Performance Analysis of Coarray-Based MUSIC and the Cram er-Rao - PowerPoint PPT Presentation

CSSIP Performance Analysis of Coarray-Based MUSIC and the Cram´ er-Rao Bound Mianzhi Wang, Zhen Zhang, and Arye Nehorai Preston M. Green Department of Electrical & Systems Engineering Washington University in St. Louis March 8, 2017 1

CSSIP Outline • Measurement model and coarray-based MUSIC • Mean-square error of coarray-based MUSIC • Cram´ er-Rao bound • Conclusions and future work 2

CSSIP Notations A H = Hermitian transpose of A A ∗ = Conjugate of A A † = ( A H A ) − 1 A H , pseudo inverse of A Π A = AA † , projection matrix onto the range space of A Π ⊥ A = I − AA † , projection matrix onto the null space of A ⊗ = Kronecker Product ⊙ = Khatri-Rao Product vec( A ) = Vectorization of A R ( A ) = Real part of A I ( A ) = Imaginary part of A 3

CSSIP Measurement Model • We consider a far-field narrow-band measurement model of sparse linear arrays: y ( t ) = A ( θ ) x ( t ) + n ( t ) , (1) where A = [ a ( θ 1 ) a ( θ 2 ) · · · a ( θ K )] , with the i -th element of a ( θ k ) being e j ¯ d i φ k , ¯ d i = d i /d 0 , φ k = (2 πd 0 sin θ k ) /λ , and λ denotes the wavelength. ULA: Co-prime array: Nested array: Sparse linear arrays MRA: Figure 1: Examples of sparse linear arrays. 4

CSSIP Measurement Model (cont.) • We consider the stochastic (unconditional) model [1], where the sources signals are assumed random and unknown. • Assumptions: 1. The source signals are temporally and spatially uncorrelated. 2. The noise is temporally and spatially uncorrelated Gaussian that is also uncorrelated from the source signals. 3. The K DOAs are distinct. • The sample covariance matrix is given by R = E [ yy H ] = AP A H + σ 2 n I , (2) where P = diag( p 1 , p 2 , . . . , p L ) is the source covariance matrix. 5

CSSIP Coarray-based MUSIC • Vectorizing R leads to r = vec R = A d p + σ 2 n i , (3) where p = [ p 1 , p 2 , . . . , p K ] T , i = vec( I ) , and e j ( ¯ d 1 − ¯ e j ( ¯ d 1 − ¯  d 1 ) φ 1 d 1 ) φ k  · · · . . ... . .   . .   A d = A ∗ ⊙ A =  e j ( ¯ d m − ¯ e j ( ¯ d m − ¯  d n ) φ 1 d n ) φ k . (4) · · ·     . . ... . .   . .   e j ( ¯ d M − ¯ e j ( ¯ d M − ¯ d M ) φ 1 d M ) φ k · · · • Observation: A d embeds a steering matrix of an difference coarray whose sensor locations are given by D co = { d m − d n | 1 ≤ m, n ≤ M } . ⇒ We can construct a virtual ULA model from (3). 6

CSSIP Coarray-based MUSIC (cont.) Example 1. An illustration of the relationship between the physical array and the difference coarray. d 0 (a) ULA of 2 M v − 1 sensors (b) − M v d 0 M v d 0 (c) 1 st subarray of size M v Figure 2: A co-prime array with sensors located at [0 , 2 , 3 , 4 , 6 , 9] λ/ 2 and its coarray: (a) physical array, (b) coarray, (c) virtual ULA part of the coarray. 7

CSSIP Coarray-based MUSIC (cont.) Definition 1. The array weight function [2] ω ( n ) : Z �→ Z is defined by ω ( l ) = |{ ( m, n ) | ¯ d m − ¯ d n = l }| , where |A| denotes the cardinality of the set A . Definition 2. Let 2 M v − 1 denote the size of the central virtual ULA. We introduce the transform matrix [3] F as a real matrix of size (2 M v − 1) × M 2 , whose elements are defined by � , ¯ d p − ¯ 1 d q = m − M v , ω ( m − M v ) F m,p +( q − 1) M = (5) 0 , otherwise , for m = 1 , 2 , . . . , M v , p = 1 , 2 , . . . , M, q = 1 , 2 , . . . , M . ⇒ We can express the measurement vector of the virtual ULA model by z = F r = A c p + σ 2 n F i . (6) 8

CSSIP Coarray-based MUSIC (cont.) • To construct the augmented sample covariance matrix, the virtual ULA is divided into M v overlapping subarrays of size M v [2], [4]. • We denote the output of the i -th subarray by z i = Γ i z for i = 1 , 2 , . . . , M v , where Γ i = [ 0 M v × ( i − 1) I M v × M v 0 M v × ( M v − i ) ] . ULA of 2 M v − 1 sensors − M v d 0 0 M v d 0 1 st subarray 2 nd subarray ... M v -th subarray Figure 3: M v overlapping subarrays. 9

CSSIP Coarray-based MUSIC (cont.) • We can then construct an augmented covariance matrix R v from z i to provide enhanced degrees of freedom and apply MUSIC to R v . • Two commonly used methods: ◮ MUSIC with directly augmented covariance matrix (DA-MUSIC) [4]: R v1 = [ z M v z M v − 1 · · · z 1 ] . (7) ◮ MUSIC with spatially smoothed covariance matrix (SS-MUSIC) [2]: M v 1 � z i z H R v2 = i . (8) M v i =1 • R v1 and R v2 are related via the following equality [2]: 1 1 M v R 2 M v ( A v P A H v + σ 2 n I ) 2 , R v2 = v1 = (9) where A v corresponds to the steering matrix of a ULA whose sensors are located at [0 , 1 , . . . , M v − 1] d 0 . 10

CSSIP Outline • Measurement model and coarray-based MUSIC • Mean-square error of coarray-based MUSIC • Cram´ er-Rao bound • Conclusions and future work 11

CSSIP Mean-Square Error of Coarray-Based MUSIC We derive the closed-form MSE expressions for DA-MUSIC and SS-MUSIC: Theorem 1. Let ˆ θ (DA) and ˆ θ (SS) denote the estimated values of θ k using DA-MUSIC k k and SS-MUSIC, respectively. Let ∆ r = vec( ˆ R − R ) . Then [3] I ( ξ T ∆ r ) − θ k . − θ k . λ θ (DA) ˆ = ˆ θ (SS) = − , (10) k k β H 2 πd 0 p k cos θ k k β k where . = denotes asymptotic equality (first-order) and ξ k = F T Γ T ( β k ⊗ α k ) , 1 ] T , Γ = [ Γ T M v Γ T M v − 1 · · · Γ T α T k = − e T k A † D = diag(0 , 1 , . . . , M v ) , v , β k = Π ⊥ A v Da v ( θ k ) . Theorem 2. The asymptotic MSE expressions of DA-MUSIC and SS-MUSIC have the same form. Denote the asymptotic MSE of the k -th DOA by ǫ ( θ k ) . We have [3]: k ( R ⊗ R T ) ξ k λ 2 ξ H ǫ ( θ k ) = , ∀ k ∈ { 1 , 2 , . . . , K } . (11) k cos 2 θ k 4 π 2 Nd 2 0 p 2 � β k � 4 2 12

CSSIP Mean-Square Error of Coarray-Based MUSIC (cont.) Theorem 1 and Theorem 2 have the following implications: • DA-MUSIC and SS-MUSIC have the same asymptotic MSE, and they are both asymptotically unbiased. • ǫ ( θ k ) depends on both the physical array geometry and the coarray geometry (as illustrated in Fig. 4). 0.16 Nested (5, 6) Nested (5, 6) 0.25 Nested (2, 12) Nested (2, 12) 0.14 RMSE (deg) RMSE (deg) Nested (3, 9) Nested (3, 9) 0.12 Nested (1, 18) Nested (1, 18) 0.2 0.1 0.08 0.15 -10 0 10 20 -10 0 10 20 SNR (dB) SNR (dB) Figure 4: RMSE vs. SNR for four different nested array configurations. The four arrays share the same virtual ULA. Left: K = 8 . Right: K = 20 . 13

CSSIP Mean-Square Error of Coarray-Based MUSIC (cont.) Corollary 1. Assume all sources have the same power p . Let SNR = p/σ 2 n denote the common SNR. Then ǫ ( θ k ) decreases monotonically as SNR increases, and λ 2 � ξ H k ( A ⊗ A ∗ ) � 2 2 SNR →∞ ǫ ( θ k ) = lim . (12) k cos 2 θ k 4 π 2 Nd 2 0 p 2 � β k � 4 2 Specifically, 1. when K = 1 , the above expression is exactly zero; 2. when K ≥ M the above expression is strictly greater than zero. Implication: Corollary 1 analytically explains the “saturation” behavior of SS-MUSIC in high SNR regions observed in previous studies. 14

CSSIP Mean-Square Error of Coarray-Based MUSIC (cont.) MSE vs. number of sensors: Coprime Coprime Nested Nested 10 -2 10 -2 O ( M − 4 . 4 ) O ( M − 4 . 4 ) MSE (deg 2 ) MSE (deg 2 ) 10 -4 10 -4 10 -6 10 -6 4 14 50 4 14 50 M M (a) K = 1 (b) K = 3 Figure 5: MSE vs. number of sensors. SNR = 0dB , and N = 1000 . The solid lines denote analytical results, while crosses denote numerical results. A dashed black trend line is included for comparison. The co-prime arrays were generated by the co-prime pairs ( m, m + 1) , and the nested arrays were generated by the parameter pairs ( m + 1 , m ) , where we varied m from 2 to 12. Observation: the MSE of coarray-based MUSIC decreases faster than O ( M − 3 ) , the asymptotic MSE of classical MUSIC for ULAs when M → ∞ . 15

CSSIP Mean-Square Error of Coarray-Based MUSIC (cont.) Resolution analysis: The analytical resolution limit is determined by � � ǫ ( θ − ∆ θ/ 2) + ǫ ( θ − ∆ θ/ 2) ≥ ∆ θ (13) Figure 6: Resolution probability of different arrays for Figure 7: Resolution probability of different arrays for different N with SNR fixed to 0dB, obtained from 500 different SNRs with N = 1000 , obtained from 500 trials. trials. The red dashed line is the analytical resolution limit. The red dashed line is the analytical resolution limit. Observation: our analytical expression predict the resolution limit well. 16

CSSIP Outline • Measurement model and coarray-based MUSIC • Mean-square error of coarray-based MUSIC • Cram´ er-Rao bound • Conclusions and future work 17

CSSIP Cram´ er-Rao Bound • The CRB of the DOAs for general sparse linear arrays with under the assumption of uncorrelated sources is given by [3], [5], [6]: CRB θ = 1 N ( M H θ Π ⊥ M s M θ ) − 1 , (14) where M θ = ( R T ⊗ R ) − 1 / 2 ˙ A d P , (15a) M s = ( R T ⊗ R ) − 1 / 2 � A d i � , (15b) A ∗ ⊙ A + A ∗ ⊙ ˙ A d = ˙ ˙ A , (15c) ˙ A = [ ∂ a ( θ 1 ) /∂θ 1 , · · · , ∂ a ( θ K ) /∂θ K ] , (15d) and A d , i follow the same definitions as in (3). • The CRB can be valid even if the number of sources exceeds the number of sensors. This is because the invertibility of the FIM depends on the coarray structure, which appears in [ ˙ A d P A d i ] . The can remain full column rank of [ ˙ A d P A d i ] even if K ≥ M . 18