Dept. E.E./ESAT-STADIUS, KU Leuven homes.esat.kuleuven.be/~moonen/ Digital Audio Signal Processing Lecture-3: Microphone Array Processing - Adaptive Beamforming - Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen/

Overview Lecture 3: Adaptive beamforming Lecture 2 Introduction & beamforming basics Fixed beamforming Lecture 3: Adaptive beamforming Introduction Review of “Optimal & Adaptive Filters” LCMV beamforming Frost beamforming Generalized sidelobe canceler

Introduction + Beamforming = `Spatial filtering’ Classification based on microphone directivity patterns and microphone array configuration Classification Fixed beamforming: data-independent, fixed filters Fm Example: delay-and-sum, filter-and-sum Adaptive beamforming: data-dependent adaptive filters Fm Example: LCMV-beamformer, Generalized Sidelobe Canceler + :

Introduction Data model & definitions Microphone signals Output signal after `filter-and-sum’ is Array directivity pattern Array gain =improvement in SNR for source at angle 

Overview Lecture 3: Adaptive beamforming Lecture 2 Introduction & beamforming basics Fixed beamforming Lecture 3: Adaptive beamforming Introduction Review of “Optimal & Adaptive Filters” LCMV beamforming Frost beamforming Generalized sidelobe canceler

Optimal & Adaptive Filters Review 1/6

Optimal & Adaptive Filters Review 2/6 Norbert Wiener

Optimal & Adaptive Filters Review 3/6

Optimal & Adaptive Filters Review 4/6

Optimal & Adaptive Filters Review 5/6 (Widrow 1965 !!)

Optimal & Adaptive Filters Review 6/6

Overview Lecture 3: Adaptive beamforming Lecture 2 Introduction & beamforming basics Fixed beamforming Lecture 3: Adaptive beamforming Introduction Review of “Optimal & Adaptive Filters” LCMV beamforming Frost beamforming Generalized sidelobe canceler

LCMV-beamforming + Adaptive filter-and-sum structure: : Aim is to minimize noise output power, while maintaining a chosen response in a given look direction (and/or other linear constraints, see below). This is similar to operation of a superdirective array (in diffuse noise), or delay-and-sum (in white noise), cfr (**) Lecture 2 p.21&29, but now noise field is unknown ! Implemented as adaptive FIR filter : + :

LCMV-beamforming LCMV = Linearly Constrained Minimum Variance f designed to minimize power (variance) of total output (read on..) z[k] : To avoid desired signal cancellation, add (J) linear constraints Example: fix array response in look-direction ψ for sample freqs wi, i=1..J (**)

LCMV-beamforming LCMV = Linearly Constrained Minimum Variance With (**) (for sufficiently large J) constrained total output power minimization approximately corresponds to constrained output noise power minimization (why?) Solution is (obtained using Lagrange-multipliers, etc..):

Overview Lecture 3: Adaptive beamforming Lecture 2 Introduction & beamforming basics Fixed beamforming Lecture 3: Adaptive beamforming Introduction Review of “Optimal & Adaptive Filters” LCMV beamforming Frost beamforming Generalized sidelobe canceler

Frost Beamforming Frost-beamformer = adaptive version of LCMV-beamformer If Ryy is known, a gradient-descent procedure for LCMV is : in each iteration filters f are updated in the direction of the constrained gradient. The P and B are such that f[k+1] statisfies the constraints (verify!). The mu is a step size parameter (to be tuned

Frost Beamforming Frost-beamformer = adaptive version of LCMV-beamformer If Ryy is unknown, an instantaneous (stochastic) approximation may be substituted, leading to a constrained LMS-algorithm: (compare to LMS formula)

Overview Lecture 3: Adaptive beamforming Lecture 2 Introduction & beamforming basics Fixed beamforming Lecture 3: Adaptive beamforming Introduction Review of “Optimal & Adaptive Filters” LCMV beamforming Frost beamforming Generalized sidelobe canceler

Generalized Sidelobe Canceler (GSC) GSC = alternative adaptive filter formulation of the LCMV-problem : constrained optimisation is reformulated as a constraint pre-processing, followed by an unconstrained optimisation, leading to a simpler adaptation scheme LCMV-problem is Define `blocking matrix’ Ca, ,with columns spanning the null-space of C Define ‘quiescent response vector’ fq satisfying constraints Parametrize all f’s that satisfy constraints (verify!) I.e. filter f can be decomposed in a fixed part fq and a variable part Ca. fa

Generalized Sidelobe Canceler (GSC) GSC = alternative adaptive filter formulation of the LCMV-problem : constrained optimisation is reformulated as a constraint pre-processing, followed by an unconstrained optimisation, leading to a simpler adaptation scheme LCMV-problem is Unconstrained optimization of fa : (MN-J coefficients)

Generalized Sidelobe Canceler GSC (continued) Hence unconstrained optimization of fa can be implemented as an adaptive filter (adaptive linear combiner), with filter inputs (=‘left- hand sides’) equal to and desired filter output (=‘right-hand side’) equal to LMS algorithm :

Generalized Sidelobe Canceler GSC then consists of three parts: Fixed beamformer (cfr. fq ), satisfying constraints but not yet minimum variance), creating `speech reference’ Blocking matrix (cfr. Ca), placing spatial nulls in the direction of the speech source (at sampling frequencies) (cfr. C’.Ca=0), creating `noise references’ Multi-channel adaptive filter (linear combiner) your favourite one, e.g. LMS +

Generalized Sidelobe Canceler A popular GSC realization is as follows Note that some reorganization has been done: the blocking matrix now generates (typically) M-1 (instead of MN-J) noise references, the multichannel adaptive filter performs FIR-filtering on each noise reference (instead of merely scaling in the linear combiner). Philosophy is the same, mathematics are different (details on next slide). Postproc y1 yM

Generalized Sidelobe Canceler Math details: (for Delta’s=0) select `sparse’ blocking matrix such that : =input to multi-channel adaptive filter =use this as blocking matrix now

Generalized Sidelobe Canceler Blocking matrix Ca (cfr. scheme page 24) Creating (M-1) independent noise references by placing spatial nulls in look-direction different possibilities (a la p.38) (broadside steering) Griffiths-Jim Problems of GSC: impossible to reduce noise from look-direction reverberation effects cause signal leakage in noise references adaptive filter should only be updated when no speech is present to avoid signal cancellation! Walsh