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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/

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 2 Overview Lecture 2 Introduction & beamforming basics Fixed beamforming Lecture 3: Adaptive beamforming Introduction Review of Optimal & Adaptive Filters LCMV beamforming Frost beamforming Generalized sidelobe canceler

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 3 Introduction Beamforming = `Spatial filtering based on microphone directivity patterns and microphone array configuration Classification Fixed beamforming: data-independent, fixed filters F m Example: delay-and-sum, filter-and-sum Adaptive beamforming: data-dependent adaptive filters F m Example: LCMV-beamformer, Generalized Sidelobe Canceler + :

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 4 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

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 5 Overview Lecture 2 Introduction & beamforming basics Fixed beamforming Lecture 3: Adaptive beamforming Introduction Review of Optimal & Adaptive Filters LCMV beamforming Frost beamforming Generalized sidelobe canceler

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 6 Optimal & Adaptive Filters Review 1/6

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 7 Optimal & Adaptive Filters Review 2/6 Norbert Wiener

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 8 Optimal & Adaptive Filters Review 3/6

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 9 Optimal & Adaptive Filters Review 4/6

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 10 Optimal & Adaptive Filters Review 5/6 (Widrow 1965 !!)

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 11 Optimal & Adaptive Filters Review 6/6

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 12 Overview Lecture 2 Introduction & beamforming basics Fixed beamforming Lecture 3: Adaptive beamforming Introduction Review of Optimal & Adaptive Filters LCMV beamforming Frost beamforming Generalized sidelobe canceler

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 13 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 : + :

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 14 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 for sample freqs wi, i=1..J (**)

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 15 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..):

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 16 Overview Lecture 2 Introduction & beamforming basics Fixed beamforming Lecture 3: Adaptive beamforming Introduction Review of Optimal & Adaptive Filters LCMV beamforming Frost beamforming Generalized sidelobe canceler

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 17 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

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 18 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) Frost Beamforming

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 19 Overview Lecture 2 Introduction & beamforming basics Fixed beamforming Lecture 3: Adaptive beamforming Introduction Review of Optimal & Adaptive Filters LCMV beamforming Frost beamforming Generalized sidelobe canceler

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 20 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 C a,,with columns spanning the null-space of C Define quiescent response vector f q satisfying constraints Parametrize all fs that satisfy constraints (verify!) I.e. filter f can be decomposed in a fixed part f q and a variable part C a. f a

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 21 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 f a : (MN-J coefficients)

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 22 GSC (continued) Hence unconstrained optimization of f a 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

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 23 Generalized Sidelobe Canceler GSC then consists of three parts: Fixed beamformer (cfr. f q ), satisfying constraints but not yet minimum variance), creating `speech reference Blocking matrix (cfr. C a ), placing spatial nulls in the direction of the speech source (at sampling frequencies) (cfr. C.C a =0), creating `noise references Multi-channel adaptive filter (linear combiner) your favourite one, e.g. LMS +

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 24 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

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 25 Generalized Sidelobe Canceler Math details: (for Deltas=0) =input to multi-channel adaptive filter select `sparse blocking matrix such that : =use this as blocking matrix now

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Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 26 Blocking matrix C a (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) Generalized Sidelobe Canceler 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! Griffiths-Jim Walsh