Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be

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/

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

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 Fm

Example: delay-and-sum, filter-and-sum

– Adaptive beamforming:

data-dependent adaptive filters Fm

Example: LCMV-beamformer, Generalized Sidelobe Canceler

+:

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

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

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

* SYFZ HHM

mmm dFYF

),().()(

),(),(

dFH

S

ZH

)().().(

),().(),(

2

FΓF

dF

noiseH

H

input

output

SNR

SNRG

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

Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 6

Optimal & Adaptive Filters Review 1/6

Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 7

Optimal & Adaptive Filters Review 2/6

Nor

bert

Wie

ner

Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 8

Optimal & Adaptive Filters Review 3/6

Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 9

Optimal & Adaptive Filters Review 4/6

Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 10

Optimal & Adaptive Filters Review 5/6

(Widrow 1965 !!)

Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 11

Optimal & Adaptive Filters Review 6/6

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

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 :

]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

+:

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 ψ for sample freqs wi, i=1..J (**)

fRfff

][min][min 2 kkzE yyT

JJMNMNT bCfbfC ,, .

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..):

bCRCCRf111 ][][

kk yyT

yyopt

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

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

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

TTI CCCCP 1

bCCCB 1T

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

ByfPf ][][][]1[ kkzkk

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

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

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’ 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 ,,

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 fa : (MN-J coefficients)

bfCfRff

Tyy

T k ,][min

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

aaq ka

fCfRfCff

JJMNMN bCf ,,

Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 22

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

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. 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

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

Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 25

Generalized Sidelobe Canceler

• 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

Digital Audio Signal Processing Version 2014-2015 Lecture-3: Microphone Array Processing p. 26

• 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)

Generalized Sidelobe Canceler

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