Adaptive beamforming. Slide 2: Chapter 7: Adaptive array processing. Slide 3: Delay-and-sum. Slide 4: Delay-and-sum, continued

Transcription

1 INF Adaptive beamforming p Adaptive beamforming Sven Peter Näsholm Department of Informatics, University of Oslo Spring semester 202 svenpn@ifiuiono Office phone number: Slide 2: Chapter 7: Adaptive array processing Minimum variance beamforming Generalized sidelobe canceler Signal coherence Spatial smoothing to de-correlate sources Eigenanalysis methods Signal/noise subspaces Eigenvector method MUSIC Slide 3: Delay-and-sum stacking adjustment of 0 M Slide 4: Delay-and-sum, continued w 0 w w 2 w 3

2 INF Adaptive beamforming p2 z(t) = M m=0 w m e jω m y m (t) w m : weight on signal m shading = apodization Conventional D-A-S: w independent of recorded signal data Slide 5: Delay-and-sum on vector form Monochromatic source: y m (t) = e j(ωt k x m) Delayed signal: y m (t m ) = y m (t)e jω m Define: Weights: w w 0 w w M Received signals: Ỹ(t) y 0(t) y (t) y M (t) Delayed Ỹ(t) :Y(t) y 0(t)e jω 0 y (t)e jω y M (t)e jω M M Beamf output: z(t) = w m e jω m y m (t) = w H Y(t) Power of z(t): m=0 P(z(t)) E { z(t) 2} = E { (w H Y)(w H Y) H} = E { w H YY H w } = w H E { YY H} w = w H Rw Slide 5: Delay-and-sum on vector form, continued Spatial covariance matrix in case of steering R E { YY H} E { } y 0(t)e jω 0 y0(t)e jω 0 E { } y 0(t)e jω 0 y(t)e jω E { } y 0(t)e jω 0 ym (t)ejω M ] E { } y (t)e jω y0 = (t)ejω 0 E { } y (t)e jω y (t)ejω E { } y (t)e jω ym (t)ejω M ] E{yM (t)e jω M y0 (t)ejω 0 } E{yM (t)e jω M y (t)ejω } E{yM (t)e jω M ym (t)ejω M } Slide 6: Delay-and-sum on vector form, continued Define: Steering vector: e e j k x 0 e j k x e j k x M z(t) = w H Y Power of z(t): P(e) E { z(t) 2} (e) = w H R(e)w if wm=, m ========= H R(e)

3 w INF Adaptive beamforming p3 Slide 7: Delay-and-sum on vector form, continued About e: Steering vector e e j k x 0 e j k x e j k x M Contains delays to focus in specific direction Represents unit amplitude signal, propagating in k direction In conventional D-A-S: w independent of received signal data Slide 8: Estimation of spatial covariance matrix Averaging in time Averaging in space = spatial smoothing = subarray averaging Slide 9: Minimum variance beamforming Assume narrow-band signals w 0 w w 2 w 3 Adaptive method [Latin: adaptare to fit to ] Allow w m to also be complex and/or negative Sensor weights defined not only as function of problem geometry, but also as function of received signals

4 INF Adaptive beamforming p4 Slide 0: Minimum variance beamforming, continued Minimum variance = Capon = Maximum likelihood (beamforming) Constrained optimization problem New steering direction e new calculation of element weights w m Minimum-variance key equations ( w 0 min w H Rw ), w w w M constraint: w H e = w = R e e H R e P(e) = e H R e [beampattern] [steered response] Slide : Minimum variance beamforming, continued Example Uniform linear array, M = 0 Comparing: Delay-and-sum Minimum-variance Investigating: Steered responses Beampatterns Slide 2: Comparison: delay-and-sum / minimum-variance source

5 INF Adaptive beamforming p5 Slide 3: Comparison: delay-and-sum / minimum-variance 2 sources

6 INF Adaptive beamforming p6 Slide 4: Comparison: delay-and-sum / minimum-variance 2 closely spaced sources

7 INF Adaptive beamforming p7 Slide 5: Example: Ultrasound imaging Experimental setup Transducer Reflectors Slide 6: Example: Ultrasound imaging, continued Resulting images

8 INF Adaptive beamforming p8 Slide 7: Example: Ultrasound imaging, continued Experimental data, heart phantom Slide 8: Sound example: Microphone array Slide 9: Sound example, continued Single microphone Delay-and-sum mixwav gaute_daswav

9 INF Adaptive beamforming p9 Minimum-variance gaute_adaptivewav Slide 20: Minimum-variance beamformer: Uncorrelated signals required for optimum functioning Correlated signals may give signal cancellation Although constraint is fulfilled: signal propagating in the look-direction may be canceled by a correlated interferer Slide 2: Minimum-variance / D-A-S example 2 correlated sources Slide 22: Generalized sidelobe canceler Conventional D-A-S weights w c : steering in assumed propagation direction Adaptive/canceler portion: removal of other signals Blocking matrix B, blocks signal from assumed propagation direction from coming into canceler part w a : adaptive weights to emphasize what is to be removed Minimize total power Unconstrained minimization: ( min wc B H ) H ( w a R wc B H ) w a wa = ( BRB H) BRwc w a Full details: D&J pp

10 INF Adaptive beamforming p0 Slide 23: Eigenanalysis and Fourier analysis (Chapter 73) Eigenvalues / Eigenvectors of R Rv i = λ i v i } R Hermitian (self-adjoint) R = R H R Positive semidefinite x H Rx 0, x 0 Eigenvalues real & positive: λ i 0, i Eigenvectors v i : orthogonal For V v v 2 v M : V is unitary matrix V H V = I V = V H Slide 24: Eigenvalue decomposition Eigendecomposition of R & R : The spectral theorem R = M λ i v i vi H = VΛV H i= R = M i= λ i v i v H i = VΛ V H Note above: Same eigenvectors, but inverse eigenvalues Slide 25: [Signal + noise] & [noise] subspaces R = VΛV H = V s Λ s Vs H +V n Λ n Vn H }{{}}{{} from from signal noise Slide 26: Beamformer output powers

11 INF Adaptive beamforming p Minimum variance, ˆR = E { YY H} : full R estimate P MV (e) = e H V s Λ s Vs H }{{ e } signal-plus-noise subspace +e H V n Λ n VH n }{{ e = } noise subspace N s i= λ i e H v i 2 + M i=n s+ λ i e H v i 2 Eigenvector method, ˆR = P EV (e) = MUSIC, ˆR = M i=n s+ e H V n Λ M i=n s+ λ i v i vi H : noise-only R estimate n Vn H e = M i=n s+ λ i e H v i 2 v i vi H : normalized noise-only R estimate P MUSIC (e) = e H V n Vn He = M e H v i 2 i=n s+ Slide 27: [Signal + noise], & [noise] subspaces, continued R always Hermitian eigenvectors orthogonal N s largest eigenvectors: span the [signal+noise subspace] M N s smallest span the [noise subspace] Signal steering vectors [signal+noise subspace], [noise subspace] [Largest eigenvectors] [signal vectors] However: signal vectors are linear combinations of [largest eigenvectors] Slide 28: EV / MUSIC (Inverse of) projection of possible steering vectors onto the noise subspace MUSIC: Dropping λ i in ˆR noise whitening EV & MUSIC: peaks at Directions-Of-Arrival (DOAs) No amplitude preservation Require knowledge of # signals, N s How to estimate N s?

12 INF Adaptive beamforming p2 Slide 29: MV / EV / MUSIC comparison Slide 30: Situation with signal coherence R = SCS H + K n signal coherence rank deficient signal matrix SCS H For minimum-variance: May cause signal cancellation Gives signal+noise-subspace consisting of linear combinations of signal vectors EV & MUSIC: may fail to produce peaks at DOA locations Want to reduce cross-correlation terms in C Cure: Spatial smoothing (subarray averaging)