Digital Audio Signal Processing DASPdspuser/dasp/material...2 Digital Audio Signal Processing...

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

marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen/

Digital Audio Signal Processing

Lecture-5: Noise Reduction-III Adaptive Beamforming

Digital Audio Signal Processing Version 2016-2017 Lecture-5: Adaptive Beamforming 2 / 34

Overview

•  Beamforming / Recap

•  Adaptive Beamforming –  Adaptive beamforming goal & set-up –  LCMV beamformer –  Generalized sidelobe canceler

•  Multi-channel Wiener filter for multi-microphone noise reduction –  Multi-microphone noise reduction problem –  Multi-channel Wiener filter (=spectral+spatial filtering)

F1(ω)

F2 (ω)

FM (ω)

+ : dM

dM cosθ

Recap 1/4

Filter-and-sum Beamforming: Classification:

Fixed beamforming: data-independent, fixed filters Fm e.g. matched filtering, superdirective, … Adaptive beamforming: data-dependent filters Fm e.g. LCMV-beamformer, generalized sidelobe canceler

=This lecture

=Previous lecture

Recap 2/4 Data model: source signal

•  Microphone signals are filtered versions of source signal S(ω) at angle θ

•  Stack all microphone signals (m=1..M) in a vector d is `steering vector’

[ ]TjM

j MeHeH )()(1 ).,(...).,(),( 1 θωτθωτ θωθωθω −−=d

)( ..),(),(shift phase dep.-pos.

pattern dir.

ωθωθω θωτ SeHY mjmm

!"!#$!"!#$−=

)().,(),( ωθωθω SdY =

Recap 3/4

Data model: source signal + noise •  Microphone signals are corrupted by additive noise

Noise correlation matrix is

•  Output signal after `filter-and-sum’ is

[ ]TMNNN )(...)()()( 21 ωωωω =N

})().({)( Hnoise E ωωω NNΦ =

)()().,(),( ωωθωθω NdY += S

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

* ωθωωθωωθωωθω SYFZ HHM

mmm dFYF ===∑

Recap 4/4 Definitions:

•  Array directivity pattern = `transfer function’ for source at angle θ

•  Array Gain = improvement in SNR for source at angle θ ( -π<θ< π )

White Noise Gain =array gain for spatially uncorrelated noise

Directivity =array gain for diffuse noise

),().()(),(),( θωω

ωθω

θω dFHSZH ==

)().().(),().(

ωωω

θωωθω

FΓFdF

noiseH

output

SNRSNR

Iwhitenoise =Γ

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=Γ

cddf ijsdiffuse

)(sinc)(

Overview

Adaptive beamforming Goal & Set-up

•  Adaptive filter-and-sum structure: –  Aim is to minimize noise output power, while maintaining a chosen

response for a given steering angle (and/or other linear constraints, see below). –  This is similar to operation of a delay-and-sum beamformer (in white

noise) or superdirective beamformer (in diffuse noise), see Lecture-4, (**) p.22 & 30, but now noise field is unknown & adapted to!

F1(ω)

F2 (ω)

FM (ω)

+ : yM [k]

Adaptive beamforming Goal & Set-up

•  Adaptive filter-and-sum structure: –  Implemented as adaptive FIR filter :

[ ] ]1[...]1[][][ T+−−= Nkykykyk mmmmy

T kkkz1

][][][ yfyf

[ ]TTM

TT kkkk ][][][][ 21 yyyy …=

[ ]TTM

TT ffff …21=[ ] ... T

1,1,0, −= Nmmmm ffff

yM [k]

F1(ω)

F2 (ω)

FM (ω)

+ : yM [k]

Overview

LCMV beamformer

LCMV = Linearly Constrained Minimum Variance –  f designed to minimize output power (variance) (read on..) z[k] :

–  To avoid desired signal cancellation, add (J) linear constraints

Example: fix array response for steering angle ψ & sample freqs ωi

For sufficiently large J constrained output power minimization approximately corresponds to constrained output noise power minimization (if source angle is equal to steering angle) (why?)

Ryy[k]= E{y[k].y[k]T}{ } fRf

ff⋅⋅= ][min][min 2 kkzE yy

FT (ωi ).d(ωi,ψ) =Lecture4-p18

dT (ωi,ψ).f =1 i =1..J

JJMNMNT ℜ∈ℜ∈ℜ∈= × bCfbfC ,, .

LCMV beamformer

LCMV = Linearly Constrained Minimum Variance –  f designed to minimize output power (variance) (read on..) z[k] :

–  To avoid desired signal cancellation, add (J) linear constraints

–  Solution is (obtained using Lagrange-multipliers, etc..):

Ryy[k]= E{y[k].y[k]T}{ } fRf

ff⋅⋅= ][min][min 2 kkzE yy

( ) bCRCCRf 111 ][][ −−− ⋅⋅⋅⋅= kk yyT

JJMNMNT ℜ∈ℜ∈ℜ∈= × bCfbfC ,, .

Overview

Generalized sidelobe canceler (GSC)

GSC = Adaptive filter formulation of the LCMV-problem Constrained optimisation is reformulated as a constraint pre-processing,

followed by an unconstrained optimisation, as follows: –  LCMV-problem is

–  Define `blocking matrix’ Ca,, 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

f = fq −Ca.fa

)( JMNMNa

−×ℜ∈C

JJMNMN ℜ∈ℜ∈ℜ∈ × bCf ,,

CT .Ca = 0

fq =C.(CT .C)−1b

fa ∈ℜ(MN−J )

Generalized sidelobe canceler (GSC)

GSC = Adaptive filter formulation of the LCMV-problem Constrained optimisation is reformulated as a constraint pre-processing,

followed by an unconstrained optimisation, as follows: –  LCMV-problem is

–  Unconstrained optimization of fa : (MN-J coefficients)

bfCfRff

=⋅⋅⋅ Tyy

T k ,][min JJMNMN ℜ∈ℜ∈ℜ∈ × bCf ,,

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

aaq ka

fCfRfCff −−

f = fq −Ca.fa

fa ∈ℜ(MN−J )

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][~][

qaaqyyT

aakkEk fCyfyfCfRfCf

!"!#$!"!#$ΔΔ==

−==−−

][~ ky

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

kkkkkk a

fCyyfyCffyy!"!#$"#$!"!#$ −+=+ µ

Generalized sidelobe canceler GSC then consists of three parts: •  Fixed beamformer (cfr. fq ), satisfying constraints (‘LC’) but not yet

minimum variance (‘MV’) ), 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 MN-J `noise references’

•  Multi-channel adaptive filter (linear combiner) your favourite one, e.g. LMS

][~ ky

A popular & significantly cheaper 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

•  Math details: (for Delta’s=0)

[ ][ ]

[ ]][.~][~

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

~...00:::0...~00...0~

][...][][][

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

][.][.][~

:1:1:1

permuted,

:1:1:1permuted

permutedpermuted,

kykykyk

+−−=

⎥⎥⎥⎥⎥

⎢⎢⎢⎢⎢

+−−=

=input to multi-channel adaptive filter

select `sparse’ blocking matrix such that :

=use this as blocking matrix now

•  Blocking matrix Ca (for scheme page 19)

–  Creating (M-1) independent noise references by placing spatial nulls in look-direction

–  different possibilities

(broadside steering)

⎥⎥⎥

⎢⎢⎢

100101010011

⎥⎥⎥

⎢⎢⎢

−−

111111111111

20004000

60008000 0 45 90 135 180

Angle (deg)

Frequency (Hz)

Griffiths-Jim

f Tq = 1 1 1 1!"

•  Problems of GSC:

–  Impossible to reduce noise from steering angle direction

–  Reverberation effects cause signal leakage in noise references

–  Adaptive filter should only be updated when no speech is present to avoid signal cancellation Hence requires voice activity detection (VAD)

Overview

ym[k]= sm[k]+ nm[k],m =1...M

Multi-Microphone Noise Reduction Problem

(some) speech estimate

speech source

noise source(s)

microphone signals ][kn

? ][ks⌢

speech part noise part

compare to Lecture 3

Will estimate speech part in microphone 1 (*) (**)

? ][1 ks⌢

(*) Estimating s[k] is more difficult, would include dereverberation (**) This is similar to single-microphone model (Lecture-3), where additional microphones (m=2..M) help to get a better estimate

ym[k]= sm[k]+ nm[k],m =1...M

•  Data model: Frequency domain.. See Lecture-4 on multi-path propagation

(with q left out for conciseness)

Hm(ω) is complete transfer function from speech source position to m-the microphone

⎥⎥⎥⎥

⎢⎢⎢⎢

⎥⎥⎥⎥

⎢⎢⎢⎢

⎥⎥⎥⎥

⎢⎢⎢⎢

)()().( )()()(

ωωω

Overview

Multi-Channel Wiener Filter (MWF)

•  Data model:

•  Will use linear filters to obtain speech estimate

•  Wiener filter (=linear MMSE approach)

Note that (unlike in DSP-CIS) `desired response’ signal S1(w) is obviously unknown here (!)

)()().( )( ωωωω NdY += S

)().()().()(ˆ1

*1 ωωωωω YFH

mmm YFS ==∑

})().()({min2

1)( ωωωω YFFHSE −

•  Wiener filter solution is (see DSP-CIS)

–  All quantities can be computed ! –  Special case of this is single-channel Wiener filter formula (Lecture-3) –  In practice, use alternative to ‘subtraction’ operation (see literature)

( ))}().({)}().({.)}().({

.)}().({)}().({)(

)0)}().({(with

lationcrosscorre

ationautocorrel*

ωωωωωω

ωωωωω

!! "!! #$!! "!! #$

compute during speech+noise periods

compute during noise-only periods

•  Note that… MWF combines spatial filtering with single-channel spectral filtering (as in Lecture-3)

then…

)noise vectorsteering

)()(.)(

ωωω

⎥⎥⎥⎥

⎢⎢⎢⎢

E{N(ω).NH (ω)} =Φnoise(ω)

…then it can be shown that

①  represents a spatial filtering (*) Compare to superdirective & delay-and-sum beamforming (Lecture-4) –  Delay-and-sum beamf. maximizes array gain in white noise field –  Superdirective beamf. maximizes array gain in diffuse noise field –  MWF maximizes array gain in unknown (!) noise field. MWF is operated without invoking any prior knowledge (steering vector/noise field) ! (the secret is in the voice activity detection… (explain))

(*) Note that spatial filtering can improve SNR, spectral filtering never improves SNR (at one frequency)

F(ω) =α(ω)scalar!.Φnoise

−1 (ω).d(ω)

Φnoise−1 (ω).d(ω)

…then it can be shown that

①  represents a spatial filtering (*)

②  represents an additional `spectral post-filter’ i.e. single-channel Wiener estimate (p.9), applied to output signal of spatial filter (prove it!)

α(ω) = ... =S(ω) 2 .H1

*(ω)dH (ω).Φnoise

−1 (ω).d(ω)( ). S(ω) 2 +1

)(ωα

F(ω) =α(ω)scalar!.Φnoise

−1 (ω).d(ω)

Φnoise−1 (ω).d(ω)

Multi-Channel Wiener Filter: Implementation

•  Implementation with short-time Fourier transform (see Lecture-2/3)

•  Implementation with time-domain linear filtering:

ymT [k]= ym[k] ym[k −1] ... ym[k − L +1]"

⌢s1[k]= ymT [k].fm

∑][kn

][1 ks⌢

][1 ky

filter coefficients

ym[k]= sm[k]+nm[k]

Solution is…

}].[][{min2

1 fyf kksE T−[ ][ ]TT

kkkk ][...][][][

[ ][ ] [ ]]}[.][{]}[.][{.}][.][{

]}[.][{.}][.][{

knkEkykEkkE

kskEkkE

compute during speech+noise periods

compute during noise-only periods

Solution is… PS: Single-channel version of this is related/similar to single channel

subspace method, see Lecture-3, p.32 (check!)

}].[][{min2

1 fyf kksE T−[ ][ ]TT

kkkk ][...][][][

[ ][ ] [ ]]}[.][{]}[.][{.}][.][{

]}[.][{.}][.][{

knkEkykEkkE

kskEkkE

Digital Audio Signal Processing DASPdspuser/dasp/material...2 Digital Audio Signal Processing...

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