The Mean-Squared-Error of Autocorrelation Sampling in Coprime Arrays

Transcription

1 IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (IEEE CAMSAP 27), Curacao, Dutch Antilles, Dec. 27. The Mean-Squared-Error of Autocorrelation Sampling in Coprime Arrays Dimitris G. Chachlakis, Panos P. Markopoulos, and Fauzia Ahmad Dept. of Electrical and Microelectronic Engineering Rochester Institute of Technology, Rochester, NY 4623, USA Dept. of Electrical and Computer Engineering Temple University, Philadelphia, PA 922, USA Abstract Standard direction-of-arrival estimation using coprime arrays samples the entries of the estimated physical-array autocorrelation matrix, organizes them in a matrix structure, and conducts multiple-signal classification (MUSIC) with singular vectors of the resulting matrix. A majority of the existing literature samples the physical-array autocorrelations by selection, retaining only one of the samples that correspond to each element of the difference coarray. Other more recent works conduct averaging of all samples that relate to each coarray element. Even though the two methods coincide when applied on the nominal/true physical-array autocorrelations, their performance differs significantly when applied on finite-snapshot estimates. In this paper, we present for the first time in closed form the mean-squared-error of both selection and averaging autocorrelation sampling and clarify/establish the superiority of the latter. Index Terms Coprime arrays, DoA estimation, mean-squarederror, sparse arrays, spatial smoothing. I. INTRODUCTION Coprime arrays have demonstrated the ability to resolve significantly more directions of arrival (DoAs) than uniform linear arrays (ULAs) with the same number of elements [] [5]. This is enabled by their specific sparse structure and accomplished by thereto tailored intelligent receive processing. In the past few years, coprime arrays have also been employed in applications other than DoA estimation, such as beamforming [6] and spacetime processing [7]. DoA estimation with coprime arrays is conducted by processing of the entries of the (estimated) spatial autocorrelation matrix of the physical array. Due to the specific array structure, appropriate processing of the autocorrelations can offer an estimate of the signal subspace that corresponds to a significantly larger virtual ULA, commonly referred to as coarray. As such, DoA estimation can be performed by standard multiple-signal classification (MUSIC) on the coarray signal subspace, with the ability to resolve significantly increased number of DoAs. In the first processing step, the coprime receiver samples the estimated physical-array autocorrelations and, possibly, extrapolates them by means of compressive-sensing techniques [8] [2]. Then, these samples are spatially smoothed to form a matrix within the span of which the signal and noise subspaces are separable [5], [2]. Most works in the literature sample the autocorrelations by selection [], collecting only one sample for each element of the coarray. Other works sample by averaging all autocorrelation values that correspond to each coarray element [2]. Understandably, the two methods coincide when applied on the nominal physical-array autocorrelations. However, they differ significantly in performance when applied on finite-snapshot estimates. In this paper, we present for the first time in closed form the mean-squared-error (MSE) of both selection and averaging autocorrelation sampling and show that, for any number of snapshots, averaging offers superior MSE performance than selection. The theoretical results are validated with numerical simulation. II. SIGNAL MODEL We consider an extended coprime array of L 2M + N elements, for coprime naturals (M, N) with M < N []. The array is formed by interleaving two ULAs, one with N elements at positions {p M,i (i )Md} N i, and the other with 2M elements at positions {p N,i ind} 2M i. Here, d is the reference unit spacing, which is typically chosen as one-half wavelength. We assume that the coprime array receives narrowband signals from K < MN +M sources on carrier frequency f c and propagation speed c. Under far-field conditions, the signal from source k {, 2,..., K} impinges on the array from direction θ k ( π 2, π 2 ] with respect to broadside. Defining the element-location vector p sort([p M,,..., p M,N, p N,,..., p N,2M ] ), where sort( ) sorts the entries of its vector argument in ascending order, the array response vector s(θ k ) for source k is s(θ k ) [ v(θ k ) [p],..., v(θ k ) L] [p] C L, with v(θ) ( ) exp j2πfc c sin(θ) for every θ ( π 2, π 2 ]. Accordingly, the qth collected snapshot is of the form y q s(θ k )x q,k + n q C L, () k where x q,k CN (, d k ) is the qth symbol for source k (power-scaled and flat-fading-channel processed) and n q CN ( L, σ 2 I L ) is additive white Gaussian noise (AWGN). We make the common assumptions that the random variables are statistically independent across different snapshots and that sym-

2 bols from different sources are independent of each other and of every entry of n q. The receiver s goal is to estimate the source DoAs in Θ {θ, θ 2,..., θ K }. A. Autocorrelation Sampling and DoA Estimation Defining the source power-vector d [d, d 2,..., d K ] R+ K and the array-response matrix S [s(θ ), s(θ 2 ),..., s(θ K )] C L K, the received-signal autocorrelation matrix is given by R y {yq y H q } S diag(d) S H + σ 2 I L. Accordingly, we define r vec(r y ) a(θ i )d i + σ 2 i L C L2, (2) i where vec( ) returns the column-wise vectorization of its matrix argument, a(θ i ) s(θ i ) s(θ i ), i L vec(il ) R L2, and is the Kronecker product operator [2]. By coprime number theory [], for every n { L +, L + 2,..., L } with L MN + M, there exists a well-defined set of indices J n {, 2,..., L 2 }, such that [a(θ)] j v(θ) n j J n, (3) for every θ ( π 2, π 2 ]. We henceforth consider that J n contains all j indices that satisfy (3). In view of (3), the receiver in standard coprime DoA estimation [] selects any single index j n J n, for n { L +,..., L }, and builds the L 2 (2L ) selection sampling matrix [ ] E sel e j L,L 2, e j 2 L,L 2,..., e j L,L 2, (4) where, for any p P N +, e p,p is the pth column of I P. That is, the receiver samples the autocorrelations contained in r, discarding by selection all duplicates (i.e., every entry with index in J n \ j n, for every n), to form r sel sel r a sel (θ i )d i + σ 2 e L,2L, (5) i where, for any θ ( π 2, π 2 ], a sel(θ) sel a(θ) [ v(θ) L, v(θ) 2 L,..., v(θ) L ] C 2L. Thereafter, the receiver applies spatial-smoothing to organize the sampled autocorrelations in the matrix Z sel F(IL r sel ) C L L, (6) where F [F, F 2,..., F L ] and, for every m {, 2,..., L }, F m [L (L m), I L, L (m )]. Importantly, by the definitions in (5) and (6), it holds that Z sel S co diag(d)s H co + σ 2 I L, (7) where [S co ] m,k v(θk ) m, for every m {, 2,..., L } and k {, 2,..., K}. That is, Z sel coincides with the autocorrelation matrix of a length-l ULA with elements at locations {,,..., L }d. Therefore, standard MUSIC DoA estimation can be applied on Z sel, with the ability to resolve concurrently up to K L DoAs. Specifically, let the columns of Q sel C L K be the dominant left-hand singular vectors of Z sel, corresponding to its K L highest singular values, calculated by means of singular value decomposition (SVD). Then, span(q sel ) span(s co ). Defining [ v(θ), v(θ),..., v(θ) ] L for θ ( π 2, π 2 ], we can accurately decide θ Θ (I L P sel ) v(θ) L, where P sel Qsel Q H sel. Equivalently, we can identify the angles in Θ by the K local minima of the MUSIC spectrum P MU (θ) (I L P sel ) v(θ) 2. (8) Interestingly, among other works, [2] recently proposed replacing E sel in (5) by the L (2L ) averaging sampling matrix E avg, where, for every i {,..., 2L }, [E avg ] :,i J i L j J i L e j,l 2. (9) {, j J By (3) and the fact that [i L ] j, it holds that, for, j / J any given n { L +,..., L }, [r] j e j,l r takes the 2 same value for every j J n. Thus, r avg avg r r sel. () Therefore, MUSIC DoA estimation in the form of (8) can be equivalently conducted using the K dominant left-hand singular vectors of Z avg F(IL r avg ) Z sel. () However, in the case of practical interest where R y is unknown and estimated by a finite collection of snapshots, the two autocorrelation sampling methods no longer coincide, and consequently, neither do the corresponding MUSIC DoA estimators. III. MSE OF AUTOCORRELATION ESTIMATES SAMPLING We consider R y to be unknown to the receiver and estimated by a collection of Q snapshots, y, y 2,..., y Q, as ˆR y Q Q q y qyq H. Accordingly, r is estimated by ˆr vec( ˆR y ) Q Q yq y q. (2) q Then, from (5) and (), the selection and averaging sampled autocorrelation vectors, r sel and r avg, are respectively estimated by ˆr sel selˆr and ˆr avg avgˆr. (3) Similarly, MUSIC can be applied using the K dominant lefthand singular vectors of either Ẑ sel F(IL ˆr sel ), or Ẑavg F(I L ˆr avg ). (4) Indeed, both Ẑsel and Ẑavg have been employed before in the literature [3], [2]. However, a formal statistical performance

3 evaluation of the selection- and averaging-sampling estimates in (3) and (4), which is the focus of this paper, has not been carried out previously. First, we discuss in Lemma below the asymptotic behavior of the estimates in (3) and (4). Lemma follows straightforwardly from () and the fact that ˆr is a Maximum-Likelihood, unbiased, consistent estimate of r [22]. Lemma. As the sample support Q increases asymptotically, both ˆr sel and ˆr avg tend to r sel r avg. Similarly, both Ẑsel and Ẑ avg tend to Z sel. By Lemma, MUSIC DoA estimation on Ẑsel and Ẑavg is expected to perform identically as Q and, thus, very similarly for high values of Q. As we show below, despite their asymptotic equivalence, the MSE performance of the selectionand averaging-sampling estimates differs significantly for finite values of the sample-support Q. For notational simplicity, we define ṗ p L, ω i,j [ṗ]i [ṗ] j, for i, j {, 2,..., L 2 } and z i,j [v(θ ) ωi,j, v(θ 2 ) ωi,j,..., v(θ K ) ωi,j ] H. Then, in view of the statistical description of x q,k and n q, the following Proposition presents in closed form the MSE of estimating each entry of r sel, by means of selection and averaging. Proposition. For any n { L +,..., L } and j n J n, it holds e E { [r sel ] jn [ˆr sel ] jn 2} ( K d + σ2 ) 2, (5) Q e n [r sel] jn 2 [ˆr sel ] j J n j J n 2σ2 K d + σ4 + i,j d 2 Q J n J n 2, (6) i,j J n zh and e e n > for every value of sample-support Q. In view of (6), e e n, when J n. Also, (5) and (6) show that, as expected, e and e n tend to zero as Q increases asymptotically. The proofs of (5) and (6) are omitted due to lack of space. Using (5) and (6), we prove below the inequality e n e, which documents the MSE superiority of averagesampling over selection-sampling. We observe that, for any i, j, z H i,j d 2 K k [z i,j] k d k 2 K k [z i,j] k d k 2 ( K d)2 and, therefore, e e n β n J n J n Q (2σ2 Kd + σ 4 ). (7) Propositions 2 and 3 below derive in a straightforward manner from (3), (4) and Proposition, and conclude the MSE results. Proposition 2. For any given sample-support Q, ˆr sel and ˆr avg, as defined in (3), attain MSEs { e(ˆr sel ) E r sel ˆr sel 2} (2L )e and (8) { e(ˆr avg ) E r sel ˆr avg 2} L n L e n, (9) respectively. The corresponding MSE difference e(ˆr sel ) e(ˆr avg ) is positive, lower-bounded by L n L β n, and converges to zero, as Q increases asymptotically. Proposition 3. For any given sample-support Q, Ẑsel and Ẑavg, as defined in (4), attain MSEs { Zsel } e(ẑsel) E Ẑsel L 2 e and (2) 2 F { Zsel e(ẑavg) E Ẑavg 2 F } L m+l m n m e n, (2) respectively. The corresponding MSE difference e(ẑsel) e(ẑavg) is positive, lower-bounded by L m+l m n m β n, and converges to zero, as Q increases asymptotically. By Proposition 3, it is now formally clear that Ẑ avg is a superior estimate of Z than Ẑsel with respect to MSE. Accordingly, it is expected that the K dominant left-hand singular vectors of Ẑavg will lie closer to those of Z, offering better description to span(s co ), and, ultimately, permitting superior DoA estimation performance. Indeed, the above statements are numerically validated in the following section. IV. NUMERICAL RESULTS We consider an (M 2, N 3) extended coprime array of L 6 elements and K 7 sources, transmitting from angles Θ {θ 6, θ 2 28, θ 3 4, θ 4 5, θ 5 8, θ 6 34, θ 7 86 }. We set σ 2 and signal-to-noise ratio (SNR) to db, for all K sources; i.e., d k k. First, we let the number of snapshots Q vary from 2 to 4 in increments of. For each value of Q, we simulate estimation of Z sel by both selection-sampling (Ẑsel) and averaging-sampling (Ẑsel); then, we measure the corresponding squared-estimation errors Z sel Ẑsel 2 F and Z sel Ẑavg 2 F. We repeat this procedure for, independent symbol/noise realizations and plot the averaged squared-error in Fig.. In the same figure, we plot the theoretical MSEs e(ẑsel) and e(ẑavg), as presented in Proposition 3. We observe that the numerically and theoretically calculated values for the MSEs coincide. All MSE curves start from a high value for Q 2 and decrease monotonically as Q increases, with a trend for asymptotic convergence to, as Q. We observe that, in accordance to Proposition 3, e(ẑavg) is significantly lower than e(ẑsel), for every value of Q. Next, we set Q 3 and calculate the MSE of selection and averaging in the estimation of Z sel for different values of K. Specifically, in Fig. 2, we plot the theoretical and numerical MSEs attained by the two estimators versus the number of transmitting sources, K, 2,..., 7, i.e., for K, we

4 MSE in estimating Z sel Selection (numerical) Selection (theory) Averaging (numerical) Averaging (theory) Average SSE Selection Averaging Fig.. Theoretically and numerically calculated MSEs e(ẑsel) and e(ẑavg), versus sample-support Q. MSE in estimating Z sel Selection (numerical) Selection (theory) Averaging (numerical) Averaging (theory) Number of sources, K Fig. 2. Theoretically and numerically calculated MSEs e(ẑsel) and e(ẑavg), versus the number of transmitting sources K. consider signal transmission only from angle θ 6 in Θ, while for K 3, we consider transmission from angles θ 6, θ 2 28, and θ 3 4. We observe that the performance gap between selection and averaging increases monotonically with K. This is in accordance with the observation that the lower-bound of the gap, β n in (7), increases monotonically as K increases, for every n such that J n > (when J n, β n ). In the next two examples, we examine the association between the MSE in the estimation of Z sel and the MUSIC DoAestimation performance. We start with measuring the error in the estimation of the signal-subspace span(s co ) by the dominant singular vectors of Ẑsel and Ẑavg. Specifically, we set K 7 and let Q vary in {2, 3,..., 4}. Then, we compute over, realizations the average values of the (normalized) squared subspace errors (SSE) 2K P sel ˆQ sel ˆQH sel 2 F and 2K P sel ˆQ avg ˆQH avg 2 F, where ˆQ sel and ˆQ avg are the K dominant left-hand singular vectors of Ẑsel and Ẑavg, respectively, obtained through SVD. We plot the calculated subspace-mses in Fig. 3, versus Q. We observe that, as expected, the estimation errors in Z sel translate accurately to subspace-estimation errors. Thus, averaging autocorrelation sampling attains lower subspace estimation error than selection sampling, for every value of Q, with the maximum difference documented at the minimum tested value of Q. Finally, for the same setup (K 7, Q 2, 3,..., 4), we Fig. 3. Average squared subspace error attained by selection and averaging sampling, versus sample-support Q. RMSE (in deg) Selection Averaging Fig. 4. RMSE of coprime MUSIC DoA estimation attained by selection and averaging sampling, versus sample-support Q. conduct MUSIC DoA estimation in the form of (8), employing, instead of the ideal signal-subspace descriptor P sel, the estimated projection matrices ˆQ sel ˆQH sel (selection) and ˆQ avg ˆQH avg (sampling), calculated upon the estimates Ẑsel and Ẑavg, respectively. After, independent realizations, we compute and plot in Fig. 4 the root-mean-squared-error (RMSE) for the two K K k r (θ k ˆθ k,r ) 2, estimators, RMSE versus the sample support Q. In the RMSE expression, r is the realization index and ˆθ k,r is the kth angle, as estimated at the rth realization. We observe that, in accordance with the theoretical MSE results, averaging autocorrelation sampling allows for superior DoA estimation, compared to selection sampling, for every sample support Q. V. CONCLUSION We have presented in closed form the MSE of selection-based and averaging-based sampling of the estimated physical-array autocorrelation matrix. The theoretical results clarify that, for any value of the number of snapshots, averaging offers superior MSE performance than selection. In addition, numerical simulation results are provided which show that averaging sampling allows for superior signal-subspace estimation and, ultimately, superior DoA-estimation RMSE performance. ACKNOWLEDGMENT The work by F. Ahmad was supported by the US Office of Naval Research under grant N4-3-6.

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