Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter...
Transcript of Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter...
![Page 1: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/1.jpg)
1
Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven
[email protected] homes.esat.kuleuven.be/~moonen/
Digital Audio Signal Processing
DASP
Lecture-4: Noise Reduction-III Adaptive Beamforming
Digital Audio Signal Processing Version 2017-2018 Lecture-4: 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)
![Page 2: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/2.jpg)
2
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 3 / 34
F1(ω)
F2 (ω)
FM (ω)
z[k]
y1[k]
+ : dM
θ
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
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 4 / 34
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 =
![Page 3: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/3.jpg)
3
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 5 / 34
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 ===∑
=
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 6 / 34
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 ==
)().().(),().(
),(2
ωωω
θωωθω
FΓFdF
noiseH
H
input
output
SNRSNR
G ==
Iwhitenoise =Γ
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=Γ
cddf ijsdiffuse
ij
)(sinc)(
ωω
![Page 4: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/4.jpg)
4
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 7 / 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)
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 8 / 34
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 matched filter beamformer (in white
noise) or superdirective beamformer (in diffuse noise), see Lecture-3, (**) p.22 & p.30, but now noise field is unknown & adapted to!
F1(ω)
F2 (ω)
FM (ω)
z[k]
y1[k]
+ : yM [k]
![Page 5: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/5.jpg)
5
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 9 / 34
Adaptive beamforming Goal & Set-up
• Adaptive filter-and-sum structure: – 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
yM [k]
F1(ω)
F2 (ω)
FM (ω)
z[k]
y1[k]
+ : yM [k]
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 10 / 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)
![Page 6: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/6.jpg)
6
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 11 / 34
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
T
FT (ωi ).d(ωi,ψ) =Lecture4-p18
dT (ωi,ψ).f =1 i =1..J
JJMNMNT ℜ∈ℜ∈ℜ∈= × bCfbfC ,, .
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 12 / 34
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
T
( ) bCRCCRf 111 ][][ −−− ⋅⋅⋅⋅= kk yyT
yyopt
JJMNMNT ℜ∈ℜ∈ℜ∈= × bCfbfC ,, .
![Page 7: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/7.jpg)
7
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 13 / 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)
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 14 / 34
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 )
![Page 8: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/8.jpg)
8
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 15 / 34
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 )
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 16 / 34
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
!"!#$!"!#$ΔΔ==
−==−−
][~ ky
][kd
])[..][][..(][..][]1[][~][][~
kkkkkk a
k
aT
kd
Tq
k
Taaa
T
fCyyfyCffyy!"!#$"#$!"!#$ −+=+ µ
![Page 9: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/9.jpg)
9
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 17 / 34
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
][kd
][~ ky
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 18 / 34
Generalized sidelobe canceler
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
y1
yM
![Page 10: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/10.jpg)
10
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 19 / 34
Generalized sidelobe canceler
• Math details: (for Delta’s=0)
[ ][ ]
[ ]][.~][~
]1[~...]1[~][~][~
~...00:::0...~00...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
CC
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 2017-2018 Lecture-4: Adaptive Beamforming 20 / 34
Generalized sidelobe canceler
• Blocking matrix Ca (for scheme p.19)
– Creating (M-1) independent noise references by placing spatial nulls in look-direction
– different possibilities
(broadside steering)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
−
=
100101010011
TaC
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−−
−−
=
111111111111
TaC 0
20004000
60008000 0 45 90 135 180
0
0.5
1
1.5
Angle (deg)
Frequency (Hz)
Griffiths-Jim
Walsh
f Tq = 1 1 1 1!"
#$
![Page 11: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/11.jpg)
11
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 21 / 34
Generalized sidelobe canceler
• 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)
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 22 / 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)
![Page 12: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/12.jpg)
12
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 23 / 34
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
? ][ks⌢
speech part noise part
M=
num
ber o
f mic
roph
ones
compare to Lecture-2
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 24 / 34
Multi-Microphone Noise Reduction Problem
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-2), where additional microphones (m=2..M) help to get a better estimate
][kn
][ks
ym[k]= sm[k]+ nm[k],m =1...M
![Page 13: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/13.jpg)
13
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 25 / 34
Multi-Microphone Noise Reduction Problem
• Data model: Frequency domain.. See Lecture-3 on multi-path propagation
(with q left out for conciseness)
Hm(ω) is complete transfer function from speech source position to m-the microphone
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+=
+=
)(
)()(
)(.
)(
)()(
)(
)()(
)()().( )()()(
2
1
2
1
2
1
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ωωω
ωωω
MMM N
NN
S
H
HH
Y
YY
S
!!!
NdNSY
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 26 / 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)
![Page 14: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/14.jpg)
14
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 27 / 34
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
M
mmm YFS ==∑
=
})().()({min2
1)( ωωωω YFFHSE −
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 28 / 34
Multi-Channel Wiener Filter (MWF)
• Wiener filter solution is (see DSP-CIS)
– All quantities can be computed ! – Special case of this is single-channel Wiener filter formula (Lecture-2) – In practice, use alternative to ‘subtraction’ operation (see literature)
( ))}().({)}().({.)}().({
...
.)}().({)}().({)(
*1
*1
1
)0)}().({(with
lationcrosscorre
*1
1
ationautocorrel*
1
ωωωωωω
ωωωωω
ωω
NEYEE
SEE
H
NE
H
NYYY
YYYF
S
−=
=
=
−
=
−
!! "!! #$!! "!! #$
compute during speech+noise periods
compute during noise-only periods
![Page 15: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/15.jpg)
15
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 29 / 34
• Note that… MWF combines spatial filtering with single-channel spectral filtering (as in Lecture-2)
if
then…
! "#$
%&%'(
)noise vectorsteering
2
1
)()(.)(
)(
)(
)()(
ωωω
ω
ω
ωω
Nd
Y
+=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
S
Y
YY
M
E{N(ω).NH (ω)} =Φnoise(ω)
Multi-Channel Wiener Filter (MWF)
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 30 / 34
…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)
Multi-Channel Wiener Filter (MWF)
F(ω) =α(ω)scalar!.Φnoise
−1 (ω).d(ω)
Φnoise−1 (ω).d(ω)
![Page 16: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/16.jpg)
16
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 31 / 34
…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!)
Multi-Channel Wiener Filter (MWF)
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)
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 32 / 34
Multi-Channel Wiener Filter: Implementation
• Implementation with short-time Fourier transform (see Lecture-2)
• Implementation with time-domain linear filtering:
ymT [k]= ym[k] ym[k −1] ... ym[k − L +1]"
#$%
⌢s1[k]= ymT [k].fm
m=1
M
∑][kn
][ks
][1 ks⌢
][1 ky
filter coefficients
ym[k]= sm[k]+nm[k]
![Page 17: Digital Audio Signal Processing DASP - KU Leuvendspuser/dasp...– Multi-channel Wiener filter (=spectral+spatial filtering) 2 Digital Audio Signal Processing Version 2017-2018 Lecture-4:](https://reader036.fdocuments.net/reader036/viewer/2022081406/5f17a01c3e653d7b5d0df4c1/html5/thumbnails/17.jpg)
17
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 33 / 34
Solution is…
}].[][{min2
1 fyf kksE T−[ ][ ]TT
MTT
TTM
TT
kkkk ][...][][][
...
21
21
yyyy
ffff
=
=
[ ][ ] [ ]]}[.][{]}[.][{.}][.][{
]}[.][{.}][.][{
11
1
1
1
knkEkykEkkE
kskEkkE
T
T
nyyy
yyyf
−=
=
−
−
compute during speech+noise periods
compute during noise-only periods
• Implementation with time-domain linear filtering:
Multi-Channel Wiener Filter: Implementation
Digital Audio Signal Processing Version 2017-2018 Lecture-4: Adaptive Beamforming 34 / 34
Solution is… PS: Single-channel version of this is related/similar to single channel
subspace method, see Lecture-2, p.32 (check!)
}].[][{min2
1 fyf kksE T−[ ][ ]TT
MTT
TTM
TT
kkkk ][...][][][
...
21
21
yyyy
ffff
=
=
[ ][ ] [ ]]}[.][{]}[.][{.}][.][{
]}[.][{.}][.][{
11
1
1
1
knkEkykEkkE
kskEkkE
T
T
nyyy
yyyf
−=
=
−
−
• Implementation with time-domain linear filtering:
Multi-Channel Wiener Filter: Implementation