ABSTRACT. Compressed sensing for MIMO radar can potentially en- hance spatial resolution and improve anti-jamming capability by virtue of multiple ...
2012 IEEE Statistical Signal Processing Workshop (SSP)
COMPRESSED SENSING FOR MIMO RADAR: A STOCHASTIC PERSPECTIVE Zhi Tian
Erik Blasch
Department of Electrical and Computer Engineering Michigan Technological University Houghton, MI 49931 USA
Air Force Research Laboratory Rome Research Site Rome, NY 13441 USA
ABSTRACT Compressed sensing for MIMO radar can potentially enhance spatial resolution and improve anti-jamming capability by virtue of multiple transmitter and receiver antennas, and at the same time reduces the number of samples needed by making use of the inherent sparsity property of most radar scenes. Existing work along this line adopts a deterministic model for the radar signals, which may not be effective to cope with fading propagation and signal correlation in practical scenarios. This paper takes a stochastic approach by modeling the target scenes as random processes that are possibly correlated. A new stochastic framework of compressed sensing for MIMO radar is developed for reconstructing useful statistics of the random target scenes using a small number of samples. The proposed approach directly extracts the useful statistics for estimation without reconstructing the random signals; as a result, it is computationally more efficient and requires a smaller number of samples than existing deterministic approach to compressed sensing. Index Terms— compressed sensing; MIMO RADAR; target scene estimation; second-order statistics 1. INTRODUCTION In MIMO radar, multiple transmitter-antennas are employed to shape the transmit beams for diversity, and multiple receiver-antennas are used to coherently collect the reflected signals for high signal-to-noise ratio (SNR) gain [1]. Compared with conventional radar, the use of multiple antennas and waveform shaping offers enhanced spatial resolution and improved capability in interference and jamming suppression. Further, compressed sensing techniques have been used for MIMO radar to reduce the cost of data collection, by exploiting the sparsity features inherent in most radar scenes [2, 3, 4]. In existing framework of compressed sensing for MIMO radar, the signal sources of interest are assumed to be unknown but deterministic with constant strength, and multiple number of samples are collected across multiple receiver antennas over time to identify these sources. This is a deterministic approach to radar scene modeling, which is essential
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for existing compressed sensing. It renders the measurement systems to be linear in the sparse radar scenes of interest, such that `1 -norm regularized least squares (LR-LS) formulations can be adopted to find sparse solutions in a computationally feasible manner [5, 6]. However, deterministic modeling can be overly simplistic for practical scenarios. There can be fading or dispersion in signal propagation, especially for distributed sensor systems; also, the signal sources can be correlated, either by nature or as a result of some modeling practice. In those cases, it is more appropriate to treat target sources as random and characterize them by their statistical properties [7]. However, it is not straightforward to deal with random radar scenes using existing framework of compressed sensing. This paper takes a stochastic perspective on radar scene modeling, in order to deal with fading and signal correlation. The radar scene is modeled as a random field, based on which a new framework of compressed sensing for MIMO radar is developed. The stochastic approach directly reconstructs useful statistics for target parameter estimation, without having to recover the random field itself. In fact, it aims to reconstruct the received data covariance matrix, which forms the basis for a number of powerful array processing techniques such as MVDR and MUSIC [7]. A main technical hurdle in doing so is that the desired second-order statistics do not have a linear relationship with the compressive measurements in time domain, making it impossible to directly apply LR-LS formulations. To overcome this difficulty, this paper utilizes the cross-correlations of received samples as the transformed compressive measurements, which turn out to relate linearly with the second-order statistics of interest [8, 9]. The developed stochastic approach to compressed sensing for MIMO radar considerably reduces the computational load as well as the number of samples needed. When proper transmit waveforms are designed, it is even possible to extract desired statistics of non-sparse radar scenes from compressive samples. 2. SIGNAL MODEL Consider a MIMO radar with NT transmit-antennas and NR receive-antennas. For clear exposition the antenna arrays are
assumed to be uniformly spaced, with spacing dT and dR at the transmitter and receiver sides respectively. Correspondingly, the array manifolds are given by [7] iT h 2π 2π aT/R (θ) = 1, e−j λ dT/R sin θ , . . . , e−j λ (NT/R −1)dT/R sin θ (1) where λ is the wavelength, θ is the impinging angle of arrival of the target of interest, and the subscript T /R indicates transmitter/receivers. Suppose that the m-th transmit-antenna emits the signal um (t)ej2πfc t , where fc is the carrier frequency. The n-th receiving antenna receives the signal yn (t)ejπfc t , which is the superposition of the reflections from L point targets. The l-th target impinges on the receiving antennas with Doppler frequency fD,l , delay τl and angle θl . The baseband equivalent signal received at the n-th antenna is given by yn (t) =
L X
sl (t)aTT (θl )u(t − τl )ej2πfD,l t ej
2π λ ndR
sin θl
(2)
l=1
n = 0, . . . , NR − 1 where sl (t) is the radar cross section (RCS) of the l-th target at time t, u(t) = [u0 (t), . . . , uNT −1 (t)]T stacks the transmitted signals from all NT antenna elements, and superscript T denotes transpose. Sampling yn (t) every Ts seconds for a time window of length KTs , the received sampling vector at time t = kTs across NR antenna outputs is denoted as y(k) = [y0 (kTs ), y1 (kTs ), . . . , yNR −1 (kTs )]T .
(3)
From {y(k)}k , the aim is to estimate the angle-delay-Doppler triplet (θl , τl , fD,l ) of each target. A major challenge in target parameter estimation is that the measurements in (2) are nonlinear functions of the wanted unknowns {(θl , τl , fD,l )}L l=1 ; besides, the number of targets L is also unknown and may vary over time. A neat approach to arrive at a linear measurement model is to adopt a set of G (possibly regularly spaced) 3D grid points at known positions (θ˜g , τ˜g , f˜D,g ) of the angle-delay-Doppler space, where the target(s) could be potentially located [2]. Using a sufficiently dense grid [3, 4], it is possible to capture the target scene at a prescribed resolution using a G × 1 vector x(k) = [x1 (k), . . . , xG (k)]T , where its g-th entry is nonzero only if a target resides at the corresponding grid point at time k. If all targets fall exactly on the grid, then the number of nonzero elements in x(k) is exactly the number of targets L, and the nonzero values in x(k) are given by the corresponding {sl (kTs )}l . However, to account for target presence off the preselected grid points, we allow for the unknown target signal strength sl to “spill over” grid points around g and thus render nonzero a few neighboring entries. With the grid model, the received signal in (2) can be well approximated as yn (k) =
G X
xg (k)hn,g (k; θ˜g , τ˜g , f˜G,g )
(4)
g=1
549
with hn,g (k; θ˜g , τ˜g , f˜G,g ) 2π ˜ ˜ = aT (θ˜g )u(kTs − τ˜g )ej2πfD,g kTs ej λ ndR sin θg T
where hn,g is known because its values are parameterized by the known grid locations {(θ˜g , τ˜g , f˜D,g )}G g=1 . Organizing hn,g into an NR × G matrix H(k), the grid-based measurement model becomes y(k) = H(k)x(k),
k = 0, . . . K − 1.
(5)
Here, we view y(k) as compressive measurements of the gridbased RSC signal x(k), since the data size NR is typically much smaller than the number of unknowns G. The generation of the compressive measurements involves the use of waveform signals uk to form proper H(k), which is reminiscent of the random codes used in compressive samplers [5, 6]. Our goal is to find the nonzero support of x(k), whose corresponding indices g reveal the existence of targets and the associated angle-delay-Doppler triplets (θ˜g , τ˜g , f˜D,g ) [2]. 3. COMPRESSED SENSING BASED ON DETERMINISTIC MODELING The grid-based RCS x(k) is typically sparse over the angledelay-Doppler space, because targets are sparsely located. In existing framework, the target sources are treated as deterministic and time-invariant [2, 3, 4], that is, x(k) = xG ,
∀k.
(6)
Under this assumption, all the measurements across time can be concatenated into an NR K × 1 vector to reconstruct the common RCS xG of length G, using the following model: H(0) y(0) .. .. (7) xG . = . . |
y(K − 1) {z } y
|
H(K − 1) {z } H
The dimension of H is NR K × G, which can be underdetermined if G is large while the time window KTs is short. Nonetheless, it is still possible to recover the sparse unknown xG , provided that H satisfies certain conditions on mutual incoherence. The transmitter waveform design of um (k) is essential in enabling the desired properties of H, which is one of the main advantages of MIMO radar over traditional radar [3, 4]. The reconstruction algorithm takes the form of `1 -norm regularized least squares (LR-LS) [5, 6], as follows: ˆ G = arg min ky − HxG k22 + λkxG k1 . x xG
(8)
Here, the first LS term accounts for measurement errors, the second `1 -norm term imposes a sparse solution, and the scalar weight λ balances the tradeoff in estimation variance and bias.
However, when the signal propagation experiences fading, time variation or dispersion, (6) no longer holds, and the measurement model becomes x(0) H(0) · · · 0 y(0) .. .. .. .. .. . = . . . . . |
y(K −1) {z } y
0 |
x(K −1) · · · H(K −1) {z }| {z } HD
x
(9) For (9), direct application of the existing compressed sensing framework is not necessarily effective for several reasons: i) The size of the wanted unknowns x grows linearly in time, at the same rate as the growth of the measurement size; as a result, the data collection cost grows in time, so does the computational complexity. ii) The composite measurement matrix HD is block diagonal, whose mutual incoherence property is not desirable [6]. iii) {x(k)}k has a similar sparsity structure across time k, but this property cannot be utilized in a straightforward manner when x is recovered from y.
um (k) = um (k + nKT ) for any integer n, ∀m, k. The resulting H(k) becomes periodic too, with period KT . The value of KT is chosen to balance between the computational load and the reconstruction capability. Because of the periodic transmission and the block stationary assumption on x(k), the received sample vectors y(k) are random and cyclostationary with cyclic period KT . Let us organize every KT sample vectors into yKT (n) = [yT (nKT ), . . . , yT (nKT + KT − 1)]T
which is a stationary vector sequence with autocorrelation ˜ y (ν) = E{yK (n)yT (n−ν)} = [Ry (k, νKT +k−l)](k,l) R KT T (11) of size NR KT × NR KT . It is uniquely determined by subblocks Ry (k, l) = E{y(nKT + k)yT (nKT + k − l)} of size KT × KT , where k is the time origin and l is the timelag. These autocorrelation matrices can be estimated as their finite-sample averages, as follows: ˆ y (k, l) = 1 R K
4. COMPRESSED SENSING BASED ON STOCHASTIC MODELING To cope with fading, dispersive, or correlated propagation channels, this paper takes on a stochastic perspective on the signal model for x(k). We assume that the vector sequence x(k) is a wide-sense stationary random process across time1 . As a result, what is of interest is not the set of realizations {x(k)}k , but its useful statistics such as the data covariance matrix Rx = E{x(k)xT (k)} (E{·} denotes expectation). The matrix Rx has sparse nonzero elements due to the inherent sparsity of radar scenes. In fact, the indices of the nonzero entries on the diagonal of Rx indicate the angledelay-Doppler triplets to be estimated. Meanwhile, since we allow spill-over in the grid-based RCS modeling to accommodate the finite grid resolution, Rx may have sparse off-diagonal components to indicate the correlation of target scene across nearby grid points. Essentially, the RCS reconstruction problem boils down to extracting the sparse second-order statistics of interest Rx of size G × G, from the available time-domain measurements {y(k)}k . Because there is no obvious linear relationship between the desired quantity Rx and the available data {y(k)}, a standard LR-LS formulation such as (8) is not immediately applicable. 4.1. Stationary Input and Cyclostationary Output To effectively reconstruct Rx from compressive samples {y(k)}k , we design the transmitter waveform pattern such that um (k) repeats for every KT (< K) samples, that is, 1 It
is assumed that the target scene is time-invariant within each observation time-block KTs , but may vary from block to block. This can be relaxed to accommodate time-varying channels and moving targets, for which a future work is to develop adaptive versions of the proposed algorithm.
550
(10)
bK/KT c
X
y(nKT + k)yT (nKT + k − l). (12)
n=0
Now, we aim to reconstruct Rx from the transformed ˜ y (ν)}ν (or equivalently {R ˆ y (k, l)}k,l ). measurements {R For simplicity we assume Rx (ν) = 0, ∀ν 6= 0, while the more general case of time-correlated radar scenes can be treated in a similar manner [9]. As a result, we may focus ˜y = R ˜ y (0), which contains adequate information of on R Rx = Rx (0). It is worth noting that the input x(k) is widesense stationary, while the output y(k) is cyclostationary due to the time-varying yet periodic transmit-waveforms u(k). This is advantageous to our reconstruction problem, because the number of unknowns in Rx is fixed given G, while the ˜ y grows when the number of transformed measurements in R period KT increases. 4.2. Linear Relationships The reconstruction problem can benefit from some explicit ˜ y . These linear relationships among the entries in Rx and R linear relationships form the basis for the reconstruction process, which will be developed in the ensuing subsection. Because of the assumption Rx (ν) = 0, ∀ν 6= 0, the time-varying cross-correlation of y(k) can be derived from (5) straightforwardly as H(k)Rx H(k)T , l = 0 Ry (k, l) = . (13) 0, l 6= 0 Defining HKT = diag{H(0), . . . , H(KT − 1)}, it holds that ˜ y = HK Rx HT R KT T ˜ y is block diagonal. where R
(14)
To facilitate reconstruction, we organize the elements in Rx into a G2 × 1 vector rx = vec{Rx }
(15)
where the operator vec{·} stacks all the columns. Evidently, Rx is uniquely determined by rx and vice versa. Similarly, ˜ y into a we organize all the block diagonal terms inside R KT NR2 × 1 vector ry˜, that is, T ry˜ = (vec{Ry (0, 0)})T , . . . , (vec{Ry (KT − 1, 0)})T . (16) ˜ y } and ry˜ are composed of the same set of Because vec{R autocorrelation terms, there is a one-on-one mapping between these two vectors, expressed as ˜ y} ry˜ = Pvec{R 2
2
(17) 2
where P is a specific {0, 1}KT NR ×KT NR permutation matrix for the mapping operation [8]. Vectorizing on both sides of (14), and using the property vec{aT Xb∗ } = (bH ⊗ aT )vec{X} (⊗ denotes Kronecker product), it can be derived from (17) that ry˜ = P(HKT ⊗ HKT )vec{Rx } = Arx KT NR2
(18) 2
× G . This where A = P(HKT ⊗ HKT ) is of size linear expression is made possible by employing the transformed measurements ry˜ rather than the original compressive measurements y(k). It is essential for reconstructing Rx in a computationally feasible manner. 4.3. Reconstruction of Sparse Radar Scene Having derived the linear relationship in (18), we can reconstruct the unknown rx using LR-LS to make use of its sparsity property. In practice, ry˜ as ensemble statistics is not available, but is replaced by its time-averaged estimate ˆry˜ indicated in (12). Overall, the reconstruction is performed as ˆrx = arg min kˆry˜ − Arx k22 + λkrx k1 . rx
(19)
With the estimated ˆrx , the data covariance matrix Rx can be constructed by re-stacking ˆrx into a G × G square matrix. Nonzero diagonal elements of Rx indicate the RCS of active targets, while their indices g reveal the target parameter estimates as (θ˜g , τ˜g , f˜D,g ). The reconstruction feasibility and accuracy are determined by A of size KT NR2 × G2 , which concerns the design of the transmit waveforms with respect to the grid resolution. When KT NR2 ≥ G2 , it is possible that A is full rank, such that the linear system in (18) may be solved by simple LS without imposing the sparsity constraints. This means that it is possible to reconstruct the useful statistics of a nonsparse random target scene via the stochastic approach to compressed sensing, by increasing KT to be greater than (G/NR )2 . This is an interesting observation, even though practical radar scenes almost always entail sparsity features.
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5. SUMMARY This paper provides a stochastic perspective on compressed sensing for MIMO radar, which differs considerably from the existing deterministic approach. By treating the radar scenes of interest as random processes, the proposed scheme can effectively reconstruct the useful statistics for target parameter estimation in practical scenarios where the target signals may be faded or correlated. As a key step in algorithm development, the autocorrelations of compressive samples are employed as the transformed measurements, and its linear relationship with the autocorrelations of the grid-based RCS signal of interest is derived. As future work, this approach will be extended to track time-varying target scenes via adaptive implementation. Also, the transmit waveform design of um (t) will be studied to induce desired mutual incoherence property for the transformed matrix A in (18), in order to offer desired reconstruction performance for the developed stochastic approach to compressed sensing for MIMO radar. 6. REFERENCES [1] A. M. Haimovich, R. S. Blum, and L. J. Cimini, “MIMO radar with widely separated antennas,” IEEE Signal Process. Mag., vol. 25, pp. 116129, Jan. 2008. [2] C.-Y. Chen, P. P. Vaidyanathan, “Compressed sensing in MIMO radar,” Asilomar Conference on Signals, Systems and Computers, pp. 41– 44, 2008. [3] T. Strohmer, B. Friedlander, “Compressed sensing for MIMO radar – algorithms and performance,” Asilomar Conference on Signals, Systems and Computers, pp. 464-468, 2009. [4] Y. Yu, A. P. Petropulu, and H. V. Poor, “Compressive sensing for MIMO radar,” IEEE ICASSP Conf., pp. 30173020, April 2009. [5] E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Processing Magazine, vol. 25(2), pp. 21–30, March 2008. [6] D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52(4), pp. 1289–1306, April 2006. [7] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part IV, Optimum Array Processing, John Wiley & Sons, 2002. [8] Z. Tian, Y. Tafesse, and B. M. Sadler, “Cyclic Feature Detection from Sub-Nyquist Samples for Wideband Spectrum Sensing,” IEEE Selected Topics in Signal Processing, Special Issue on Robust Measures and Tests Using Sparse Data for Detection and Estimation, February 2012. [9] G. Leus, Z. Tian, “Recovering Second-Order Statistics from Compressive Measurements,” IEEE CAMSAP Conf., pp. 337-340, Dec. 2011.
