High Order Super Nested Arrays

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1 High Order Super Nested Arrays Chun-Lin Liu 1 and P. P. Vaidyanathan 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 (Caltech) High Order Super Nested Arrays SAM / 21

2 Outline 1 Introduction (DOA, Sensor Arrays,...) 2 Review of Super Nested Arrays 3 High Order Super Nested Arrays 4 Numerical Examples 5 Concluding Remarks Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

3 Introduction (DOA, Sensor Arrays,...) Outline 1 Introduction (DOA, Sensor Arrays,...) 2 Review of Super Nested Arrays 3 High Order Super Nested Arrays 4 Numerical Examples 5 Concluding Remarks Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

4 Introduction (DOA, Sensor Arrays,...) DOA estimation in the presence of mutual coupling 1 DOA θ i Monochromatic Uncorrelated Sources Sensor arrays Mutual coupling DOA Estimators Estimated DOA θ i We will develop new sparse arrays with less mutual coupling. 1 Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

5 Introduction (DOA, Sensor 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 (Caltech) High Order Super Nested Arrays SAM / 21

6 Nested arrays 1 Introduction (DOA, Sensor Arrays,...) The nested array N 1 = 4, N 2 = 4. Difference coarray S = N 1 + N 2 = Dense ULA N 1 sensors spacing 1 Sparse ULA N 2 sensors spacing N D = {n 1 n 2 n 1, n 2 S} D = O(N 1 N 2 ) For sufficient number of snapshots, ( U 1)/2 = O(N 1 N 2 ) uncorrelated sources can be identified. (U = Central ULA part of D) 1 Pal and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

7 Review of Super Nested Arrays Outline 1 Introduction (DOA, Sensor Arrays,...) 2 Review of Super Nested Arrays 3 High Order Super Nested Arrays 4 Numerical Examples 5 Concluding Remarks Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

8 Review of Super Nested Arrays Super nested arrays 1 1 Super nested arrays have the same number of sensors as nested arrays. 2 Super nested arrays have the same difference coarrays as nested arrays. In particular, no holes. 3 Super nested arrays are more sparse than nested arrays, i.e., super nested arrays have less mutual coupling. Nested array N 1 = 13, N 2 = 5. Super nested array N 1 = 13, N 2 = 5. 1 Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

9 Review of Super Nested Arrays How to characterize mutual coupling in arrays? 1 The weight function w(m) The number of sensor pairs with separation m. Nested array, N 1 = N 2 = Super nested array, N 1 = N 2 = More sparse Less mutual coupling w(1) w(2) w(3) Liu and Vaidyanathan, IEEE Trans. Signal Proc., Nested Super Nested Super Nested Super nested nested nested Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

10 High Order Super Nested Arrays Outline 1 Introduction (DOA, Sensor Arrays,...) 2 Review of Super Nested Arrays 3 High Order Super Nested Arrays 4 Numerical Examples 5 Concluding Remarks Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

11 High Order Super Nested Arrays Goal: Desired properties of super nested arrays They should have the same number of sensors as nested arrays, S High order super nested = S Super nested = S Nested. They should have the same difference coarray as nested arrays, D High order super nested = D Super nested = D Nested. (In particular, no holes) They should be more sparse than nested arrays, w High order super nested (1) w Super nested (1) w Nested (1), w High order super nested (2) w Super nested (2) w Nested (2), Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

12 High Order Super Nested Arrays 2D representations for 1D nested arrays 1 The nested array with N 1 = N 2 = Layer 5 Layer 4 Layer 3 Sparse ULA Layer 2 Dense ULA Layer 1 1 Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

13 High Order Super Nested Arrays High order super nested arrays 2D Rep D Rep Second-order super nested array N 1 = 15, N 2 = 7, Q = 2 1D Rep. High-order super nested array N 1 = 15, N 2 = 7, Q = 3. 1D Rep. Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

14 High Order Super Nested Arrays The hierarchy of Qth-order super nested arrays 1 S (Q) = Parent nested array S (1) Dense ULA Sparse ULA ( Q q=1 X(Q) q Second-order super nested array S (2) X (2) 2 X (2) 1 Y (2) 1 Y (2) 2 Z (2) 1 Z (2) 2 ) Y (Q) q Z (Q) 1 Z (Q) 2, Rule 2 Rule 1 Rule 1 Rule 2 Rule 3 S (3) S (4) S(1) Rule 2 X (3) 3 X (3) 2 X (3) 1 Y (3) 1 Y (3) 2 Y (3) 3 Z (3) 1 Z (3) 2 Rule 1 Rule 1 Rule 1 Rule 1 Rule 2 Rule 3 X (4) 4 X (4) 3 X (4) 2 X (4) 1 Y (4) 1 Y (4) 2 Y (4) 3 Y (4) 4 Z (4) 1 Z (4) 2 S (2) S (3) S (4) 1 MATLAB routines are available at Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

15 High Order Super Nested Arrays Main properties of super nested arrays: 1) Difference coarray D Nested D (Q) Super nested D (Q) Super nested = D Nested if Q 3, N 1 and N 2 are sufficiently large. 1 Properties of D (Q) Super nested : Contiguous integers. Hole-free. 1 The lower bounds are given in the papers. Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

16 High Order Super Nested Arrays Main properties of super nested arrays: 2) Weight functions w(1) w(2) w(3) Nested N 1 N 1 1 N 1 2 { 2, if N 1 is even, Super nested High order super nested Q 3 1, if N 1 is odd. { 2, if N 1 is even, 1, if N 1 is odd, { N 1 3, N 1 1, if N 1 is even, if N 1 is odd, 2 N 1 /4 + 1, if N 1 is odd, N 1 /2 + 1, if N 1 = 8k 2, N 1 /2 1, if N 1 = 8k + 2, N 1 /2, otherwise, 3, if N 1 = 4, 6, 4, if N 1 is even N 1 8, 1, if N 1 is odd, { 5, if N 1 is even, 2, if N 1 is odd, Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

17 Numerical Examples Outline 1 Introduction (DOA, Sensor Arrays,...) 2 Review of Super Nested Arrays 3 High Order Super Nested Arrays 4 Numerical Examples 5 Concluding Remarks Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

18 Simulation procedure Numerical Examples θ 1 = 0.25 θ D = 0.25 D = 30 Uncorrelated Sources Spatial Smoothing MUSIC (SS MUSIC) 1 34 Sensors Estimated normalized DOA ˆ θi 0 db SNR, 200 snapshots, RMSE E = ( 1 D D i=1 (ˆ θi θ i ) 2 ) 1/2 1 Pal and Vaidyanathan, IEEE Trans. Signal Proc., 2010; Liu and Vaidyanathan, IEEE Signal Proc. Lett., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

19 Numerical Examples MUSIC spectra (34 sensors, 30 sources) Nested array (E = ) Coprime array (E = ) Super nested array Q = 2, E = High order super nested array Q = 3, E = Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

20 Concluding Remarks Outline 1 Introduction (DOA, Sensor Arrays,...) 2 Review of Super Nested Arrays 3 High Order Super Nested Arrays 4 Numerical Examples 5 Concluding Remarks Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

21 Concluding remarks Concluding Remarks High order super nested arrays They have the same number of sensors as (super) nested arrays. They have the same difference coarray as (super) nested arrays if N 1 and N 2 are sufficiently large. They have reduced mutual coupling than (super) nested arrays. They can be constructed recursively from (super) nested arrays. In the future, decoupling algorithms will improve the performance. 1 For more information, please go to our project website: systems.caltech.edu/dsp/students/clliu/supernested.html Thank you! 1 Friedlander and Weiss, IEEE Trans. Antennas Propag., 1991; BouDaher, Ahmad, Amin, and Hoorfar, EUSIPCO, Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 21

22 The data model (ideal) x S = D ) A i v S ( θi + ns, i=1 S: An integer set for the sensor locations, in units of λ/2. θ i = (d/λ) sin θ i : the normalized DOA ( 1/2 θ i < 1/2). A i : The complex amplitude for the ith source. v S ( θ i ) = [e j2π θ i n ] n S : steering vectors. Statistical Assumptions A i : zero mean, variance σ 2 i. n S : zero mean, covariance σ 2 I. Sources are uncorrelated: E[A i A j ] = σ2 i δ i,j. Sources are uncorrelated to the noise: E[A i n H S ] = 0. θ i is considered to be fixed but unknown. Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

23 The data model in the presence of mutual coupling 1 D x S = A i Cv S ( θ i ) + n S, i=1 C: mutual coupling matrix satisfying { c C n1,n 2 = n1 n 2, if n 1 n 2 B, 0, otherwise, n 1 and n 2 are sensor locations. 1 = c 0 > c 1 > c 2 > > c B. In this paper, we assume that c k /c l = l/k. Mutual coupling is a function of sensor separations. 1 Friedlander and Weiss, IEEE Trans. Antennas Propag., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

24 The mutual coupling models in simulations B = 100, c 1 = 0.6e j π 3, cl = c 1 l e j π 8 (l 1), for l = 2, 3,..., B Coefficients c 1 c 2 c 3 c 4 c 5 Real Imaginary Magnitudes of mutual coupling matrices, [C] i,j Nested array Coprime array Second-order super nested array Third-order super nested array Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

25 Another mutual coupling model: King s formula 1 If the sensor array is a linear dipole array, C can be written as C = (Z A + Z L )(Z + Z L I) 1, where Z A and Z L are the element/load impedance, respectively. Z n1,n is given by { 2 η0 4π ( ln(2βl) Ci(2βl) ) + jsi(2βl)), if n 1 = n 2, η 0 ( R 4π n1,n2 + j X n1,n2, if n 1 n 2. Here η 0 = µ 0 /ɛ 0 120π is the intrinsic impedance. β = 2π/λ is the wavenumber, where λ is the wavelength. l is the length of dipole antennas. R and X are R n1,n 2 = sin(βl) ( Si(u 0 ) + Si(v 0 ) + 2Si(u 1 ) 2Si(v 1 )) + cos(βl)(ci(u 0 ) + Ci(v 0 ) 2Ci(u 1 ) 2Ci(v 1 ) + 2Ci(βd n1,n 2 )) X n1,n 2 = sin(βl) ( Ci(u 0 ) + Ci(v 0 ) + 2Ci(u 1 ) 2Ci(v 1 )) + cos(βl)( Si(u 0 ) Si(v 0 ) + 2Si(u 1 ) + 2Si(v 1 ) 2Si(βd n1,n 2 )) + ( ) 2Ci(u 1 ) + 2Ci(v 1 ) 4Ci(βd n1,n 2 ), ( ) 2Si(u 1 ) + 2Si(v 1 ) 4Si(βd n1,n 2 ). where d n1,n 2 = n 1 n 2 λ/2 is the distance between sensors. The parameters u 0, v 0, u 1, and v 1 are ( ) ( ) u 0 = β d 2 n 1,n + l 2 l, v 2 0 = β d 2 n 1,n + l 2 + l, 2 ( ) ( ) u 1 = β d 2 n 1,n l 2 0.5l, v 2 1 = β d 2 n 1,n l l. 2 Here Si(u) = u 0 sin t t dt and Ci(u) = u cos t dt are sine/cosine integrals. t 1 King, IRE Trans. Antennas Propag., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

26 Properties of the weight functions w(m) 1 The weight function w(m) The number of sensor pairs with separation m. For any linear array with N sensors, weight functions satisfy 1 w(0) equals the total number of sensors, i.e., w(0) = N. 2 The sum of the weight functions is purely dependent on N. w(m) = N 2. m D 3 Weight functions are symmetric. w(m) = w( m), for m D. 1 Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

27 Performance over SNR 1 (a) RMSE ULA MRA Nested Coprime Super Nested, Q = 2,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = 16,N 2 = SNR (db) 34 sensors, 20 equal-power sources, 500 snapshots, dipole model, Z A = Z L = 50, l = λ/2, θ i = (i 1)/(D 1), 1000 runs. 1 Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

28 Performance over Snapshots 1 (b) RMSE ULA MRA Nested Coprime Super Nested, Q = 2,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = 16,N 2 = Snapshots K 34 sensors, 20 equal-power sources, 0dB SNR, dipole model, Z A = Z L = 50, l = λ/2, θ i = (i 1)/(D 1), 1000 runs. 1 Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

29 Performance over Number of sources 1 (c) RMSE ULA MRA Nested Coprime Super Nested, Q = 2,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = 16,N 2 = Number of sources D 34 sensors, equal-power sources, 0dB SNR, 500 snapshots, dipole model, Z A = Z L = 50, l = λ/2, θ i = (i 1)/(D 1), 1000 runs. 1 Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

30 Performance over two closely spaced sources ULA MRA Nested Coprime Super Nested, Q = 2,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = 16,N 2 = 18 RMSE θ 34 sensors, two equal-power sources at θ 1 = θ/2, θ 1 = 0.2 θ/2, 0dB SNR, 500 snapshots, dipole model, Z A = Z L = 50, l = λ/2, 1000 runs. 1 Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

31 Performance over mutual coupling models 1 (a) RMSE ULA MRA Nested Coprime Super Nested, Q = 2,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = 16,N 2 = c 1 10 sources, 34 sensors 0dB SNR, 500 snapshots, Toeplitz model, phases of c l are random. θ i = (i 1)/(D 1), 1000 runs. 1 Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

32 Performance over mutual coupling models 1 (b) RMSE ULA MRA Nested Coprime Super Nested, Q = 2,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = 16,N 2 = c 1 20 sources, 34 sensors 0dB SNR, 500 snapshots, Toeplitz model, phases of c l are random. θ i = (i 1)/(D 1), 1000 runs. 1 Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

33 Performance over mutual coupling models MRA Nested Coprime Super Nested, Q = 2,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = N 2 = 17 Super Nested, Q = 3,N 1 = 16,N 2 = 18 (c) RMSE c 1 40 sources, 34 sensors 0dB SNR, 500 snapshots, Toeplitz model, phases of c l are random. θ i = (i 1)/(D 1), 1000 runs. 1 Liu and Vaidyanathan, IEEE Trans. Signal Proc., Liu and Vaidyanathan (Caltech) High Order Super Nested Arrays SAM / 12

SUPER NESTED ARRAYS: SPARSE ARRAYS WITH LESS MUTUAL COUPLING THAN NESTED ARRAYS

SUPER NESTED ARRAYS: SPARSE ARRAYS WITH LESS MUTUAL COUPLING THAN NESTED ARRAYS Chun-Lin Liu 1 and P. P. Vaidyanathan 2 Dept. of Electrical Engineering, 136-93 California Institute of Technology, Pasadena,