Approximate multitarget matched filter for MIMO radar ... - IEEE Xplore

2014 International Radar Conference

Approximate Multitarget Matched Filter for MIMO Radar Detection via Orthogonal Matching Pursuit Olivier Rabaste

Laurent Savy

Guy Desodt

ONERA, The French Aerospace Lab BP 80100 91123 Palaiseau Cedex, France

ONERA, The French Aerospace Lab BP 80100 91123 Palaiseau Cedex, France

Thales Air Systems Limours, France

given direction then corresponds to the coherent summation of the transmitted waveform with phase shift corresponding to that direction, and consequently the waveforms received by two targets located in two different directions differ. The transmission directivity can be retrieved through a specific processing at the reception.

Abstract—In this article, we consider the problem of detecting multiple targets in MIMO radar. MIMO ambiguity functions generally present strong range/angle coupling or high sidelobe levels so that weak targets will often be buried in the sidelobes of stronger targets. We propose to solve this problem by iteratively building an approximation of the multitarget matched filter through an Orthogonal Matching Pursuit (OMP) procedure that permits to clean the received signal from the strong target sidelobes and thus to detect the low-SNR targets. However, whereas classic OMP exploits a finite discretization of the target parameter space, this procedure must be adapted here to deal with real radar targets that can be located anywhere in the range/angle/doppler cell. We propose here to solve this problem by jointly estimating at each iteration the target states in the maximum likelihood sense via a gradient descent algorithm. We also show that the stopping criteria for this iterative procedure can be set to satisfy a given false alarm probability. Simulations show that the proposed method permits to solve the multitarget detection problem in the MIMO framework and retrieve low-SNR targets buried in the high sidelobes of strong targets produced by the MIMO matched filter.

I.

This reception processing consists of a matched filter in range, doppler and one or two angle directions (depending on the application). The output of such a processing for one single target can be analysed through the computation of the so-called MIMO ambiguity function [7], [5], [8] for a given set of transmitted waveforms. Although this is rarely discussed in the literature, it appears that for any waveform families, the MIMO ambiguity function presents either a uniform spreading of the signal energy in the range/angle plane, or a strong range/angle coupling concentrated in some specific area of this range/angle plane. In both cases, these high sidelobes/coupling may be problematic in multitarget settings: for instance low SNR targets may be buried into the sidelobes generated by stronger targets, or sidelobes created by targets with similar SNR may deteriorate their state estimation. Note that this problem is induced by the fact that the matched filter is the optimal filter only in the single target case.

I NTRODUCTION

In traditional radar applications, the signal transmitted by a phased array radar is usually focused in a narrow beam so as to maximize the energy received by a potential target. This implies that the radar must rapidly illuminate different successive directions to watch over an entire surveillance area. However it may be of interest to illuminate instantaneously a wider area, for example when not enough time is allowed for the successive directions. A first possibility is then to broaden the radar beam by using specific phase laws. Unfortunately this solution leads to the loss of all angle information from the transmission side, and is possible only for small broadening factors.

Two solutions may be considered to solve the sidelobe/coupling problem. The first solution consists in replacing the matched filter by a mismatched filter optimized to present lower sidelobes [9]. Such a filter will of course lead to a loss in processing gain, but remains computationally effective. The second solution consists in computing the multitarget matched filter, i.e. a filter that directly solves the multitarget detection problem. Such a filter is very difficult to compute, because it requires to solve a difficult multiple hypothesis testing problem with an unknown number of hypotheses due to the unknown number of targets.

Another solution has recently appeared in the literature: the MIMO radar. It consists in transmitting different waveforms by the different transmitting antennas. If these waveforms are sufficiently orthogonal, it is possible to retrieve at the reception side the information carried by each signal thanks to a matched filter bank. Two different MIMO radar frameworks exist: the non coherent MIMO radar (or statistical MIMO radar) that mainly exploits the spatial diversity by using well separated or non colocated transmitting antennas [1], [2], and the coherent MIMO radar that exploits information received by colocated antennas to improve the angle resolution [3], [4], [5], [6]. In this paper, we shall consider the second case, in the framework of intrapulse coding. In such a framework, each antenna transmits different signals, so that the whole surveillance area is illuminated at the same time. The signal transmitted in a

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We propose in this article to solve the multitarget detection and estimation problem by iteratively building an approximation of the multitarget matched filter. The proposed solution leads to an algorithm similar to the well-known Matching Pursuit algorithm that permits to decompose a measured signal into the contributions of several waveforms peaked up from a predefined dictionary [10]. We thus propose here to resort to an Orthogonal Matching Pursuit (OMP) algorithm. We show that the iterative procedure of this algorithm can be stopped using a classic detection test. At first sight, the OMP permits to reject the high sidelobes induced by strong targets. However this rejection will be performant only if the target states are correctly estimated within the range/angle/doppler cell, and not only for the classic radar discretization. This is especially true for strong targets. Since real radar targets can be 1


located anywhere in the range/angle/doppler cell, we therefore propose to improve the extraction procedure by adding an estimation step that permits to jointly estimate the target states in the maximum-likelihood sense. The optimization problem that results from this estimation is solved by a gradient descent algorithm. It permits to solve efficiently the well-known gridproblem faced by compressed sensing methods in radar [11], [12]. This paper is organised as follows: in section II, we present the MIMO ambiguity function. In section III, we present the multitarget problem faced in MIMO radar, the proposed solution based on Orthogonal Matching Pursuit and its adaptatation to the grid problem. Finally in section IV, we provide some performance results based on simulations. II.

T HE MIMO

Fig. 1. Range/angle cut of the MIMO ambiguity function for a phase code family (Gold codes) containing 127 symbols. Pulse duration: Tp = 0.1 ms.

AMBIGUITY FUNCTION

In the following, we will assume that the transmitting array is composed of NT antennas or subarrays and that the receiving array contains NR antennas or subarrays. The positions of the mth antenna (center of subarray) of the transmitting array and of the nth antenna (center of subarray) of the receiving array will respectively be denoted by vectors xT,m and xR,n . The two arrays are assumed to be colocated. In the remainder of this paper, we will consider antennas for simplicity, but they can be replaced by subarrays. In a similar way, we will only consider linear arrays and one single angular dimension, the extension to a second angular dimension and 2D-arrays being straightforward.

A0 provides the MIMO ambiguity function given by [5]: NR −1

A(τ, ν, θ, θ ) =

NX T −1

NT −1 NT −1

X X

m′ =0

m=0

E where k(θ) is the wave vector, gm (θ) is the gain of transmitting antenna m in direction θ and the notation ·T represents the matrix transpose. If a target is present in direction θ0 with delay τ 0 , doppler ν 0 and complex amplitude A0 , the signal received on the q th antenna is: 0

yq (t) = A0 gqR (θ0 )ejxR,q k(θ ) s(t − τ 0 , θ0 )ej2πν t , where gqR (θ0 ) is the gain of receiving antenna q in direction θ0 . The optimal coherent MIMO processing on receive consists of a matched filter in delay, doppler and angle applied to the received signal, thus providing the output signal: Z NX R −1 T r(τ, ν, θ) = e−jxR,q k(θ) yq (t)s∗ (t − τ, θ)e−j2πνt dt.

III.

m=0

Z

sm (t)s∗m′ (t + τ )e−j2πνt dt. (2)

OMP FOR

APPROXIMATE MULTITARGET MATCHED FILTER

A. The multitarget problem We consider now a multitarget setting. In the presence of N 0 targets, the signal received by the q th antenna can be expressed as:

q=0

(1) Note that for simplicity we have omitted here and for the remainder of the paper the antenna gains.

N X 0

Inserting the expressions of yq (t) and s∗ (t − τ, θ) in (1), replacing ν −ν 0 by ν and τ −τ 0 by τ and discarding amplitude 978-1-4799-4195-7/14/$31.00©2014IEEE

0 T jxT T ,m k(θ )−jxE,m′ k(θ)

×

Depending on the waveform family considered, this MIMO ambiguity function presents either severe range/angle coupling or high sidelobe levels, as discussed in [13], that may span a very large range/angle domain. This is the case for instance when using different phase codes, or when considering chirp signals located in adjacent frequency bands. Range/angle cuts of the ambiguity function for a phase code family (Gold codes are used here) is presented in Figure 1. These range/angle coupling or sidelobe levels can present serious problems when small targets and strong targets are present. We want to emphasize here that this problem arises mainly due to the fact that the classic matched filter is intrinsically a monotarget filter and thus optimal only in that particular setting. In the presence of multiple targets, a multitarget matched filter should be used.

T

0

e

!

This function of four parameters (delay, doppler, target true angle, target tested angle) represents the output of the matched filter processing for any target direction θ0 . It would be a function of six parameters if two directions were considered. Note that the cut of this MIMO ambiguity function for two different target directions θ0 differ; indeed signals received in these two different directions are the result of a different summation in phase of the transmitted signals sm (t) and therefore their autocorrelation shape differ. Properties of the MIMO ambiguity function have extensively been studied in [7], [5], [8].

E gm (θ)ejxT ,m k(θ) sm (t),

T

e

0 jxT R,q (k(θ )−k(θ))

q=0

In this specific MIMO framework, each antenna of the transmitting array transmits its own specific waveform, different from the one transmitted by the other antennas. Let us denote by sm (t) the waveform transmitted by the mth antenna. With this notation, the overall signal transmitted in a given direction θ can be expressed as s(t, θ) =

X

0

yq (t) =

p=1

2

T

0

0

A0p ejxR,q k(θp ) s(t − τp0 , θp0 )ej2πνp t + nq (t),


where τp0 , θp0 , νp0 and A0p represent respectively the range, direction angle, doppler and complex amplitude of the pth target, and nq (t) represents the noise on reception antenna q. We can now sample the received signals yq (t) at sampling period Ts . The received signal can then be written: N X

the hypothesis testing problem is decided, in which case the algorithm stops. Interestingly the proposed procedure can be assimilated to the Matching Pursuit (MP) algorithm [10]. We will therefore use here the Orthogonal Matching Pursuit (OMP) algorithm [14] (a simple improvement of the MP) to solve the multitarget detection problem. Starting with y1 = y, the proposed algorithm follows an iterative procedure where the k th iteration can be summarized by the two following steps:

0

yq =

A0p sq (Θp ) + nq ,

p=1

where Θp = (τp0 , θp0 , νp0 ) denotes the parameters of target p, and sq (Θp ), yq and nq are vectors containing the sampled T 0 0 versions of continuous signals ejxR,q k(θp ) s(t − τp0 , θp0 )ej2πνp t , yq (t) and nq (t). We denote here by Ns the number of time samples contained by these vectors; Ns spans the full observation time that corresponds to the maximum range surveyed by the MIMO radar, which means that signal vectors sp are non zero only for samples corresponding to the interval [τp0 , τp0 + Tp ] where Tp is the pulse duration. Finally, concatenating all the signals received by the antenna array T T into a single column vector y = [y1T , y2T , . . . , yN ] and R applying the same procedure to produce the signal vector s(Θp ) = [sT1 (Θp ), sT2 (Θp ), . . . , sTNR (Θp )]T and the noise vector n = [nT1 , nT2 , . . . , nTNR ]T , the overall signal can then be written N0 X y= A0p s(Θp ) + n, where the noise vector n will be assumed in the following to be circular complex white Gaussian with variance σ 2 .

−1 H rejection step: yk+1 = (INR Ns − Sk (SH Sk )y, k Sk ) where Sk = [s(Θ1 ), . . . , s(Θk )] and IN is the identity matrix of size N .

In classic OMP, the iterations are stopped when a certain stopping criteria, usually set on heuristic considerations, is satisfied. In the proposed setting, this stopping criteria can be defined based on the detection theory. S Recall that the first iteration consists in testing H0 against +∞ n=1 Hn . In the NeymanPearson detection framework [15], the detection threshold for such a test is set in order to guarantee a predefined false alarm rate. Assuming that the classic GLRT test

A0p s(Θp ) + n,

p=1

for n ≥ 0. Hypothesis H0 corresponds then here to the case with no target at all. Solving this problem would consist in testing all possible hypotheses, i.e. all possible numbers of targets and for each target number, all target parameters, and choosing the most likely one. This would correspond to form for each number of targets n the multitarget matched filter to the 3n-parameter (n targets P with three parameters (delay, angle, doppler) each) signal np=1 s(Θp ), which is absolutely untractable. Instead, we will build here an approximation of such a multitarget matched filter.

max Θ

|s(Θ)H y|2 H1 ≷ τ, s(Θ)H s(Θ) H0

(3)

is a good approximation of the Neyman-Pearson test for testing S+∞ H0 against n=1 Hn , the detection threshold can be computed on the sole knowledge of the statistics on that test under H0 , thus providing the classic expression for the detection threshold τ = −σ 2 ln(PF A ). Similarly, assuming perfect rejectionSof the first estimated targets, the k th iteration (Hk +∞ against n=k+1 Hn ) can be solved considering the new set of hypotheses defined by

B. Multitarget solution via Orthogonal Matching Pursuit We propose in this paper to solve this difficult multitarget detection problem iteratively: we first consider the hypothesis S+∞ testing problem H0 against n=1 Hn , i.e. we test the hypothesis “no target present” against the hypothesis “at least one target is present”. If this last hypothesis is decided, S we consider the new hypothesis testing problem H1 against +∞ n=2 Hn , i.e. the hypothesis “one target is present” against the hypothesis “at least two targets are present”. This procedure can be iterated - at iterationS k, we consider the hypothesis testing +∞ problem Hk against n=k+1 Hn - until the first hypothesis of 978-1-4799-4195-7/14/$31.00©2014IEEE

•

Θ

C. Stopping criteria

Since the number of targets and their respective parameters are unknown, the multitarget detection problem consists in the multiple composite hypothesis testing problem defined by all hypotheses of the form: n X

detection step: find Θk = arg max

Note that the main difference between the MP algorithm and the OMP algorithm lies in the rejection step: for classic MP, the residual vector yk+1 is obtained by subtracting the new detected vector from the previous residual vector yk , while in OMP the residual vector yk+1 is obtained by the orthogonal projection of the measurement vector y onto the subspace orthogonal to the subspace defined by all extracted signal vectors (s(Θk′ ))k′ =1,...,k . Finally note that the OMP algorithm will extract first the strongest targets and produce a cleaned residual vector that should be free of any of the detected target sidelobes. Thus we expect this algorithm to manage to retrieve the low-SNR targets buried in the sidelobes of stronger targets.

p=1

Hypothesis Hn : y =

|s(Θ)H yk |2 ; s(Θ)H s(Θ)

•

Hypothesis Hnk : yk =

n X

A0p−k+1 s(Θp−k+1 )+n, for n ≥ 0.

p=1

Then S+∞ the hypothesis testing problem of testing Hk against becomes the hypothesis testing problem of testing n=k+1 Hn S +∞ k H0 against n=1 Hnk that can be solved using the GLRT (3) and replacing y by yk . Again the detection threshold is then given by τ = −σ 2 ln(PF A ). Finally our stopping criteria is simply to compare the output of the GLRT test which is computed for the detection step of the OMP algorithm, and 3


compare it with this GLRT detection threshold that will insure a false alarm rate PF A . Of course this is only an approximation since it implies perfect rejection of the sidelobes produced by the estimated targets. However it still represents a performant criteria, funded on a classic radar background. Note finally that this detection criteria could be replaced by any other radar detection criteria, for instance a criteria insuring CFAR (Constant False Alarm Rate) when dealing with heterogeneous environments.

only the last component. This choice is motivated by the fact that target not yet detected can deteriorate the estimation of strongest targets. Computing the joint estimation at each iteration permits to correct possible precedent estimation errors. Besides this strategy seems more appropriate in the OMP framework, while the last component estimation strategy would fit more to a classic MP algorithm.

D. Adaptation of the OMP algorithm to the grid problem

Finally we provide some comments regarding the use of the OMP algorithm for multitarget detection in radar applications. The OMP algorithm provides some interesting features that makes it quite appropriate for the considered application:

E. Some remarks on the proposed algorithm

The detection step of the proposed OMP procedure requires the maximization of the GLRT metric over the target parameter space Θ0 = (τ 0 , θ0 , ν 0 ). In any MP or compressive sensing applications, this is usually done by discretizing this parameter space into a finite grid where each node of the grid corresponds to a different parameter hypothesis. However in the MIMO framework, as well as in many radar framework, targets never lie on the grid and the rejection of signal corresponding to the closest neighbour on the grid does not permit to perfectly eliminate the target contribution and therefore generally leads to additional detections corresponding to strong residue of this target. This is particularly problematic for high-SNR targets and/or settings presenting strong sidelobes such as MIMO radar.

•

it is based on classic matched filtering operations, as applied usually in radar systems; if one would like to change this specific processing by another classic radar processing, it would be easy to adapt it in the OMP framework.

•

as we have seen, the OMP algorithm permits to solve the grid problem easily and in an intuitively simple way;

•

the problem considered in this paper can be viewed as the estimation of a sparse vector (there are only a few targets in the observation window) from a set of discrete measurements (the received signal samples), as considered in compressive sensing applications. Indeed, discritizing the target parameter space (τ 0 , θ0 , ν 0 ) into a finite grid of NG nodes where each node of the grid corresponds to a different parameter hypothesis, and assuming that all targets lie on this grid, the multitarget received signal can be rewritten as: y = Ax + n,

We therefore propose to modify the OMP procedure in order to correctly estimate off-grid targets. The proposed procedure starts with the result of the detection step at iteration k that provides the optimal value Θk from the GLRT test on the residual signal yk , and locally computes the maximum likelihood (ML) estimate of all already extracted target parameters Θ1:k = [Θ1 , Θ2 , . . . , Θk ] and all target amplitudes A1:k = [A1 , A2 , . . . , Ak ]T at iteration k. The ML estimate of [Θ1:k , A1:k ] is provided by ˆ 1:k , A ˆ 1:k ] = arg min ky − Sk (Θ1:k )A1:k k2 [Θ Θ1:k A1:k

(4)

where the matrix A can be decomposed on each reception antenna in the form A = [A1 , A2 , . . . , ANR ] and the nth column of the matrix Aq corresponds to the signal backscattered by a target located on the nth grid node and received by the reception antenna q, and where the nth entry of x is the corresponding target amplitude. In that setting, the vector x is clearly sparse and could be reconstructed by many reconstruction methods developped for compressive sensing applications. However we must notice that matrix A is of size NR Ns × NG and may then be extremely large. The advantage of OMP here is that it manages to solve the sparse reconstruction problem without the need for expliciting the matrix A.

where S1:k = [s(Θ1 ), . . . , s(Θk )] is a matrix of size NR Ns × k, where we have eliminated the explicit dependence on Θ1:k to lighten the notation. Minimization of (4) over amplitude parameters A1:k is simple and classic and leads to −1 H A1:k = (SH S1:k y. 1:k S1:k )

Replacing A1:k by its ML expression in (4) permits to transform problem (4) into the following optimization problem over Θ1:k : ˆ 1:k = arg min ky − S1:k (SH S1:k )−1 SH yk2 . Θ 1:k 1:k Θ1:k

Such an optimization over the 3k-dimensional parameter Θ1:k can be performed with efficient descent algorithms, for instance a quasi-Newton method such as the BFGS algorithm [16] that does not require the exact computation of the hessian. Of course the iterative descent algorithm is initialized with the k − 1 estimates obtained at previous OMP iteration and the current on-grid value found for Θk . Such an initialization generally leads to a convex criteria so that convergence of the descent algorithm to the optimal solution is guaranteed.

•

Note that we have chosen here to compute the joint estimate of all parameters at each iteration, instead of estimating 978-1-4799-4195-7/14/$31.00©2014IEEE

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the grid problem is especially problematic in the presence of strong targets: strong targets located offgrid that are not properly removed will result in additional detections. The proposed adaptation to solve the grid problem relies on a good quality estimator of the parameters of the strong targets so that they are properly removed from the measurements. Fortunately, since the estimation performance generally increases with the SNR, parameters for strong targets can be estimated with a sufficiently good quality to be properly eliminated.


IV.

S IMULATIONS

AND RESULTS

To simplify and speed up the simulations as well as the representations, we considered in the simulation only the range/angle ambiguity function, thus assuming that all targets are located at zero doppler. Note that the worst sidelobes are located in that particular range/angle domain, where many code families present strong coupling or sidelobes. This will therefore permit to show that the proposed algorithm can correctly eliminate strong sidelobes created by high-SNR targets and recover low-SNR targets. The transmission array is assumed to be a linear array composed of NT = 10 antennas that are assumed to be omnidirectional and separated by half the wavelength. Note that we could add some sub-array gains but this would not change the conclusions regarding the ability of OMP to detect weak targets. The reception array is composed of only NR = 1 antenna, which represents a very difficult case because the use of reception beamforming permits to severely decrease all sidelobes in any direction different from the direction of interest. The family code used in the simulation is the wellknown Gold code [17] which provides, for sequences of length of the form 2n − 1 with n integer (except for multiples of 4), several binary phase codes that present very good periodic autocorrelations and crosscorrelations, and satisfying aperiodic ones. The Gold codes used here are of length Nm = 127. The pulse duration is set to Tp = 0.1 ms. The bandwidth of the transmitted signals are therefore equal to B = Nm /Tp = 1.27 MHz. One different code is transmitted per antenna. The carrier frequency is set to 3 GHz the pulse repetition period to Tr = 10Tp and the sampling frequency to fs = 2B. For simplicity, we will consider the transmission of only one pulse. However the size of the observation window is given by the distance ambiguity computed as if a train pulse was transmitted and is therefore approximately equal to Tr , i.e. the received signals contain approximately Tr fs = 2540 samples. The range/angle parameter space is divided in the following way: the number of range hypotheses is set to be equal to the number of received samples, that is Nr = 2540 samples, while the angle domain, considered in terms of sin(θ) ∈] − 1, 1], is divided in steps of size 1/NT , leading to 20 angle hypotheses. Note that with such values, the size of matrix A would be 2540 × 50800 which is already untractable. Fortunately, as stated before, the OMP algorithm does not require to form this matrix A.

Fig. 2. Range/angle output of the MIMO processing at reception. The transmitted signals are binary phase codes of length 127 from the Gold family. Two targets are present in the observation window, one at 40 dB, the other at 15 dB. Blue × represent real positions of the targets, magenta ◦ represent targets extracted with the OMP algorithm, and black ⋄ represent targets extracted with the modified OMP algorithm.

Fig. 3. Range/angle output of the MIMO processing at reception. The transmitted signals are binary phase codes of length 127 from the Gold family. Four targets are present in the observation window, at random SNR in [20, 40] dB. Blue × represent real positions of the targets, magenta ◦ represent targets extracted with the OMP algorithm, and black ⋄ represent targets extracted with the modified OMP algorithm.

in the observation window. The resulting estimated parameters are shown in Figure 3. Since drawn targets are not so strong, the OMP algorithm provides only 8 detections when the modified OMP properly recovers the 4 targets. In both cases, we see that the modified OMP managed to recover all targets, while the OMP algorithm often provides duplicated peaks for strong targets due to the grid problem.

For the simulation, we compared a classic processing performing the MIMO matched filter with the OMP algorithm and the modified OMP algorithm that solves the grid problem. First for illustration purpose we consider two scenarios. The first one contains two targets, a strong one with SNR equal to 40 dB and a weaker one with SNR equal to 15 dB that is buried in the sidelobes of the first target, since the ambiguity function for the chosen family presents sidelobes that spans the whole ambiguity domain with a mean level around −20 dB. The resulting estimated parameters are shown in Figure 2 together with the representation of the range/angle received signal: two targets are present, the OMP algorithm provides 14 detections due to badly rejected strong target while the modified OMP properly recovers the 2 targets. The second scenario contains four targets of random SNR uniformly drawn in the interval [20, 40] dB, and with position uniformly drawn 978-1-4799-4195-7/14/$31.00©2014IEEE

We performed Monte Carlo simulations to estimate the detection performance of the proposed algorithm for a weak target buried in the sidelobes of a stronger target. For this purpose, we considered a scenario with two targets, one strong target with SNR equal to 40 dB and a weak target with SNR ranging from 5 to 20 dB. For each SNR value, NMC = 100 Monte Carlo simulations were run, and we applied the classic detection processing, the OMP and the modified OMP. The detection threshold is computed for a theoretical false alarm probability of 10−6 for each hypothesis. Results are stated in terms of detection probability for the weak target (the strong 5


Fig. 4. Detection probability versus SNR for a weak target with SNR spanning the interval [5, 20] dB, buried into the sidelobes of a strong target at SNR=40 dB.

Fig. 5. Mean number of false alarms on the whole observation window versus SNR for a weak target with SNR spanning the interval [5, 20] dB, buried into the sidelobes of a strong target at SNR=40 dB.

one was always detected) and of false alarm rate. The measured detection probability is presented in Figure 4 and the false alarm rate in Figure 5 as the mean number of false alarms over the whole observation window. We can see that the weak target is indeed detected by both OMP algorithm while the classic processing cannot detect it because it is buried in the sidelobe of the strong target. However we can also see that the OMP algorithm provides a much higher false alarm rate than the modified OMP, which can be explained by the fact that the OMP algorithm does not reject well the strong target so that some artefacts remain. We can also remark that the number of false alarms remains relatively constant with the SNR for the modified OMP algorithm, which is expected if both targets are correctly rejected by the algorithm. V.

[2] A. De Maio and M. Lops. Design principles of mimo radar detectors. Aerospace and Electronic Systems, IEEE Transactions on, 43(3):886 –898, 2007. [3] J. Li and P. Stoica. MIMO Radar - Diversity means Superiority. In 14th Annual Workshop on Adaptive Sensor Array Processing, 2006. [4] J. Li and P. Stoica. MIMO Radar with Colocated Antennas. IEEE Signal Processing Magazine, pages 106–114, 2007. [5] C.Y. Chen and P.P. Vaidyanathan. MIMO Radar Ambiguity Properties and Optimization Using Frequency-Hopping Waveforms. IEEE Trans. on Signal Processing, 56(12):5926–59368, 2008. [6] Y.I. Abramovich, G.J. Frazer, and B.A. Johnson. Noncausal Adaptive Spatial Clutter Mitigation in Monostatic MIMO radar: Fundamental Limitations. IEEE Journal of Selected Topics in Signal Processing, 4(1):40–54, 2010. [7] G. San Antonio, D.R. Fuhrmann, and F.C. Robey. MIMO Radar Ambiguity Functions. IEEE Journal of Selected Topics in Signal Processing, 1(1):167–177, 2007. [8] C.-Y. Chen. Signal Processing Algorithms for MIMO Radar. PhD thesis, California Institute of Technology, 2009. [9] O. Rabaste and L. Savy. Quadratically Constrained Quadratic Programs for Mismatched Filter Optimization with Radar Applications. submitted to IEEE on Aerospace and Electronic Systems. [10] S. Mallat and Z. Zhang. Matching Pursuits with Time-Frequency Dictionaries. IEEE Trans. on Signal Processing, 41(12):3397–3415, 1993. [11] Y. Chi, A. Pezeshki, L. Scharf, and R. Calderbank. Sensitivity to basis mismatch in compressed sensing. In Proc. of 2010 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 3930–3933, 2010. [12] R. Jagannath, G. Leus, and R. Pribic. Grid Matching for Sparse Signal Recovery in Compressive Sensing. In Proc. of the 9th European Radar Conference,, pages 111–114, 2012. [13] O. Rabaste, L. Savy, M. Cattenoz, and J.-P. Guyvarch. Signal Waveforms and Range/Angle Coupling in Coherent Colocated MIMO Radar. In 2013 International Conference on Radar, 2013. [14] G. Davis, S. Mallat, and Z. Zhang. Adaptive time-frequency decompositions with matching pursuits. Optical Engineering, 1994. [15] S.M. Kay. Fundamentals of Statistical Signal Processing. Detection Theory. Prentice Hall, 1998. [16] M. Minoux. Programmation mathmatique, thorie et algorithmes, Tome 1. Dunod, 1983. [17] R. Gold. Optimal binary sequences for spread spectrum multiplexing. IEEE Trans. on Information Theory, 13(4):619–621, 1967. [18] L. Aouchiche, G. Desodt, L. Ferro-Famil, R. Kassab, and O. Rabaste. Adaptive Grid OMP technique for the detection of low reflectivity moving objects in the presence of Doppler ambiguities. In Radar Conference,, 2014.

C ONCLUSION

We have considered in this article the problem of detecting multiple targets in MIMO radar. Since the MIMO ambiguity function presents strong sidelobes that may span part or the totality of the ambiguity domain, a classic simple matched filter processing does not permit to detect weak targets buried in the sidelobes of stronger ones. We have therefore proposed a specific processing to solve this problem. The proposed solution consists in approximating the multitarget matched filter thanks to a procedure similar to the OMP algorithm. We have shown that the stopping criteria for this iterative procedure can be set in order to approximately satisfy a given false alarm rate. We have also proposed a modification of this algorithm that permits to deal with the grid problem. Simulations permitted to show that the proposed method can indeed detect weak targets masked by stronger ones. The proposed method could be adapted not only to the MIMO framework, but also to other applications where strong sidelobes or ambiguities can arise, for instance doppler ambiguities [18]. Note that this paper presents first results to validate the principle of OMP approach to MIMO radar processing. Future work will require to study the robustness of the approach against model mismatch, high density of targets and very close targets. R EFERENCES [1]

E. Fishler, A. Haimovich, R.S. Blum, Jr. Cimini, L.J., D. Chizhik, and R.A. Valenzuela. Spatial diversity in radars-models and detection performance. Signal Processing, IEEE Transactions on, 54(3):823 – 838, 2006.

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