Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom

Transcription

1 Nested Arrays: A Novel Approach to Array Processing with Enhanced Degrees of Freedom Xiangfeng Wang OSPAC May 7, 2013

2 Reference Reference Pal Piya, and P. P. Vaidyanathan. Nested arrays: a novel approach to array processing with enhanced degrees of freedom. IEEE Transactions on Signal Processing. 58.8(2010):

3 Outline Outline Motivation and Goal Co-array Perspective Nested Array: Optimization DOA Estimation Numerical Examples Conclusion

4 Motivation and Goal Motivation and Goal Antenna arrays Direction-of-arrival (DOA) estimation Uniform linear arrays (ULA) Degree of freedom (DOF) is N 1 for ULA Goal: nested arrrays obtain O(N 2 ) DOF from only O(N) physical sensors

5 Co-array Perspective Signal Model A N element possibly nonuniform linear antenna array a(θ) R N 1 be the steering vector a(θ) i = e j 2π λ d i sin θ where d i denotes the position of the i-th sensor Assume D narrowband sources from directions {θ i, i = 1, 2,, D} with powers {σi 2, i = 1, 2,, D} Received signal x[k] = A s[k] + n[k] where A = [a(θ 1 ) a(θ 2 ) a(θ D )] denotes the array manifold matrix and s[k] = [s 1 [k] s 2 [k] s D [k]] T, and n[k] denotes temporally and spatially white, and uncorrelated from the sources.

6 Co-array Perspective Difference Co-Array R xx = E [xx H] = AR ss A H + σni 2 = A σ 2 1 σ 2 2. σ 2 D AH + σni 2 [ D ] z = vec(r xx ) = vec σi 2 (a(θ i )a H (θ i )) + σn1 2 n i=1 = (A A) p + σ 2 n1 n

7 Co-array Perspective Difference Co-Array A A behave like the manifold of a array whose sensor locations are given by the distinct values in the set {x i x j, 1 i, j N} Difference Co-Array: Let us consider an array of N sensors, with x i denoting the position vector of the i-th sensor. Define the set D = {x i x j }, i, j = 1, 2,, N, where D u denotes the distinct elements of the set D. The difference co-array is defined as the array which has sensors located at positions given by the set D u.

8 Co-array Perspective Difference Co-Array Weight Function: Define an integer valued function w : D u N + such that w(d) = no. of occurences of d in D, d D u. w(0) = N, 1 w(d) N 1, d D u \{0}. w(d) = w( d), d D u,d 0 w(d) = N(N 1) Cardinality of D u gives the defrees of freedom DOF max = N(N 1) + 1

9 Co-array Perspective Difference Co-Array Summary: If we use second-order statistics, then by exploiting the degrees of freedom (DOF) of the difference co-array, there is a possibility that we can get O(N 2 ) degrees of freedom using only O(N) physical elements Examples: an N element ULA 2N 1 elements ULA

10 Nested Array Nested Array Idea: Use a possibly nonuniform array its difference co-array has significantly more degrees of freedom than the original array. Nested Array: can be generated very easily in a systematic fashion and degrees of freedom of its co-array can be exactly predict. Two Level Nested Array K Levels Nested Array

11 Nested Array Two Level Nested Array A concatenation of two ULAs Inner ULA has N 1 elements with spacing d 1 S inner = {md 1, m = 1, 2,, N 1 } Outer ULA has N 2 elements with spacing d 2 = (N 1 + 1)d 1 S outer = {n(n 1 + 1)d 1, n = 1, 2,, N 2 } Difference co-array: a ULA with 2N 2 (N 1 + 1) 1 elements S ca = {nd 1, n = M,, M, M = N 2 (N 1 + 1) 1} 2N 2 (N 1 + 1) 1 freedoms in the co-array using only N 1 + N 2 elements

12 Nested Array Two Level Nested Array N 1 = 3, d 1 = d, N 2 = 3, d 2 = 4d, M = 11

13 Nested Array Two Level Nested Array max N 1,N 2 2N 2 (N 1 + 1) 1 s.t. N 1 + N 2 = N N optimal N 1, N 2 DOF even N 1 = N 2 = 1 2 N N N 2 odd N 1 = N 1, N 2 2 = N+1 2 obtain little over half of the maximum N N

14 Nested Array K Levels Nested Array Parameters: K, N i, i = 1, 2,, K N + S 1 = {nd, n = 1, 2,, N 1 } i 1 S i = {nd (N j + 1), n = 1, 2,, N i }, i = 2,, K j=1

15 Nested Array K Levels Nested Array Degrees of freedom in the corresponding difference co-array DOF K = 2 {[N 2 (N 1 + 1) 1] + [(N 3 1)(N 1 + N 2 + 1) +(N 1 + 1)] + + [(N K 1)(N 1 + N 2 + N N K 1 + 1) + (N 1 + N N K 2 + 1)} + 1 K DOF K = 2 K i=1 j=i+1 N i N j + N K 1 + 1

16 Nested Array K Levels Nested Array max max K N + N 1,,N K N + subject to DOF K K N i = N i=1 Theorem 1. Given a number N of sensors, the optimal number of nesting levels K and the number of sensors per nesting level are given by K = N 1, { 1, i = 1, 2,, K 1, N i = 0, i = K.

17 Nested Array K Levels Nested Array ( K ) 2 1 DOF K = 2 N i 2 = N 2 i=1 K i=1 K Ni 2 + 2N K 1 i=1 N 2 i + N K 1 1 j K 1 break N j into the sum of two smaller integers N j1 and N j2, i.e., N j = N j1 + N j2 (K + 1) levels of nested with {N 1, N 2,, N j 1, N j1, N j2, N j+1,, N K } sensors DOF K+1 = N 2 Ni 2 + Nj1 2 + Nj N K 1 i j,i=1

18 Nested Array K Levels Nested Array DOF = DOF K+1 DOF K = N 2 j N 2 j1 N2 j2 = 2N j1n j2 0 breaking up always increases the degrees of freedom similar analysis for j = K DOF = DOF K+1 DOF K = 2N K1 (N K2 1) 0 The corresponding difference array is a nonuniform linear array with degrees of freedom given by DOF opt = N(N 1) + 1 which is same as the upper bound. Structure of the Optimally Nested Array: the optimum nested array has sensors located at the positions given by the set S opt = {d, 2d, 4d, 8d,, 2 N 1 d} The optimally nested array is one with exponential spacing

19 Application Spatial Smoothing Based DOA Estimation Spatial smoothing works only for a ULA and we shall focus on the two-level nested array or any array whose difference co-array is a filled ULA. 2-level nested array, N 2 sensors in each level A A R N2 D with (N 2 2)/2 + N distinct rows

20 Application Spatial Smoothing Based DOA Estimation Theorem 2 : The matrix R ss as defined as where R ss = ˆR 2 1 ˆR = (A 11 ΛA H N N 11 + σni 2 ) 2 has the same form as the covariance matrix of the signal received by alonger ULA consisting of N 2 /4 + N/2 sensors and hence by applying MUSIC on R ss, uptp N 2 /4 + N/2 1 sources can be identified.

21 Numerical Examples Environment 6 sensor array (N = 6) 8 narrowband sources (D = 8) directions of arrival { 60, 45, 30, 0, 15, 30, 45, 60 } noise is assumed to be spatially and temporally white 2 level nested array with 3 sensors in each level N 2 /4 + N/2 1 = 11 SS-method: proposed spatial smoothing based technique QS-method: KR product based MUSIC in [1] which requires quasi stationarity

22 Numerical Examples MUSIC Spectrum with different numbers of snapshots

23 Numerical Examples MUSIC Spectrum with different numbers of snapshots

24 Numerical Examples Optimally Nested Array vs 2 Level Nested Array

25 Numerical Examples Optimally Nested Array vs 2 Level Nested Array

26 Conclusion Conclusion A novel nested array structure is proposed which can realize significantly more degrees of freedom Optimum nested array structure was found through solving a combinatorial optimization problem An alternative spatial smoothing based approach to underdetermined DOA estimation Future research?

27 Conclusion References W.K. Ma, T.H. Hsieh, and C.Y. Chi. DOA estimation of quasi-stationary signals via Khatri-Rao subspace. ICASSP, April 2009: