Beamspace Adaptive Beamforming and the GSC

Transcription

1 April 27, 2011

2 Overview The MVDR beamformer: performance and behavior. Generalized Sidelobe Canceller reformulation. Implementation of the GSC. Beamspace interpretation of the GSC. Reduced complexity of beamspace MVDR.

3 Repetition: MVDR Beamforming 1 Model: Signal, (spatially white) noise, and interference for M-element array x = A { d + n, E x x H} = R s + R n, R s = A 2 d d H, R n = R i + σw 2 I (1) For spatially white noise only, the DAS beamformer is optimal (in the sense of minimum noise power in output): y DAS = w H DAS x for w DAS = 1 M d (2) For spatially non-white interference, the Minimum Variance beamformer minimizes interference-plus-noise power in the beamformer output: { w w mv = argmin w E H x 2} = argmin w w H R w (3) In other words: { w H E MV n 2} { w E H n 2} for all weight vectors w (4) Note: This is actually minimum power, not minimum variance. Subtle difference, theoretical equivalence. Exchange R for R n...

4 Repetition: MVDR Beamforming 2 MV weight vector for the case of one single interfering source with power σi 2 and propagation vector d i : w MV = ΛM ( ) Mσi 2 w σw 2 das,s ρ si w σw 2 + Mσi 2 das,i for w das,s = vs M, w das,i = v i M (5) Interpretation: DAS beamformer steered towards signal minus (scaled) DAS beamformer steered towards interference. Scaling depends on INR, i.e. σ 2 i σ 2 w Magnitude (db) Magnitude (db) Angle (deg) Angle (deg)

5 Generalized Sidelobe Canceller Constrained minimization is sometimes difficult to implement and analyze. However: MVDR can be reformulated as unconstrained minimization. First suggested by Griffiths and Jim (1982) as an alternative implementation of Frosts Linearly Constrained MV (LCMV) beamformer (1972). This implementation is usually referred to as the Griffiths-Jim beamformer or the GSC. Given a matrix B C M,M 1 such that d H B = 0. Then the constrained optimization problem: min w w H R w s.t. w H d = 1 (6) is identical to the unconstrained optimization problem: min( 1 β M d Bβ) H R( 1 M d Bβ) (7) with solution: β = ( ) 1 B H RB B H R d M (8)

6 GSC Implementation From now on: Assume d = 1, i.e. signal arriving from broadside. Can be implemented as a transversal adaptive filter (using e.g. LMS or RLS algorithm). Looks like a Wiener filter, in that it subtracts adaptively filtered noise from desired signal......however, it does not produce MMSE output. Note: Sensitive to signal-interference correlation, just like MVDR. 30 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-30, NO. 1, JANUAR where X and A are the M - 1 dimensional signal and cient vectors. The overall output of the generalized sidelobe can structure y (k) is Fig. 4. Generalized sidelobe canceling form of linearly constrained adaptive array processing algorithm. Because y ~(k) contains no desired signal terms, the res of the processor to the desired signal s(k)l is that pro only by y, (k). Thus from (22)-(25) the output due presence of only the desired signal satisfies the const defined by (9), regardless of W,. In addition, since contains only noise and interference terms, finding the filter coefficients Al (k) which minimize the power con in y(k) is equivalent to finding the minimum variance early constrained beamformer. The unconstrained leastsquare (LMS) algorithm [ 121 can be employed to adap filter coefficients to the desired solution, fixed nonadaptive coefficients. In typical applications the weights W, are chosen so as to trade off the relationship be- The step size 1-1 is normalized by the total power contain tween array beamwidth and average sidelobe level [ l l]. the X (k - Carl-Inge Colombo Nilsen (One widely (carlingn), 2) using methods analogous to those des used method UiO employs Beamspace Chebyshev Adaptive polynomials Beamforming and the GSC

7 GSC Interpretation: Beamspace Blocking matrix B suggested by Griffiths and Jim: B = What signals are processed by the lower branch of the GSC? (9) x m = x m x m+1 for m = 0, 1,, M 2 (10) Two-element endfire beams. Good for broadband; broadside nulling for all frequencies Magnitude (db) Angle (deg)

8 GSC Interpretation: Beamspace D = 1 M [ d0,, ] [ ] d M 1, dm = e j 2πm M n (11) n Special case: ULA with DFT matrix B = D without first column ( d 0 = w DAS for broadside arrival). Invertible transformation: D H D = DD H = 1 M I. 5 Beampattern magnitude (db) Signal beam 1st Left Noise beam 5th Right Noise beam Angle (degrees)

9 GSC Interpretation: Beamspace The constrained optimization problem becomes: { ( w H D )(D x) H min E w min w BS E 2} s.t. w H 1 = 1 { w H BS x BS 2 } = w H BSR BS w BS s.t. w BS e 0 = 1 (12) Beamspace solution: w BS = R 1 BS e0 e H 0 R 1 BS e0 (13)

10 GSC Interpretation: Beamspace Beamspace beamformer output: M 1 y BS = x BS,0 + wbs,mx BS,m (14) m=1 Interpretation is more obvious: DAS beamformer minus weighted set of other DAS beamformers. Note that there is only information about the signal in x BS,0. Certain beams x BS,m contain more information about interference than others. Idea: Remove those x BS,m that are expected to contain little or no information about interference. Result: x BS becomes smaller, i.e. R BS becomes smaller and easier to invert. Inversion of R is O(M 3 ). Reduced-dimension beamspace presents a way of using a priori knowledge in the adaptive beamformer......but is this realistic knowledge?

11 Beamspace Example: Single Beam Try B = d m, base weights on single-snapshot noise cov. matrix R n = n n H : β = Yields beamspace MV weight vector: ( d H m R n dm ) d H m R n w das = y p,dy p,das y p,d 2 + σ2 w M (15) w bs = w das β d m (16) Yields interference in output: y p,d y p,bs = y p,das (1 2 y p,d 2 + σ2 w M ) (17) Is the choice of d m arbitrary? What is the impact on the white noise gain? What is the impact of spatially white noise?

12 GSC Interpretation: Beamspace Special case: Adaptive Sidelobe Reduction (from Synthetic Aperture Radar). Summary: Only use x BS,m for m = 0, 1, M 1 and set w BS,0 = 1, w BS,1 = w BS,M 1 = α. Complexity reduced from determining M weights to 1 weight. Additionally, solution is on the form: [ w ASR ] m = 1 + α cos ( ) 2πm M which corresponds to a known family of windows (including Hamming and Hann). (18)

13 Conclusions GSC interpretation of MVDR beamformer yields unconstrained optimization problem. Unconstrained optimization problems are often easier to analyze and implement. Beamspace interpretation of GSC can give reduced complexity. Example: Ultrasound imaging. Reduction from 64-dimensional element space to 3-dimensional beamspace with similar performance. In general: O(M) O(3).