digital audio signal processing lecture-2 microphone array processing marc moonen dept....
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![Page 1: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/1.jpg)
Digital Audio Signal Processing
Lecture-2
Microphone Array Processing
Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven
homes.esat.kuleuven.be/~moonen/
![Page 2: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/2.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 2
Overview
• Introduction & beamforming basics– Data model & definitions
• Fixed beamforming– Filter-and-sum beamformer design – Matched filtering
• White noise gain maximization• Ex: Delay-and-sum beamforming
– Superdirective beamforming • Directivity maximization
– Directional microphones (delay-and-subtract)
• Adaptive Beamforming– LCMV beamformer– Generalized sidelobe canceler
![Page 3: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/3.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 3
Introduction
• Directivity pattern of a microphone– A microphone (*) is characterized by a `directivity pattern which
specifies the gain & phase shift that the microphone gives to a
signal coming from a certain direction (i.e. `angle-of-arrival’) – In general the directivity pattern is a function of frequency (ω) – In a 3D scenario `angle-of-arrival’
is azimuth + elevation angle– Will consider only 2D scenarios for
simplicity, with one angle-of arrival (θ),
hence directivity pattern is H(ω,θ) – Directivity pattern is fixed and defined
by physical microphone design
(*) We do digital signal prcessing, so this includes front-end filtering/A-to-D/..
|H(ω,θ)| for 1 frequency
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Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 4
Introduction
• Virtual directivity pattern – By weighting or filtering (=freq. dependent weighting) and then
summing signals from different microphones, a (software controlled) virtual directivity pattern (=weigthed sum of individual patterns) can be produced
– This assumes all microphones receive the same signals (so are all in the same position). However…
M
mmm HFH
1virtual ),().(),(
01000
20003000
045
90135
180
0
0.5
1
Angle (deg)
+:
![Page 5: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/5.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 5
+:
Introduction
• However, in a microphone array different microphones are in different positions/locations, hence also receive different signals
• Example : uniform linear array i.e. microphones placed on a line & uniform inter-micr. distances (d) & ideal micr. characteristics (p.9) For a far-field source signal (plane waveforms), each microphone receives the same signal, up to an angle-dependent delay… fs=sampling rate c=propagation speed
][][ 1 mm kyky
sm
m fc
d cos)(
dmdm )1(
![Page 6: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/6.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 6
+:
Introduction
• Beamforming = `spatial filtering’ based on
microphone characteristics (directivity patterns)
AND microphone array configuration (`spatial sampling’)
• Classification:
Fixed beamforming: data-independent, fixed filters Fm
e.g. delay-and-sum, filter-and-sum
Adaptive beamforming: data-dependent filters Fm
e.g. LCMV-beamformer, generalized sidelobe canceler
![Page 7: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/7.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 7
Introduction
• Background/history: ideas borrowed from antenna array design and processing for radar & (later) wireless communications
• Microphone array processing considerably more difficult than antenna array processing: – narrowband radio signals versus broadband audio signals– far-field (plane wavefronts) versus near-field (spherical wavefronts)– pure-delay environment versus multi-path environment
• Applications: voice controlled systems (e.g. Xbox Kinect), speech communication systems, hearing aids,…
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Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 8
Data model and definitions
Data model: source signal in far-field (see p.13 for near-field)
• Microphone signals are filtered versions of source signal S() at angle
• Stack all microphone signals (m=1..M) in a vector
d is `steering vector’• Output signal after `filter-and-sum’ is
TjM
j MeHeH )()(1 ).,(...).,(),( 1 d
)()}.,().({),().(),().(),(1
* SYFZ HHM
mmm dFYF
H instead of T for convenience (**)
)( ..),(),(
shift phase dep.-pos.
)(
pattern dir.
SeHY mjmm
)().,(),( SdY
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Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 9
Data model: source signal in far-field
• If all microphones have the same directivity pattern Ho(ω,θ), steering vector can be factored as…
• Will often consider arrays with ideal omni-directional microphones : Ho(ω,θ)=1 Example : uniform linear array, see p.5
Data model and definitions
)().,(),( SdY
positions spatial
)()(
pattern dir.
0 ...1.),(),( 2Tjj MeeH d
microphone-1 is used as a reference (=arbitrary)
![Page 10: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/10.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 10
Data model and definitions
Definitions: (1)
• In a linear array (p.5) : =90o=broadside direction = 0o =end-fire direction
• Array directivity pattern (compare to p.3) = `transfer function’ for source at angle ( -π<< π )
• Steering direction = angle with maximum amplification (for 1 freq.)
• Beamwidth (BW) = region around max with (e.g.) amplification > -3dB (for 1 freq.)
),().()(
),(),(
dFH
S
ZH
),(maxarg)(max H
![Page 11: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/11.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 11
Data model and definitions
Data model: source signal + noise• Microphone signals are corrupted by additive noise
• Define noise correlation matrix as
• Will assume noise field is homogeneous, i.e. all diagonal elements of noise correlation matrix are equal :
• Then noise coherence matrix is
TMNNN )(...)()()( 21 N
})().({)( Hnoise E NNΦ
iΦΦ noiseii , )()(
1....
....
....1
)(.)(
1)(
noise
noisenoise ΦΓ
)()().,(),( NdY S
![Page 12: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/12.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 12
Data model and definitions
Definitions: (2) • Array Gain = improvement in SNR for source at angle ( -π<< π )
• White Noise Gain =array gain for spatially uncorrelated noise
(e.g. sensor noise)
ps: often used as a measure for robustness• Directivity =array gain for diffuse noise (=coming from all directions)
DI and WNG evaluated at max is often used as a performance criterion
c
ddf ijsdiffuseij
)(sinc)(
skip this formula)(.).(
),().(),(
2
FF
dFdiffusenoise
H
H
DI
)().(
),().(),(
2
FF
dFH
H
WNG Iwhite
noise
|signal transfer function|^2
|noise transfer function|^2)().().(
),().(),(
2
FΓF
dF
noiseH
H
input
output
SNR
SNRG
PS: ω is rad/sample ( -Π≤ω≤Π ) ω fs is rad/sec
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Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 13
• Far-field assumptions not valid for sources close to microphone array– spherical wavefronts instead of planar waveforms– include attenuation of signals– 2 coordinates ,r (=position q) instead of 1 coordinate (in 2D case)
• Different steering vector (e.g. with Hm(ω,θ)=1 m=1..M) :
PS: Near-field beamforming
TjM
jj Meaeaea )()(2
)(1
21),( qqqqd ),( d
s
mref
m fc
pqpqq
)(
with q position of source pref position of reference microphone pm position of mth microphone
m
ref
mapq
pq
e
e=1 (3D)…2 (2D)
![Page 14: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/14.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 14
PS: Multipath propagation
• In a multipath scenario, acoustic waves are reflected against walls, objects, etc..
• Every reflection may be treated as a separate source (near-field or far-field)
• A more realistic data model is then..
with q position of source and Hm(ω,q), complete transfer function from source position to m-the microphone (incl. micr. characteristic, position, and multipath propagation)
`Beamforming’ aspect vanishes here, see also Lecture-3
(`multi-channel noise reduction’)
)()().,(),( NqdqY S
TMHHH ),(...),(),(),( 21 qqqqd
![Page 15: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/15.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 15
Overview
• Introduction & beamforming basics– Data model & definitions
• Fixed beamforming– Filter-and-sum beamformer design – Matched filtering
• White noise gain maximization• Ex: Delay-and-sum beamforming
– Superdirective beamforming • Directivity maximization
– Directional microphones (delay-and-subtract)
• Adaptive Beamforming– LCMV beamformer– Generalized sidelobe canceler
![Page 16: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/16.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 16
Filter-and-sum beamformer design
• Basic: procedure based on page 10
Array directivity pattern to be matched to given (desired) pattern over frequency/angle range of interest
• Non-linear optimization for FIR filter design (=ignore phase response)
• Quadratic optimization for FIR filter design (=co-design phase response)
),().()(
),(),(
dFH
S
ZH
),( dH
1
0,1 .)( , )(...)()(
N
n
jnnmm
TM efFFF F
ddHH dNnMmf nm ),(),(min
2
1..0,..1,,
ddHH dNnMmf nm ),(),(min
2
1..0,..1,,
![Page 17: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/17.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 17
Filter-and-sum beamformer design
• Quadratic optimization for FIR filter design (continued)
With
optimal solution is
ddHH dNnMmf nm ),(),(min
2
1..0,..1,,
),(
).1(
.0
1
11,0,
),(.)(.),(.)(...)(),().(),(
... , ...
d
dfddF
ffffNMxM
jN
j
MxMTH
MHH
TTM
TTNmmm
e
e
IFFH
ff
ddHdd d
Hoptimal ),().,( , ),().,( , . *1 dpddQpQf
Kronecker product
![Page 18: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/18.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 18
Filter-and-sum beamformer design
• Design example
M=8Logarithmic arrayN=50fs=8 kHz
01000
20003000
045
90135
180
0
0.5
1
Angle (deg)
Frequency (Hz)
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Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 19
Matched filtering: WNG maximization
• Basic: procedure based on page 12
• Maximize White Noise Gain (WNG) for given steering angle ψ
• A priori knowledge/assumptions:– angle-of-arrival ψ of desired signal + corresponding steering vector– noise scenario = white
})().(
),().(arg{max)},(arg{max)(
2
)()(MF
FF
dFF FF H
H
WNG
![Page 20: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/20.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 20
Matched filtering: WNG maximization
• Maximization in
is equivalent to minimization of noise output power (under
white input noise), subject to unit response for steering angle (**)
• Optimal solution (`matched filter’) is
• [FIR approximation]
),(.),(
1)( 2
MF
dd
F
1),().( s.t. ),().(min )( dFFFFHH
})().(
),().(arg{max)(
2
)(MF
FF
dFF F H
H
dNnMmf nm
2MF1..0,..1, )()(min
,FF
![Page 21: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/21.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 21
Matched filtering example: Delay-and-sum
• Basic: Microphone signals are delayed and then summed together
• Fractional delays implemented with truncated interpolation filters (=FIR)
• Consider array with ideal omni-directional micr’s
Then array can be steered to angle :
Hence (for ideal omni-dir. micr.’s) this is matched filter solution
1),( H
d
cos)1( dm
d
2
m
1
M
1
M
eF
mj
m
)(
M
mmm ky
Mkz
1
][.1
][
)(mm M
),()(
dF
![Page 22: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/22.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 22
Matched filtering example: Delay-and-sum
• Array directivity pattern H(ω,θ):
=destructive interference
=constructive interference• White noise gain :
(independent of ω)
For ideal omni-dir. micr. array, delay-and-sum beamformer provides
WNG equal to M for all freqs. (in the direction of steering angle ψ).
)( 1),(
1),(
),().,(1
),(
max
H
HM
H H dd
MWNGH
H
..)().(
),().(),(
2
FF
dF
ideal omni-dir. micr.’s
![Page 23: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/23.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 23
02000
40006000
8000 045
90135
180
0.2
0.4
0.6
0.8
1
Angle (deg)Frequency (Hz)
• Array directivity pattern H(,) for uniform linear array:
H(,) has sinc-like shape and is frequency-dependent
Matched filtering example: Delay-and-sum
)2/sin(
)2/sin(
),(
2/
2/1
)cos(cos)1(
j
jM
M
m
fc
dmj
e
Me
eHs
-20
-10
0
90
270
180 0
Spatial directivity pattern for f=5000 Hz
M=5 microphones
d=3 cm inter-microphone distance
=60 steering angle
fs=16 kHz sampling frequency
=endfire
1),( H
=60wavelength=4cm
ideal omni-dir. micr.’s
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Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 24
0
2000
4000
6000
8000 050
100150
0.20.40.60.8
1
Angle (deg)
Matched filtering example: Delay-and-sum
For an ambiguity, called spatial aliasing, occurs.
This is analogous to time-domain aliasing where now the spatial sampling (=d) is too large. Aliasing does not occur (for any ) if
M=5, =60, fs=16 kHz, d=8 cm
)cos1.(
d
cf
2.2min
max
f
c
f
cd
s
)cos1.(.. and 0for occurs 2 then
2 if 3)
)cos1.(.. and for occurs 2 then
2 if 2)
) all(for for 0 )1
integer for 2 1),(
Details...
d
cfπ
d
cfπ
pπ.pγiffH
ideal omni-dir. micr.’s
)cos1.(
d
cf
![Page 25: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/25.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 25
Matched filtering example: Delay-and-sum
• Beamwidth for a uniform linear array:
hence large dependence on # microphones, distance (compare p.24 & 25) and frequency (e.g. BW infinitely large at DC)
• Array topologies:– Uniformly spaced arrays– Nested (logarithmic) arrays (small d for high , large d for small )– 2D- (planar) / 3D-arrays
with e.g. =1/sqrt(2) (-3 dB)
d
2d
4d
sec
)1(96
dMcBW
ideal omni-dir. micr.’s
![Page 26: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/26.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 26
Super-directive beamforming : DI maximization
• Basic: procedure based on page 12
• Maximize Directivity (DI) for given steering angle ψ
• A priori knowledge/assumptions:– angle-of-arrival ψ of desired signal + corresponding steering vector– noise scenario = diffuse
})(.).(
),().(arg{max)},(arg{max)(
2
)()(SD
FF
dFF FF diffuse
noiseH
H
DI
![Page 27: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/27.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 27
• Maximization in
is equivalent to minimization of noise output power (under
diffuse input noise), subject to unit response for steering angle (**)
• Optimal solution is
• [FIR approximation]
})(.).(
),().(arg{max)(
2
)(SD
FF
dFF F diffuse
noiseH
H
1),().( s.t. ),().().(min )( dFFFFHdiffuse
noiseH
),(.)}(.{1
)( 1
),(.1)}(.{),(
SD
dFdd
diffusenoisediffuse
noiseH
dNnMmf nm
2SD1..0,..1, )()(min
,FF
Super-directive beamforming : DI maximization
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Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 28
• Directivity patterns for end-fire steering (ψ=0):
Superdirective beamformer has highest DI, but very poor WNG (at low frequencies, where diffuse noise coherence matrix becomes ill-conditioned)
hence problems with robustness (e.g. sensor noise) !
-20
-10
0
90
270
180 0
Superdirective beamformer (f=3000 Hz)
-20
-10
0
90
270
180 0
Delay-and-sum beamformer (f=3000 Hz)
M=5 d=3 cmfs=16 kHz
Maximum directivity=M.M obtained for end-fire steering and for frequency->0 (no proof)
ideal omni-dir. micr.’s
0 2000 4000 6000 80000
5
10
15
20
25
Frequency (Hz)
Dir
ecti
vit
y (
lin
ear)
SuperdirectiveDelay-and-sum
0 2000 4000 6000 8000-60
-50
-40
-30
-20
-10
0
10
Frequency (Hz)
Whit
e n
ois
e g
ain
(d
B)
SuperdirectiveDelay-and-sum
WNG=M= 5
PS: diffuse noise ≈white noise for high frequencies(cfr. ωΠ and c/fs=λmin/2≈min(dj-di) in diffuse noise coherence matrix)
DI=WNG=5
DI=M 2=25
Super-directive beamforming : DI maximization
![Page 29: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/29.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 29
• First-order differential microphone = directional microphone 2 closely spaced microphones, where one microphone is delayed (=hardware) and whose outputs are then subtracted from each other
• Array directivity pattern:
– First-order high-pass frequency dependence– P() = freq.independent (!) directional response– 0 1 1 : P() is scaled cosine, shifted up with 1
such that max = 0o (=end-fire) and P(max )=1
d
+
_
)cos
(1),( c
dj
eH
d/c <<, <<
cd /1
cos)1()( 11 P
Differential microphones : Delay-and-subtract
![Page 30: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/30.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 30
Differential microphones : Delay-and-subtract
• Types: dipole, cardioid, hypercardioid, supercardioid (HJ84)
=endfire
=broadside
Dipole:
1= 0 (=0) zero at 90o
DI=4.8dB
Cardioid:
1= 0.5 zero at 180o
DI=4.8dB
Supercardioid:
1=
zero at 125o, DI=5.7 dB highest front-to-back ratio
35.02/)13(
Hypercardioid:
1= 0.25
zero at 109o highest DI=6.0dB
![Page 31: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/31.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 31
Overview
• Introduction & beamforming basics– Data model & definitions
• Fixed beamforming– Filter-and-sum beamformer design – Matched filtering
• White noise gain maximization• Ex: Delay-and-sum beamforming
– Superdirective beamforming • Directivity maximization
– Directional microphones (delay-and-subtract)
• Adaptive Beamforming– LCMV beamformer– Generalized sidelobe canceler
![Page 32: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/32.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 32
LCMV beamformer
• 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. (**) p.20 & 27, 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+:
![Page 33: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/33.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 33
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 in look-direction ψ for sample freqs ωi
For sufficiently large J constrained output power minimization approximately corresponds to constrained output noise power minimization(why?)
– Solution is (obtained using Lagrange-multipliers, etc..):
fRfff
][min][min 2 kkzE yyT
JJMNMNT bCfbfC ,, .
bCRCCRf111 ][][
kk yyT
yyopt
![Page 34: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/34.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 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, leading to a simple 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 ,,
![Page 35: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/35.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 35
Generalized sidelobe canceler
GSC = Adaptive filter formulation of the LCMV-problem
Constrained optimisation is reformulated as a constraint pre-processing, followed by an unconstrained optimisation, leading to a simple adaptation scheme– LCMV-problem is
– Unconstrained optimization of fa : (MN-J coefficients)
bfCfRff
Tyy
T k ,][min JJMNMN bCf ,,
).].([.).(min aaqyyT
aaq ka
fCfRfCff
![Page 36: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/36.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 36
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
![Page 37: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/37.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 37
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
![Page 38: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/38.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 38
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 39: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/39.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 39
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
![Page 40: Digital Audio Signal Processing Lecture-2 Microphone Array Processing Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven marc.moonen@esat.kuleuven.be homes.esat.kuleuven.be/~moonen](https://reader038.vdocuments.site/reader038/viewer/2022103022/56649f4f5503460f94c71c8c/html5/thumbnails/40.jpg)
Digital Audio Signal Processing Version 2015-2016 Lecture-2: Microphone Array Processing p. 40
Generalized sidelobe canceler
• Blocking matrix Ca (cfr. scheme page 24)
– Creating (M-1) independent noise references by placing spatial nulls in look-direction
– different possibilities
(broadside steering)
• 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!
1111TC
1001
0101
0011TaC
1111
1111
1111TaC 0
20004000
60008000 0 45 90 135 180
0
0.5
1
1.5
Angle (deg)
Griffiths-Jim
Walsh