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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011
Sparsity-Based Multi-Target Tracking Using OFDM Radar Satyabrata Sen and Arye Nehorai, Fellow, IEEE
Abstract—We propose a sparsity-based approach to track multiple targets in a region of interest using an orthogonal-frequency-division multiplexing (OFDM) radar. We observe that in a particular pulse interval the targets lie at a few points on the delay-Doppler plane and hence we exploit that inherent sparsity to develop a tracking procedure. The use of an OFDM signal not only increases the frequency diversity of our system, as different scattering centers of a target resonate variably at different frequencies, but also decreases the block-coherence measure of the equivalent sparse measurement model. In the tracking filter, we exploit this block-sparsity property in developing a block version of the compressive sampling matching pursuit (CoSaMP) algorithm. We present numerical examples to show the performance of our sparsity-based tracking approach and compare it with a particle filter (PF) based tracking procedure. The sparsity-based tracking algorithm takes much less computational time and provides equivalent and sometimes better, tracking performance than the PF-based tracking. Index Terms—Block compressive sampling matching pursuit (CoSaMP), block sparse, delay-Doppler sparsity, multitarget tracking, orthogonal-frequency-division multiplexing (OFDM) radar.
I. INTRODUCTION For a number of years, the problem of simultaneous detection and tracking of multiple targets has been one of the most relevant and challenging issues in a wide variety of military and civilian systems [1], [2]. Multi-target tracking is of primary interest in many applications, such as radar tracking of airborne or ground moving vehicles, sonar tracking of submarines, tracking of people for security purposes and mobile robotics. The situation becomes even more complicated when the tracks of two targets cross each other. In this work (see also [3]), we look into the multitarget tracking problem from a different perspective. We observe that a multitarget scene is generated by keeping track of the range and velocity (delay and Doppler, respectively) of each target over time. Suppose we discretize the delay-Doppler plane into 2 grid points. Then, , the target scene will be sparse if number of targets in the delay-Doppler plane. This enables us to efficiently track the targets by solving a simple sparse-recovery algorithm through a linear program, e.g., 1 -minimization [4] or second-order cone programming (SOCP) [5], or by using a greedy pursuit, e.g., orthogonal matching pursuit (OMP) [6] or compressive sampling matching pursuit (CoSaMP) [7]. First, in Section II, we present a state model describing the dynamic behavior of the targets. Then, we develop a parametric measurement model considering an orthogonal frequency division multiplexing (OFDM) radar. We use an OFDM radar for two reasons: i) it increases the frequency diversity of the system, because different scattering centers of the targets resonate at different frequencies and
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Fig. 1. A schematic representation of a multitarget tracking scenario (not drawn to scale).
ii) it decreases the block-coherence value of the system matrix of the equivalent sparse-measurement model. Next, by exploiting the sparsity on the delay-Doppler plane, we convert the OFDM measurement model to an equivalent sparse measurement model in Section III. The nonzero components of the sparse vector correspond to the scattering coefficients of the targets. However, due to the use of a multicarrier OFDM signal, each target produces multiple scattering coefficients, not a single value. Hence, our sparse vector exhibits an additional structure in the form of the nonzero coefficients occurring in clusters. Such vectors are referred to as block-sparse [8], [9]. For efficient sparse-recovery, we evaluate the block-coherence measure [10] of the associated measurement matrix. We also prove that the minimum value of block-coherence is attainable when equal amounts of energy are transmitted over the available OFDM subcarriers and further prove that the minimum value is inversely proportional to the number of OFDM subcarriers. Hence, this reconfirms the advantage of using OFDM signals. As pointed out in [8], the conventional CoSaMP algorithm provides two benefits. It ensures speedy and robust recovery and provides tight error bounds by including the ideas from the combinatorial algorithms [7], [11]. Further, it has a simple, iterative greedy structure that can be modified easily to incorporate the block-sparsity nature of the sparse vector, instead of treating it as a conventional sparse vector and thereby ignoring the additional structure in the problem. Therefore, in Section IV, we propose to employ a block version of the CoSaMP algorithm, termed BCoSaMP, in the tracking filter. To illustrate the potential of our sparsity-based tracking method, we present several numerical examples in Section V. At each pulse interval, we dynamically partition a smaller portion of the delay-Doppler plane, depending on the predicted state parameters. We compare the resulting tracking performance with that of a particle filter (PF) based tracking procedure. Our results show that the sparsity-based tracking algorithm not only takes much less time (about one order less) than the PF-based tracking procedure, but also achieves equal (and sometimes better) tracking performance. Conclusions and possible future work are presented in Section VI. II. PROBLEM DESCRIPTION AND MODELING
Manuscript received February 23, 2010; revised June 16, 2010, December 08, 2010; accepted December 14, 2010. Date of publication December 30, 2010; date of current version March 09, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Visa Koivunen. This work was supported by the Department of Defense under the AFOSR MURI Grant FA9550-05-1-0443 and the ONR Grant N000140810849. The authors are with the Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO 63130 USA (e-mail:
[email protected];
[email protected]). 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/TSP.2010.2103064
Fig. 1 presents a schematic representation of the problem scenario. We consider an OFDM radar system that overlooks a region of interest containing multiple moving targets. We assume that the targets are at far-field with respect to the radar, i.e., the relative distance between any two targets is much smaller than their individual distances with respect to the radar. Hence, at a particular pulse interval all the targets can be associated with the same direction-of-arrival (DOA) unit vectors ^ . In the following, we first present a dynamic state model for target tracking. Then, we develop a parametric OFDM radar signal model accounting for different measurements over multiple frequencies.
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011
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A. Target Dynamic Model
static clutter returns, co-channel interference and measurement noise at baseband. Here, denotes the number of temporal samples per pulse transmission, covering the range of delays corresponding to a region of 1 1 1 N 01 ). The sampling interval interest ( p 0 1 or time resolution 1 i 0 i01 depends on the bandwidth of the signal transmitted by the radar, i.e., 1 = 1 (2 ) = p (2( + 1)). Furthermore, assuming p (1 + km ) p , the indicator function m km + p ] in the discrete-time domain. [ ] ( n ) is nonzero over [ k Hence, for each target there will be s p 1 = 2( +1) temporal samples corresponding to target returns plus clutter/interference and noise. For example, in (5) the th target responses will be found at = m m +1 . . . m + s 01, where m = d km 1 e0 0 1 . Stacking the measurements of all subchannels and temporal points into a long column vector of length , we obtain the OFDM measurement model at the th pulse interval as follows:
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