New Cramér-Rao Bound Expressions for Coprime and Other Sparse Arrays

Transcription

1 New Cramér-Rao Bound Expressions for Coprime and Other Sparse Arrays Chun-Lin Liu 1 P. P. Vaidyanathan 2 1,2 Dept. of Electrical Engineering, MC California Institute of Technology, Pasadena, CA 91125, USA cl.liu@caltech.edu 1, ppvnath@systems.caltech.edu 2 SAM 2016 Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

2 Outline 1 DOA Estimation using Sparse Arrays 2 Cramér-Rao Bounds for Sparse Arrays 3 Numerical Examples 4 Concluding Remarks Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

3 DOA Estimation using Sparse Arrays Outline 1 DOA Estimation using Sparse Arrays 2 Cramér-Rao Bounds for Sparse Arrays 3 Numerical Examples 4 Concluding Remarks Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

4 DOA Estimation using Sparse Arrays Direction-of-arrival (DOA) estimation 1 θ i Monochromatic Uncorrelated Sources Sensor Arrays DOA Estimators Estimated DOA θ i 1 Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

5 DOA Estimation using Sparse Arrays ULA and sparse arrays ULA (not sparse) Identify at most N 1 uncorrelated sources, given N sensors. 1 Can only find fewer sources than sensors. Sparse arrays 1 Minimum redundancy arrays 2 2 Nested arrays 3 3 Coprime arrays 4 4 Super nested arrays 5 Identify O(N 2 ) uncorrelated sources with O(N) physical sensors. More sources than sensors! 1 Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, Moffet, IEEE Trans. Antennas Propag., Pal and Vaidyanathan, IEEE Trans. Signal Proc., Vaidyanathan and Pal, IEEE Trans. Signal Proc., Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

6 Coprime arrays 1 DOA Estimation using Sparse Arrays The coprime array with (M, N) = 1 is the union of 1 an N-element ULA with spacing Mλ/2 and 2 a 2M-element ULA with spacing Nλ/2. ULA (1) ULA (2) Physical array S (M = 3, N = 4): Difference coarray D = {n 1 n 2 n 1, n 2 S} Holes 1 Vaidyanathan and Pal, IEEE Trans. Signal Proc., Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

7 DOA Estimation using Sparse Arrays DOA estimation using sparse arrays Multiple data vectors in the physical array S S = O(N) Step 1 A single correlation vector in the difference array D D = O(N 2 ) Step 2: DOA estimation using this single correlation vector. Spatial smoothing MUSIC (SS MUSIC), 1 to name a few 2,3. 1 Pal and Vaidyanathan, IEEE Trans. Signal Proc., Empirically, SS MUSIC can identify up to ( U 1)/2 uncorrelated sources with sufficient snapshots. Why? D U 2 Abramovich, Spencer, and Gorokhov, IEEE Trans. Signal Proc., 1998, Zhang, Amin, and Himed, IEEE ICASSP, 2013; Pal and Vaidyanathan, IEEE Trans. Signal Proc., 2015; Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

8 Cramér-Rao Bounds for Sparse Arrays Outline 1 DOA Estimation using Sparse Arrays 2 Cramér-Rao Bounds for Sparse Arrays 3 Numerical Examples 4 Concluding Remarks Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

9 Cramér-Rao Bounds for Sparse Arrays Fisher Information Matrices and Cramér-Rao Bounds x: random vectors with distribution p(x; α). α: Real-valued, unknown, and deterministic parameters. FIM I(α), based on p(x; α) and α. (under some regularity conditions) If I(α) is nonsingular, then CRB(α) I 1 (α). If α(x) is a unbiased estimator of α, then Cov( α(x)) CRB(α). CRB depend only on p(x; α) and α, NOT on the estimators α(x). CRB pose fundamental limits to the estimation performance of all unbiased estimators. Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

10 Related work Cramér-Rao Bounds for Sparse Arrays 1 Deterministic CRB 1 assumes A i (k) are deterministic. 2 Stochastic CRB 2 becomes invalid for D > S since VS HV S is singular. CRB = pn 2K Re [ H Q T ] 1 H = U H S [I V S(VS HV S) 1 VS H]U S. 3 Abramovich et al. 3 plotted the CRB curves numerically. 4 Jansson et al. s CRB expressions 4 are undefined if D = S. x S (k) = D A i (k)v S ( θ i )+n S (k) i=1 V S = [v S ( θ i )] C S D SS MUSIC Sources are stochastic and known to be uncorrelated. (E[A i (k 1 )A j (k 2)] = p i δ i,j δ k1,k 2 ) Resolve more sources than sensors (D S ). 1 Stoca and Nehorai, IEEE Trans. Acoustics, Speech and Signal Proc., Stoca and Nehorai, IEEE Trans. Acoustics, Speech and Signal Proc., Abramovich, Gray, Gorokhov, and Spencer, IEEE Trans. Signal Proc., Jansson, Göransson, and Ottersten, IEEE Trans. Signal Proc., Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

11 Cramér-Rao Bounds for Sparse Arrays Augmented Coarray Manifold matrices (Proposed) A c = [ diag(d)v D V D e 0 ] D = { 4, 3, 1, 0, 1, 3, 4}. D = 2 sources θ 1 and θ 2. D D 1 ( 4)e j2π θ 1 ( 4) ( 3)e j2π θ 1 ( 3) ( 1)e j2π θ 1 ( 1) (0)e j2π θ 1 (0) (1)e j2π θ 1 (1) (3)e j2π θ 1 (3) (4)e j2π θ 1 (4) ( 4)e j2π θ 2 ( 4) ( 3)e j2π θ 2 ( 3) ( 1)e j2π θ 2 ( 1) (0)e j2π θ 2 (0) (1)e j2π θ 2 (1) (3)e j2π θ 2 (3) (4)e j2π θ 2 (4) e j2π θ 1 ( 4) e j2π θ 1 ( 3) e j2π θ 1 ( 1) e j2π θ 1 (0) e j2π θ 1 (1) e j2π θ 1 (3) e j2π θ 1 (4) e j2π θ 2 ( 4) e j2π θ 2 ( 3) e j2π θ 2 ( 1) e j2π θ 2 (0) e j2π θ 2 (1) e j2π θ 2 (3) e j2π θ 2 (4) D diag(d)v D V D e 0 Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

12 Cramér-Rao Bounds for Sparse Arrays The proposed CRB expression for sparse arrays 1 A c has full column rank. (rank(a c ) = 2D + 1) if and only if FIM is nonsingular. (CRB exists) CRB( θ) = CRB( θ 1,..., θ ( ) D ) = 1 4π 2 K G H 0 Π 1 MW D G 0. { θ i } D i=1 : normalized DOAs. θ i = 0.5 sin θ i [ 0.5, 0.5]. S: The physical array. D: The difference coarray. p i : The ith source power. p n: The noise power. K: Number of snapshots. V D = [v D ( θ 1 ),..., v D ( θ D )]. W D = [V D, e 0 ]. P = diag(p 1,..., p D ). G 0 = Mdiag(D)V D P. M = (J H (R T S R S) 1 J) 1/2 0. R S = D i=1 p i v S ( θ i )vs H ( θ i ) + p ni. J {0, 1} S 2 D, J :,m = vec(i(m)), m D. I(m) n1,n 2 = { 1, if n 1 n 2 = m, 0, otherwise, n 1, n 2 S. 1 Liu and Vaidyanathan, Digital Signal Proc.,Elsevier, 2016; Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

13 Cramér-Rao Bounds for Sparse Arrays Implications drawn from the CRB expressions ACM matrices depend on The difference coarray D, the normalized DOA θ = [ θ 1,..., θ D ] T, and the number of sources D. These factors influence the existence of CRB. A c = [ ] diag(d)v D V D e 0 CRB( θ) = 1 4π 2 K (G H 0 Π MWD G 0 ) 1 CRB depend on The physical array S, the normalized DOA θ = [ θ 1,..., θ D ] T, the number of sources D, the number of snapshots K, and the SNR p 1 /p n,..., p D /p n. These parameters control the values of CRB. If rank(a c ) = 2D + 1, then lim K CRB( θ) = 0. Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

14 Cramér-Rao Bounds for Sparse Arrays Asymptotic CRB expressions for large SNR 1 1 A c has full column rank. 2 D uncorrelated sources have equal SNR p/p n. 3 Large SNR (p/p n ). Theorem (rank(v S ) < S ) CRB( θ) = D < S 1 4π 2 K(p/p n ) S 1 1, where S 1 is positive definite and independent of SNR. lim CRB( θ) = 0. p/p n Theorem (rank(v S ) S ) Typically D S CRB( θ) = 1 4π 2 K S 1 2, where S 2 is positive definite and independent of SNR. lim CRB( θ) 0. p/p n 1 These phenomena were observed by Abramovich et al. in 1998, which can be proved using our CRB expressions. Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

15 Numerical Examples Outline 1 DOA Estimation using Sparse Arrays 2 Cramér-Rao Bounds for Sparse Arrays 3 Numerical Examples 4 Concluding Remarks Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

16 Numerical Examples CRB vs SNR, fewer sources than sensors (D < S = 4) CRB( θ1) D = 1 D = 2 D = 3 D = 1, Asymptotic D = 2, Asymptotic D = 3, Asymptotic SNR (db) S = {1, 2, 3, 6}. D = { 5,..., 5}. θ i = (i 1)/D. The number of snapshots K = 200. Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

17 Numerical Examples CRB vs SNR, more sources than sensors (D S = 4) CRB( θ1) D = 4 D = 5 D = 4, Asymptotic D = 5, Asymptotic SNR (db) S = {1, 2, 3, 6}. D = { 5,..., 5}. θ i = (i 1)/D. The number of snapshots K = 200. Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

18 Numerical Examples CRB vs D, coprime array, 10 sensors 1 S ( U 1)/2 ( D 1)/ D rank(a c ) = 2D + 1 Cannot judge rank(a c ) < 2D + 1 CRB( θ1) Number of sources D M = 3, N = 5, S = {0, 3, 5, 6, 9, 10, 12, 15, 20, 25}. D = {0, ±1,..., ±17, ±19, ±20, ±22, ±25}. θ i = (i 1)/D for i = 1,..., D. 20dB SNR, 500 snapshots. 1 Liu and Vaidyanathan, Digital Signal Proc.,Elsevier, 2016; Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

19 Concluding Remarks Outline 1 DOA Estimation using Sparse Arrays 2 Cramér-Rao Bounds for Sparse Arrays 3 Numerical Examples 4 Concluding Remarks Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20

20 Concluding remarks Concluding Remarks The proposed CRB expressions for sparse arrays explain why more sources than sensors (D S ) can be identified, and why the CRB stagnates for large SNR and D S. More details can be found in our journal and website. 1 Other recent papers: Koochakzadeh and Pal 2 and by Wang and Nehorai. 3 Future: study the optimal array geoemtry for CRB. Work supported by Office of Naval Research. Thank you! 1 Liu and Vaidyanathan, Digital Signal Proc.,Elsevier, 2016; 2 Koochakzadeh and Pal, IEEE Signal Proc. Letter, Wang and Nehorai, arxiv: [stat.ap], Liu and Vaidyanathan CRB for Sparse Arrays SAM / 20