The Gaussian random variable has the probability density
function
(152)
which yields
(153)
(144)
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224 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 4, NO. 1, FEBRUARY 2010
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 the
cumulative 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 Gaussian
random variable . Thus,
(156)
This was solved with numerical integration. The inverse was
accomplished via a simple search algorithm. The cdf is a non-
decreasing function; therefore, comparison of the received cdf
for some test point against the desired cdf determines
in which direction to increment the search term. For these tests,
the stopping criteria was
(157)
We use this inverse function to compute the threshold under
the null hypothesis by calling
(158)
where computes the numerical inverse described
above. is then used to compute the detection probability
(159)
APPENDIX IV
CALCULATION OF THE NOMINAL PERFORMANCE FOR
TR-MIMO AND S-MIMO
Using the statistical properties of the Chi-squared distribution
and 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 Davis
in 2003.
From 2003 to 2004, he was a Visiting Re-
searcher with the University of California at Santa
Cruz. From 2004 to 2008, he was a Post-Doctoral
Research Fellow, then Project Scientist, with the
Department of Electrical and Computer Engineering,
Carnegie Mellon University, Pittsburgh, PA. Since August 2008, he has been
an Assistant Professor with Department of Engineering and Aviation Sciences
at the University of Maryland Eastern Shore, Princess Anne. His research
interests are in the general area of statistical signal and image processing, with
applications in radar/sonar, biomedical imaging, structure health monitoring,
and wireless communications. He has published over 40 technical Journal and
Conference 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, Sigma
Xi, he American Society for Engineering Education (ASEE), and the American
Association for Cancer Research (AACR).
José M. F. Moura (S’71–M’75–SM’90–F’94)
received the engenheiro electrotécnico degree from
the Instituto Superior Técnico (IST), Lisbon, Por-
tugal, and the M.Sc., E.E., and D.Sc. degrees in
electrical engineering and computer science from
the Massachusetts Institute of Technology (MIT),
Cambridge.
He is a Professor of Electrical and Computer
Engineering 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 at
the University of Southern California, Los Angeles. He is a founding codirector
of the Center for Sensed Critical Infrastructures Research (CenSCIR) and
manages a large education and research program between Carnegie Mellon
and Portugal (www.icti.cmu.edu). His research interests include statistical
and algebraic signal processing, image, bioimaging, and video processing,
and digital communications. He has published over 380 technical journal and
conference papers, is the coeditor of two books, holds six patents, and has
given 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 Processing
Society (SPS) and vice-chair of the IEEE Publications Services and Products
Board (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 IEEE
PROCEEDINGS, the IEEE Signal Processing Magazine, and the ACM Trans-
actions on Sensor Networks. He chaired the IEEE Transactions Committee
that 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 Processing
Society Meritorious Service Award and in 2000 the IEEE Millennium Medal.
In 2007 he received the CMU’s College of Engineering Outstanding Research
Award and in 2008 the Philip L. Dowd Fellowship Award for contributions to
engineering 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 Villanova
University, Villanova, PA, in 2006, and the M.S.
degree in electrical and computer engineering from
Carnegie Mellon University, Pittsburgh, PA, in
2008. He is currently pursuing the Ph.D. degree
in electrical and computer engineering at Carnegie
Mellon 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. Lynch
Award from the Villanova University Engineering Alumni Society, and the 2006
Computer 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 of
several IEEE societies, the Acoustical Society of America, Tau Beta Pi, and Eta
Kappa Nu.
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