Direction of Arrival Estimation: Subspace Methods. Bhaskar D Rao University of California, San Diego

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1 Direction of Arrival Estimation: Subspace Methods Bhaskar D Rao University of California, San Diego brao@ucsdedu

2 Reference Books and Papers 1 Optimum Array Processing, H L Van Trees 2 Stoica, P, & Moses, R L (2005) Spectral analysis of signals (Vol 1) Upper Saddle River, NJ: Pearson Prentice Hall 3 Schmidt, R (1986) Multiple emitter location and signal parameter estimation IEEE transactions on antennas and propagation, 34(3), Roy, Richard, and Thomas Kailath ESPRIT-estimation of signal parameters via rotational invariance techniques IEEE Transactions on acoustics, speech, and signal processing 377 (1989): Rao, Bhaskar D, and K S Arun Model based processing of signals: A state space approach Proceedings of the IEEE 802 (1992):

3 Narrow-band Signals Assumptions D 1 x(ω c, n) = V(ω c, k s )F s [n] + V(ω c, k l )F l [n] + Z[n] l=1 F s [n], F l [n], l = 1,, D 1, and Z[n] are zero mean E( F s [n] 2 ) = p s, and E( F l [n] 2 = p l, l = 1,, D 1, and E(Z[n]Z H [n]) = σ 2 z I All the signals/sources are uncorrelated with each other and over time: E(F l [n]f m[p]) = p l δ[l m]δ[n p] and E(F l [n]f s [p]) = 0 The sources are uncorrelated with the noise: E(Z[n]F l [m]) = 0

4 Adaptive Beamforming MVDR, MPDR, and LMMSE beamformers and their relationship Generalized Sidelobe Canceller Transmit and Receive Beamforming MVDR Power Spectrum Estimation

5 Direction of Arrival (DOA) Estimation D 1 x[n] = V l F l [n] + Z[n], n = 0, 1,, L 1 l=0 Here we have chosen to denote V s by V 0 for simplicity of notation Note V l = V(k l ) We will refer to the L vector measurements as L snapshots Goal: Estimate the direction of arrival k l or equivalently (θ l, φ l ), l = 0,, D 1 Can also consider estimating the source powers p l, l = 0, 1,, D 1, and noise variance σ 2 z ULA : N 1 j V(ψ) = [e 2 ψ,, e j N 1 2 ψ ] T N 1 j = e 2 ψ [1, e jψ, e j2ψ,, e j(n 1)ψ ] T Note for a ULA, V(ψ) = JV (ψ), where J is all zeros except for ones along the anti-diagonal Also note J = J 2

6 ULA and Time Series x 0 [n] x 1 [n] x N 1 [n] D 1 = l=0 1 e jψ l e j(n 1)ψ l e j N 1 2 ψ l F l [n]+z[n], n = 0, 1,, L 1 Considerable similarity with a time series problems composed of complex exponentials x[n] = D 1 l=0 c le jω l n + z[n], n = 0, 1,, (M 1) In vector form x[0] 1 z[0] x[1] e jω l z[1] x[m 1] D 1 = l=0 e j(m 1)ω l c l + z[m 1] The time series, sum of exponentials, problem with M samples is equivalent to a ULA with M sensors and one (vector) measurement or one snapshot

7 Estimation of the Frequencies of the Exponential In array processing we have a limited number of sensors N but can compensate potentially with L snapshots The time series problem is equivalent to one snapshot but a potentially large array if the number of samples M is large One complex exponential (Noise free): x[n] = c 0 e jω0n, n = 0, 1,, M 1 Interested in the frequency ω 0 Potential solution: Take the FFT and look at the peak Another option is to note that x[n] = c 0 e jω0(n 1) e jω0 = e jω0 x[n 1] In the noise free case, 2 samples is enough Can solve a least squares problem min a M 1 n=1 x[n] ax[n 1] 2 and find ω 0 from the estimated a

8 More than one exponentials Suppose we have two exponentials: x[n] = c 0 e jω0n + c 1 e jω1n, n = 0, 1,, M 1 A FFT based approach may fail to resolve the frequencies if they are close and the number of samples M is small Thumb rule ω 0 ω 1 > 2π M Can show x[n] = a 1 x[n 1] + a 2 x[n 2] and so one can solve a 1 and a 2 using a least squares technique In the absence of noise we just need 4 samples to set up a system of equations to solve for the parameters [ ] [ ] [ ] x[n] x[n 1] x[n 2] a1 = x[n 1] x[n 2] x[n 3] a 2 How to get the frequencies from the parameters a 1 and a 2? Can show that the zeros of the filter H(z) = 1 a 1 z 1 a 2 z 2 are located at e jω0 and e jω1

9 D exponentials Suppose we have a series composed of D exponentials: x[n] = D 1 l=0 c le jω l n, n = 0, 1,, M 1 A FFT based approach can be used with drawbacks as before: resolution and large number of samples may be needed Can show x[n] = D k=1 a kx[n k] and so one can solve a k, k = 1,, D using a least squares technique In the absence of noise we just need 2D samples to set up a system of equations to solve for the parameters x[n] x[n 1] x[n D] a 1 x[n 1] x[n 2] x[n D 1] a 2 x[n D + 1] = x[n D] x[n 2D 1] How to get the frequencies from the parameters a k, k = 1, D? Can show that the zeros of the filter H(z) = 1 D k=1 a kz k are located at e jω l, l = 0,, D 1 a D

10 Relationship of Zeros of Filters to e jω l Claim: A signal with D exponentials, ie x[n] = D 1 l=0 c le jω l n, is exactly predictable from its D past samples, ie x[n] = D k=1 a kx[n k] or x[n] D k=1 a kx[n k] = 0 Consider a FIR filter H(z) = 1 P k=1 b kx[n k], P D, with input x[n] and denote its output by e[n] e[n] = x[n] P b k x[n k] = k=1 If we minimize n e[n] 2, what is the result? D H(e ω k )c k e jω k n The output will be zero This is achieved by making H(e jω k ) = 0 at the D frequencies Since the filter is of order P D, we can select the filter zeros to meet this criteria and minimize the output The excess (P D) zeros can be placed arbitrarily Can be a source of confusion when determining the frequency estimates k=1

11 Subspace Methods D 1 x(n) = V(k l )F l [n] + Z[n] l=0 = [V 0, V 1,, V D 1 ] = VF[n] + Z[n] where V C N D, and F[n] C D 1 F s [n] F 1 [n] F D 1 [n] Note that E(F[n]) = 0 D 1, E(F[n]Z H [n]) = 0 D N + Z[n]

12 Covariance Matrix Source Covariance matrix S F = E(F[n]F H [n]) = S H F = S H s S H 1 S H D 1, where S F C D D, S H s C 1 D and the diagonal elements are p s, p 1,, p D 1 For uncorrelated sources S F is a diagonal matrix, ie S F = diag(p s, p 1,, p D 1 ) and S H s = [p s, 0,, 0] Data Covariance Matrix S x = S s + σ 2 z I = VS F V H + σ 2 z I S x C N N, S s C N N, S F C D D, and V C N D

13 Signal and Noise Subspaces Subspace methods: Based on eigen-decomposition of a covariance matrix Eigen-decomposition of S s : Note S s = VS F V H S s has rank D, is Hermitian symmetric and positive semi-definite Hence S s has D non-zero eigenvalues and corresponding orthornomal eigenvectors S s = [q 1, q 2,, q D ] λ s λ s 2 0 q H 1 q H 2 = E sλ s E H s 0 0 λ s D q H D where E s C N D, and Λ s = diag(λ i ) Note E H s E s = I D D and P s = E s E H s is an orthogonal projection matrix onto R(E s ), ie P s = P H s and P s = P 2 s Important Observation: R(V) = R(E s ) and R(E s ) is referred to as Signal subspace

14 Subspaces Cont d Choose E n = [q D,, q N ] such that E H n E n = I (N D) (N D) and E H n E s = 0 (N D) D Defining E = [E s, E n ] C N N, we have E H E = EE H = I N N Important Observation: R(E n ) R(V) = R(V s ) R(E n ) is referred as Noise subspace [ ] [ ] [ ] S s = E s Λ s E H Λs 0 E H s = [E s, E n ] s Λs E H = E E H n 0 0 σz 2 I = σz 2 EE H = E (σz 2 I) E H = E σz σz σz 2 EH In fact E can be replaced by any orthonormal matrix in the expansion for σ 2 z I For subspace methods, choosing a matrix compatible with S s is more useful

15 Eigen decomposition of Data Covariance Matrix S x = S s + σ 2 z I [ Λs 0 = E 0 0 ] E H + E σz σz σz 2 [ ] Λs + σz = E 2 I D D 0 0 σz 2 E H I (N D) (N D) = E s (Λ s + σz 2 I D D ) E H s + σz 2 E n E H n D N = (λ s l + σz 2 )q l q H l + σz 2 q l q H l l=1 l=(d+1) EH

16 Observations S x = S s + σz 2 I = E s (Λ s + σz 2 I D D ) E H s + σz 2 E n E H n D N = (λ s l + σz 2 )q l q H l + σz 2 q l q H l l=1 l=(d+1) S x has eigenvalues λ l = λ s l + σ 2 z, l = 1,, D, and λ l = σ 2 z, l = (D + 1),, N The number of sources D can be identified by examining the multiplicity of the smallest eigenvalue which is expected to be σ 2 z with multiplicity (N D) The eigenvectors corresponding to the signal subspaces and noise subspaces are readily available from the eigen decomposition of S x, ie E s and E n are easily extracted

17 MUltiple SIgnal Classification (MUSIC) MUSIC uses the noise subspace spanned by the noise eigenvectors E n Since R(V) = R(E s ) R(E n ), we have V l R(E n ), l = 0, 1,, (D 1) or V H l E n = 0, l = 0, 1,, (D 1) Many options to use this orthogonality property All methods start with S x, usually estimated from the data MUSIC Algorithm (also known as Spectral-MUSIC) Perform an eigen-decomposition of S x Estimate D and the signal subspace E s and the noise subspace E n Compute the spatial MUSIC spectrum as follows: P MUSIC (k) = 1 V H (k)e n 2 Note that V H (k)e n 2 = V H (k)e n E H n V(k) = V H (k)p n V(k) where P n = E n E H n The peaks in P MUSIC (k) gives us the DOA

18 MUSIC Example

19 ULA and MUSIC N 1 j ULA : V(ψ) = e 2 ψ [1, e jψ, e j2ψ,, e j(n 1)ψ ] T N 1 j Will drop e 2 ψ for this discussion for simplicity and because it does not matter V H (ψ)e n 2 = N l=d+1 V H (ψ)q l 2 = N l=d+1 q l (e jψ ) 2 where q l (e jψ ) = V H (ψ)q l = N 1 m=0 q l(m)e jmψ, with q l = [q l [0], q l [1],, q l [N 1]] T q l (e jψ ) is the Fourier transform of the impulse response of a filter formed from the eigenvector q l Defining B(e jψ ) = N l=d+1 q l(e jψ ) 2 The MUSIC spectrum is P MUSIC (ψ) = 1 B(e ψ ), and it will have peaks at ψ l, l = 0, 1,, (D 1) corresponding to the DOA Then B(e jψ l ) = 0, l = 0, 1,, (D 1)

20 Root-MUSIC With B(e jψ ) = N l=d+1 q l(e jψ ) 2, in the z-domain we have B(z) = N l=d q l(z)q l ( 1 z ) Alternately B(z) = N l=d+1 B l(z) where B l (z) = q l (z)q l ( 1 z ) = N 1 l= (N 1) r q l [m]z m, Each of the B l (z) represents a Moving Average Spectrum and the sum B(z) is also a Moving Average spectrum B(z) = N 1 l= (N 1) r q [m]z m = H rm (z)h rm( 1 z ), where r q [m] is an autocorrelation sequence and H rm (z) is obtained by spectral factorization B(e jψ l ) = 0, l = 0, 1,, (D 1) implies H rm (z) will have D of it s zeros at e jψ l, where ψ l corresponds to the DOA The remaining (N 1 D) zeros will be inside the unit circle assuming we use a minimum-phase spectral factor

21 Root-MUSIC Algorithm Perform an eigen-decomposition of S x Estimate D and the signal subspace E s and the noise subspace E n Compute q l (z) = N 1 m=0 q l[m]z m, l = D + 1,, N 1 Compute B(z) = N l=d+1 B l(z) where B l (z) = q l (z)q l ( 1 z ) = N 1 l= (N 1) r q l [m]z m Perform a spectral factorization to find a minimum phase factor H rm (z), ie B(z) = H rm (z)h rm( 1 z ) The D roots on/close to the unit circle are identified and their argument used to estimate the DOA

22 Root-MUSIC and Spectral-MUSIC Example