On Waveform Design for MIMO Radar with Matrix Completion

Transcription

1 On Waveform Design for MIMO Radar with Matrix Completion Shunqiao Sun and Athina P. Petropulu ECE Department, Rutgers, The State University of New Jersey, Piscataway, NJ, {shunq.sun, Abstract In matrix completion based MIMO radars, the data matrix coherence, and consequently the performance of matrix completion depend on the transmit waveforms. It was recently shown that for uniform linear arrays and orthogonal waveforms, the optimal choice for waveforms are white-noise type functions. This paper deals with the design of optimal transmit waveforms. The design is formulated as an optimization problem on the complex Stiefel manifold, which is a non-convex set, and the optimal waveforms are found via a steepest descent algorithm with nonmonotone line search methods. Numerical results show that for large dimensional data matrices and for a large number of antennas, the objective function converges to its global minimum and the matrix coherence corresponding to the optimal waveforms is asymptotically optimal, thus resulting in very good target estimation performance. Index Terms MIMO radar, matrix completion, waveform design, optimization on manifolds, orthogonal constraints I. INTRODUCTION The idea of using low-rank matrix completion techniques in MIMO radars (MIMO-MC radars) was first proposed in [], [2] as means of reducing the volume of data required for accurate target detection and estimation. As compared to MIMO radars based on sparse signal recovery [3], [4], [5], MIMO-MC radars achieve similar performance but without requiring a target space grid. In standard MIMO radars, each receive antenna samples the target returns and forwards the obtained samples to a fusion center. Based on the data received, the fusion center formulates a matrix, referred to here as the data matrix, which can then be used in standard array processing methods for target detection and estimation. For a small number of targets relative to the number of transmission and reception antennas, the aforementioned data matrix is lowrank [] and thus can be recovered from a small subset of uniformly at random sampled entries via matrix completion (MC) techniques[6], [7], [8]. By exploiting the latter fact, the MIMO-MC radar receive antennas obtain a small number of samples at uniformly random sampling times, and forward the samples to the fusion center, which, based on knowledge of the sampling instances of all antennas, populates the data matrix in a uniformly sparse fashion. Subsequently, the fusion center recovers the full matrix via MC. This is referred to as Scheme II in [2]. Alternatively, the receive antennas can perform matched filtering with a randomly selected set of transmit This work was supported by ONR under Grant N and NSF under grant ECCS waveforms, and forward the results to the fusion center, which can then formulate the data matrix and subsequently perform MC. This is referred to as Scheme I in [2]. Details on the general topic of matrix completion and condition for matrix recovery can be found in [6], [7], [8]. The conditions for the applicability of MC on Scheme I of [2] under uniform linear arrays (ULA) can be found in [9]. In that case, and under ideal conditions, the transmit waveforms do not affect the MC performance. On the other hand, for Scheme II of [2], it was shown in [2], [0] that the transmit waveforms affect the matrix completion performance, as they directly affect the data matrix coherence [7]; a larger coherence implies that more samples need to be collected for reliable target estimation. That observation motivates the design of optimal waveforms for Scheme II of [2], which is the topic of this paper. In MIMO radars, orthogonal transmit waveforms are often used as they achieve a higher degree of freedom []. In [2], it was shown that, among the class of orthogonal waveforms, and for MIMO radars using ULAs, the optimal waveforms are white noise-type functions. In the same paper it was also shown that white-noise type waveforms result in the smallest possible matrix coherence, i.e.,, for sufficiently large number of transmit and receive antennas. This papers, building on the observations of [2], proposes a scheme for designing optimal transmit waveforms for MIMO-MC radars. In particular, the waveform design problem is formulated as optimization on matrix manifolds [3]. Due to the orthogonality constraint, the matrix manifold is the complex Stiefel manifold, which is non-convex. The solution is obtained via the steepest descent algorithm [4] with a nonmonotone line search method [5]. For the iteration, the derivative of the objective function w.r.t. the waveform matrix is obtained in a closed form. Simulation results show that the objective value converges to its global minimum for large dimensional data matrices, and that under the optimized waveforms, the matrix coherence approaches its smallest value,, as the number of antennas increases. A. MIMO-MC Radar II. SYSTEM MODEL AND COHERENCE The scenario considered here involves narrowband orthogonal transmit waveforms, with pulse repetition interval T PRI, and carrier wavelengthλ. There arek targets at anglesθ k, and ULAs equipped with M t transmit and M r receive antennas, /4/$ IEEE 473

2 respectively, are used for transmission and reception. The data matrix,x, is formulated based on the receive antenna samples, with each antenna contributing a row to it. Suppose that the receive antennas sample the target returns at Nyquist rate. It holds that X = W+J, where J is interference/noise and W = BDA T S T, () where A C Mt K is the transmit steering matrix with A mk = e j2π(m )αt k,(m,k) N + M t N + K, with αk t = d t sin(θ k ) λ and d t denoting the inter-element distance for the transmit array; B C Mr K, is the receive steering matrix, defined in a similar fashion as A; D C K K is a non-zero diagonal matrix defined as D =diag ([ ]) β ζ β 2 ζ 2 β K ζ K, with ζk = e j 2π λ 2ϑ k(q )T PRI. The sets {β k } k N +,{ϑ k } K k N + contain the K target reflection coefficients and speeds, respectively, and q denotes the pulse index; S is the sampled transmit waveform matrix, defined as S =[s(0t s ),...,s((n )T s )] T C N Mt, where s(τ)=[s (τ),...,s Mt (τ)] T, with T s and N respectively denoting the Nyquist sampling period and number of samples in one pulse. For orthogonal waveforms, it holds that S H S = I Mt [6]. When both M r and N are larger than K, the noise free data matrix W is rank-k [2] and thus can be recovered from a small number of its entries via matrix completion. This observation leds to Scheme II of [], [2], in which the receive antennas forward samples obtained at random times, thus partially filling the data matrix. B. Optimal Waveforms and Coherence Bounds Let us consider a ULA configuration with d t = λ/2. Itwas shown in [2] that, for orthogonal waveforms, the optimum waveforms should be white-noise type functions, i.e., they should sufficiently satisfy: ( ) S i α t n 2 S = (i) A (θ n ) 2 = M t 2 N,i N+ N, (2) where θ n is every angle in the angle space [ π 2, 2] π, which corresponds to αn t [ 2, 2] ; Si (αn) t is the DTFT of the i-th row of matrix S, evaluated at frequency αn; t and S (i), i N + N denotes the i-th row of S. For h {t,r}, let us define the following function sin 2 (πm h x) β ξh (M h )= sup x [ξ h, sin 2] 2, (3) (πx) where ξ h = min g ( α h i α h ) (i,j) N + K N+ K,i j j and g(x) = { x x, x x 2. Let U,V denote the left and right x x,otherwise subspaces of W and μ(u) and μ(v) the corresponding coherence (see definitions in [6]). The following Lemma holds. Lemma. [2] Under the orthogonal waveforms that meet the optimal condition of (2), the coherence μ(v) is bounded as M t μ(v) M t (K ) (4) β ξt (M t ) with β ξt (M t ), as defined in (3). Additionally, it holds that [9] μ(u) M r M r (K ) β ξr (M r ). (5) If ξ =min{ξ r,ξ t } 0, then for any fixed K, if both M t and M r are larger than K β ξ = K sin(πξ) = O(K), both (4) and (5) hold with constant β ξ which is independent of M t and M r. In the limit w.r.t. M t and M r, we have μ(v)=and μ(u)=, which are the lowest possible values. III. TRANSMIT WAVEFORM OPTIMIZATION ON A. Problem Formulation MANIFOLDS Let us discretize the angle space [ π 2, 2] π into L angles, i.e., θ l, l N + L, and let c il = S (i) A (θ l ). According to the optimal condition (2), it holds that c il 2 = Mt N for i N+ N and l N + L. Define A =[A (θ ),...,A (θ L )] and F = S A. It holds that[f F ] il = c il 2 where denotes the Hadamard product. Based on (2), let us define the objective function f (S)= F F M 2 t N N T L, (6) and formulate the waveform design problem as min f(s) s.t. S H S = I Mt. (7) The orthogonal constraint{ is on the complex Stiefel manifold S(N,M t ). i.e., the set S C N M t : S H S = I }. In the following we adopt existing results on function optimization on matrix manifolds [3]. B. Modified Steepest Descent on the Complex Stiefel Manifold To solve the optimization problem of (7) we we apply the modified steepest descent method of [4]. Let Z k T S (N,M t ) be the steepest descent at point S k S(N,M t ). Here, T S (N,M t ), is the tangent space, which is intuitively the plane that is tangent to the complex Stiefel manifold at point S S(N,M t ) [7]. The steepest algorithm starts from point S k and moves along Z k with a step size δ, i.e., S k+ = S k +δz k, (8) To preserve the orthogonality during the update steps, the new point S k+ is projected back to the complex Stiefel manifold, i.e., S k+ =Π ( S k +δz k), where Π is the projection operator. According to [4], the projection of any S onto the Steifel manifold is taken to be the point of the Stiefel manifold closest to S in the Frobenius norm. Let g ( Z k) = f ( Π ( S k +Z k)) be the local cost function for S k S(N,M t ). The gradient of g ( Z k) at Z k = 0 under the canonical inner product is [4] S f ( S k) = S f ( S k) S k( S f ( S k)) H S k, (9) and the steepest descent is Z k = S f ( S k). Here, S f (S), is the derivative of f (S), which is (see Appendix) S f (S)=2 { [(S A ) (SA) N] Y T} A H, (0) F 474

3 where N = M t N N T L, Y = AT S T. The step size δ is chosen using a nonmonotone line search method based on [5], i.e., to meet f ( Π ( S k +δz k)) C k +βδ S f ( S k),z k, () S f ( S k+),z k σ S f ( S k),z k, (2) where the canonical inner product is defined as [7] Z,Z 2 = R { tr [ ( Z H 2 I 2 SSH) ]} Z, for Z,Z 2 T S (N,M t ). Here, C k+ is the reference value and is taken to a convex combination of C k and f ( [ S k+) as C k+ = ηqk C k +f ( S k+)]/ Q k+, where Q k+ = ηq k +, C 0 = f ( S 0) and Q 0 =. The modified steepest descent algorithm is summarized in Algorithm. Algorithm Modified steepest descent algorithm with nonmonotone line search : Initialize: Choose S 0 S(N,M t ) and parameters α,η,ɛ (0,), 0 <β<σ<. Set δ =,C 0 = f ( S 0),Q 0 =,k=0. 2: Descent direction update: Compute the descent direction as Z k = S f ( S k). 3: Convergence test: If Z k,z k ɛ, then stop. 4: Line search update: Compute S k+ =Π ( S k +δz k) and S f ( S k+).iff ( S k+) βδ S f ( S k),z k + C k and S f ( S k+),z k σ S f ( S k),z k, then set δ = αδ and repeat Step 4. 5: Cost [ update: Q k+ = ηq k +, C k+ = ηqk C k +f ( S k+)]/ Q k+. 6: Set S k+ =Π ( S k +δz k), k = k +. Go to Step 2. IV. NUMERICAL RESULTS In this section, we provide some numerical results to demonstrate the performance of Algorithm when used to design MIMO-MC radar transmit waveforms. We consider a ULA transmit array with d t = λ 2, M t =30and N =32. We first take the DOA space to be [ 0,0 ], (correspondingly, αk t [ ,0.0868]) and discretize it uniformly with spacing. Correspondingly, L =2and A C In the nonmonotone line search, we chose β =0., σ =0.99 to satisfy the conditions of () and (2). These values are selected by trial and error. In addition, we set α =0.5 to adjust the step size and ɛ =0 5 for the stopping check value. The initial step size is set as δ =0 2. The iteration is initialized with Hadamard waveforms, i.e., S 0 S(32,30). The convergence of the proposed modified steepest descent algorithm for η =,0.5,0 is shown in Fig. (a). As it can be seen from Fig. (a), the value of the objective function, f, approaches its global minimal, i.e., 0. The corresponding optimal solution, S, is not unique, and depends on the initial point and the step size selection. As it will be seen later (see Fig. 2 (a) for an example), all solutions result in very similar MC recovery performance. It should be pointed out that since the complex Stiefel manifold is not a convex set, there is no guarantee for the algorithms to converge to the global minimum. Our simulations show that for the entire DOA space [ 90 : :90 ], when the dimension of S is relatively small, e.g., N =32,M t =30, the objective value gets stuck to a local minimum; however, if the spacing increases, e.g., 5, the iteration converges to the global minimum. If the dimension is relatively large, e.g., N = 52,M t = 500, even for small spacing, i.e.,, the objective value also converges to its global minimum (see Fig. (b) for η =0). Next, we look at the MC performance corresponding to the optimized waveforms as function of the portion of observed entries, p. We consider the scenario of K =2targets located at θ = 2,θ 2 =2, and take M r = 28,N=32,M t =30 and SNR = 25dB. The optimized waveforms are obtained via Algorithm by focusing on DOA space [ 0 : :0 ], i.e., L =2. The simulation results are averaged over 50 independent runs. In each run, the noise is randomly generated. In the simulations, the data matrix is recovered via the SVT algorithm of [8]. Figure 2 (a) shows the recovery error, suggesting that the optimized waveforms result in significantly better performance as compared to the Hadamard waveforms, especially for small values of p. One can see that in order to achieve an error around 5%, MC with optimized waveforms requires 40% of the matrix entries, while MC with Hadamard waveforms requires more than80% of the matrix entries. Also, although the optimized waveforms are not unique, all solutions result in almost identical MC performance. In the same figure, we also compare the optimized waveforms against Gaussian orthogonal waveforms (G-Orth). Although in this case the difference is smaller, one can see that the recovery error corresponding to the optimized waveforms reaches the noise level faster (at p =0.3) than that of the G-Oth waveforms, which reaches the noise level at p =0.4. Although the waveform design requires angle space discretization, the sensitivity due to targets falling off grids is rather low. To show this, we take the DOA space to be [ 90 :5 :90 ]. The recovery error comparison between K =2on-grid targets located at θ = 60,θ 2 =0 and off-grid at θ = 62.5,θ 2 =2.5 is shown in Fig. 2 (b). The other simulation settings are the same as in Fig. 2 (a). It can be found that the performance of off-grid targets is almost identical to that of on-grid targets when p 0.3. Under the optimized waveforms, our simulations show that for different number of targets, the coherence is always bounded by the bound of (4) and approaches to its smallest value (not necessarily in a monotone way) when M t increases. In Fig. 3 we plot for the coherence μ(v) of the matrix W and its bound defined in (4) versus the number of transmit antennas, corresponding to M t for K =4targets located at [ 0, 5,5,0 ]. The optimized waveforms for different M t are obtained via Algorithm by focusing on DOA space [ 0 : :0 ], i.e., L =2. It can be seen that the coherence μ(v) under the optimized waveforms approaches its smallest value,, asm t increases, and is always bounded 475

4 f η= η=0.5 η=0 Relative Recovery Errors Reciprocal of SNR Hadamard G Orth Opt Wave Opt Wave Iteration (a) p (a) Low dimension spacing 5 Low dimension spacing High dimension spacing Reciprocal of SNR Opt wave: On grid Opt wave: Off grid f Relative Recovery Errors Iteration x 0 4 (b) Fig. : f versus iterations: (a) DOA space [ 0 : :0 ]; (b) DOA space [ 90 : θ :90 ], θ =, p (b) Fig. 2: MC error as function of p: (a) Comparison with Hadamard and G-Orth waveforms; (b) Off-grid cases. by the bound of (4). 0 µ(v) Bound V. CONCLUSIONS In this paper, we applied the modified steepest descent algorithm [4] with nonmonotone line search of [5] to design optimal transmit waveforms for MIMO-MC radars subject to orthogonal constraints. Simulation results show that the objective value of the waveform design problem converges to its global minimum for large dimensional matrices. Under the optimized waveforms, the matrix coherence approaches to its smallest value,, as the number of transmit antennas increases. As a result, only a small portion of samples are needed for matrix recovery via MC if the optimized waveforms are used. APPENDIX Following the notation in [4], the cost function f : C N M t R can be written in the Taylor series approximation in the matrix form f (S+δZ)=f (S)+δR { tr ( Z H D S )} +O ( δ 2 ), where D S C N Mt is the derivative of f evaluated at S. Since F F and N are real-valued matrices, { the objective function can be written as f (S) = tr (F F N)(F F N) T}. To find the derivative of f we do the following expansion: f (S+δZ)=f (S)+δtr{T}+O ( δ 2), µ(v) Mt Fig. 3: μ(v) and its bound under optimized waveforms. where T = H {[(S A ) (ZA)] T +[(S A ) (ZA)] H} + { [(S A ) (ZA)]+[(S A ) (ZA)] } H T and H =[(S A ) (SA) N] R N L. Thus, it holds that tr{t} = R { 2H [( A H Z H) ( A T S T)]}. Let Y = A T S T C L N. Then, it can be shown that tr { H [( A H Z H) Y ]} =tr { Z H [( H Y T) A H]}. Consequently, it holds that tr{t} = R { Z H ( 2[(S A ) (SA) N] Y T) A H}. By coefficient comparison between above equation and the matrix form of the Taylor series, we obtain the derivative S f (S) of f (S). 476

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