Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven [email protected]

Digital Audio Signal Processing

Lecture-3: Microphone Array Processing

- Adaptive Beamforming -

Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven

[email protected]

homes.esat.kuleuven.be/~moonen/

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


Introduction

Beamforming = `Spatial filtering’ 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

)()().,(),( NdY S

)()}.,().({),().(),().(),(1

* SYFZ HHM

mmm dFYF

),().()(

),(),(

dFH

S

ZH

)().().(

),().(),(

2

FΓF

dF

noiseH

H

input

output

SNR

SNRG


Overview




Optimal & Adaptive Filters Review 1/6



Nor

bert

Wie

ner







(Widrow 1965 !!)




Overview




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 :

]1[...]1[][][ T Nkykykyk mmmmy

M

mm

Tm

T kkkz1

][][][ yfyf

TTM

TT kkkk ][][][][ 21 yyyy

TTM

TT ffff 21 ... T

1,1,0, Nmmmm ffff

+:


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 (**)

fRfff

][min][min 2 kkzE yyT

JJMNMNT bCfbfC ,, .


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..):

bCRCCRf111 ][][

kk yyT

yyopt


Overview




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

BfRfPf ][][][]1[ kkkk yy

TTI CCCCP 1

bCCCB 1T


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

ByfPf ][][][]1[ kkzkk

BfRfPf ][][][]1[ kkkk yy


Overview




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

bfCfRff

Tyy

T k ,][min

)( JMNMNa

C

JJMNMN bCf ,,


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)

bfCfRff

Tyy

T k ,][min

).].([.).(min aaqyyT

aaq ka

fCfRfCff

JJMNMN bCf ,,


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

}))..][().][({min...).].([.).(min

2][~][

aa

kd

qaaqyyT

aaq

T

aakkEk fCyfyfCfRfCf

ky

TTff

][~ ky

][kd

])[..][][..(][..][]1[

][~][][~

kkkkkk a

k

aT

kd

Tq

k

Taaa

T

fCyyfyCff

yy



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

][kd

][~ ky

qf

aC

+

][1 ky

][2 ky

][kyM

][kz

af

][kd

][~ ky



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



• Math details: (for Delta’s=0)

][.

~][~

]1[~...]1[~][~][~

~...00

:::

0...~

0

0...0~

][...][][][

]1[...]1[][][

][.][.][~

:1:1

:1:1:1

permuted,

21:1

:1:1:1permuted

permutedpermuted,

kk

Lkkkk

kykykyk

Lkkkk

kkk

MTaM

TTM

TM

TM

Ta

Ta

Ta

Ta

TMM

TTM

TM

TM

Ta

Ta

yCy

yyyy

C

C

C

C

y

yyyy

yCyCy

=input to multi-channel adaptive filter

select `sparse’ blocking matrix

such that :

=use this as blocking matrix now


• 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)


1111TC

1001

0101

0011TaC

1111

1111

1111TaC

• 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!

02000

40006000

8000 0 45 90 135 180

0

0.5

1

1.5

Angle (deg)

Griffiths-Jim

Walsh

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Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven [email protected]