digital audio signal processing dasp - ku leuvendspuser/dasp... · 2 digital audio signal...
TRANSCRIPT
1
Marc Moonen Dept. E.E./ESAT-STADIUS, KU Leuven
[email protected] homes.esat.kuleuven.be/~moonen/
Digital Audio Signal Processing
DASP
Lecture-3: Noise Reduction-II Fixed Beamforming
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 2 / 34
Overview
• Introduction & beamforming basics • Data model & definitions
• 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)
2
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 3 / 34
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
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 4 / 34
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 ),().(),( θωωθω
01000
20003000 0
4590
135180
0
0.5
1
Angle (deg)
Frequency (Hz)
F1(ω)
F2 (ω)
FM (ω)
z[k]
y1[k]
y2[k]
yM [k]
+ :
Fm (ω) = fm,n.e− jωn
n=0
N−1
∑
3
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 5 / 34
F1(ω)
F2 (ω)
FM (ω)
z[k]
y1[k]
+ : dM
θ
dM cosθ
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.10) For a far-field source signal (i.e. plane wave), each microphone receives the same signal, up to an angle-dependent delay… fs=sampling rate c=propagation speed
][][ 1 mm kyky τ+=
Hvirtual (ω,θ ) = Fm (ω).e− jωτm (θ )
m=1
M
∑
sm
m fc
d θθτ
cos)( =dmdm )1( −=
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 6 / 34
F1(ω)
F2 (ω)
FM (ω)
z[k]
y1[k]
+ : dM
θ
dM cosθ
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
=This lecture
=Next lecture
4
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 7 / 34
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 waves) versus near-field (spherical waves) – pure-delay environment versus multi-path environment
• Applications: voice controlled systems (e.g. Xbox Kinect), speech communication systems, hearing aids,…
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 8 / 34
Overview
• Introduction & beamforming basics • Data model & definitions
• 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)
5
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 9 / 34
Data model & definitions 1/5 Data model: source signal in far-field (see p.14 for near-field)
• 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’ • Output signal after `filter-and-sum’ is
[ ]TjM
j MeHeH )()(1 ).,(...).,(),( 1 θωτθωτ θωθωθω −−=d
)()}.,().({),().(),().(),(1
* ωθωωθωωθωωθω SYFZ HHM
mmm dFYF ===∑
=
H instead of T for convenience (**)
)( ..),(),(shift phase dep.-pos.
)(
pattern dir.
ωθωθω θωτ SeHY mjmm
!"!#$!"!#$−=
)().,(),( ωθωθω SdY =
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 10 / 34
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 • Will use microphone-1 as reference (e.g. defining input SNR):
Data model & definitions 2/5
)().,(),( ωθωθω SdY =
[ ]!!!!! "!!!!! #$!"!#$positions spatial
)()(
pattern dir.
0 ...1.),(),( 2Tjj MeeH θωτθωτθωθω −−=d
microphone-1 is used as a reference (=arbitrary)
d1(ω,θ )=1
6
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 11 / 34
Data model & definitions 3/5
Definitions: (1) • In a linear array (p.5) : θ =90o=broadside direction θ = 0o =end-fire direction
• Array directivity pattern (compare to p.3) = `transfer function’ for source at angle θ ( -π<θ< π )
• Steering direction = angle θ with maximum amplification (for 1 freq.)
• Beamwidth (BW) = region around θmax with amplification > (max.amplif - 3dB) (for 1 freq.)
),().()(),(),( θωω
ωθω
θω dFHSZH ==
),(maxarg)(max θωωθ θ H=
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 12 / 34
Data model & definitions 4/5
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 ωωωω =N
})().({)( Hnoise E ωωω NNΦ =
iΦΦ noiseii ∀= , )()( ωω
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
==
1............1
)(.)(
1)( !ωωφ
ω noisenoise
noise ΦΓ
)()().,(),( ωωθωθω NdY += S
7
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 13 / 34
Data model & definitions 5/5
Definitions: (2) • Array Gain = improvement in SNR for source at angle θ ( -π<θ< 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 θmax is often used as a performance criterion
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=Γ
cddf ijsdiffuse
ij
)(sinc)(
ωω
skip this formula )(.).(
),().(),(
2
ωω
θωωθω
FFdFdiffusenoise
H
H
DIΓ
=
)().(),().(
),(2
ωω
θωωθω
FFdF
H
H
WNG =Iwhite
noise =Γ
|signal transfer function|^2 (with micr-1 used as reference: d1 =1)
|noise transfer function|^2 )().().(),().(
),(2
ωωω
θωωθω
FΓFdF
noiseH
H
input
output
SNRSNR
G ==
PS: ω is rad/sample ( -Π≤ω≤Π ) ω fs is rad/sec
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 14 / 34
• Far-field assumptions not valid for sources close to microphone array – spherical waves instead of plane waves – include attenuation of signals – 2 coordinates θ,r (=position q) instead of 1 coordinate θ (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 ωτωτωτω −−−= …),( θωd
smref
m fc
pqpqq
−−−=)(τ
with q position of source pref position of reference microphone pm position of mth microphone
m
refma pq
pq−
−=
e
e=1 (3D)…2 (2D)
8
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 15 / 34
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-5 (`multi-channel noise reduction’)
)()().,(),( ωωωω NqdqY += S
[ ]TMHHH ),(...),(),(),( 21 qqqqd ωωωω =
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 16 / 34
Overview
• Introduction & beamforming basics • Data model & definitions
• 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)
9
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 17 / 34
Filter-and-sum beamformer design
• Basic: procedure based on page 11 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)
),().()(),(),( θωω
ωθω
θω dFHSZH ==
),( θωdH
[ ] ∑−
=
−==1
0,1 .)( , )(...)()(
N
n
jnnmm
TM efFFF ωωωωωF
θωθωθωθ ω
ddHH dNnMmf nm ),(),(min
2
1..0,..1,, ∫ ∫ −−==
( ) θωθωθωθ ω
ddHH dNnMmf nm ),(),(min 2
1..0,..1,, ∫ ∫ −−==
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 18 / 34
Filter-and-sum beamformer design • Quadratic optimization for FIR filter design (continued)
With optimal solution is
θωθωθωθ ω
ddHH dNnMmf nm ),(),(min
2
1..0,..1,, ∫ ∫ −−==
[ ] [ ]
[ ]!!!! "!!!! #$
!!! %!!! &'
(
),(
).1(
.0
1
11,0,
),(.)(.),(.)(...)(),().(),(
... , ...
θω
ω
ω
θωθωωωθωωθω
d
dfddF
ffffNMxM
jN
j
MxMTH
MHH
TTM
TTNmmm
e
eIFFH
ff
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⊗===
==
−
−
∫ ∫∫ ∫ === −
θ ωθ ωθωθωθωθωθωθω ddHdd d
Hoptimal ),().,( , ),().,( , . *1 dpddQpQf
Kronecker product
10
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 19 / 34
Filter-and-sum beamformer design • Design example
M=8 Logarithmic array N=50 fs=8 kHz
01000
20003000 0
4590
135180
0
0.5
1
Angle (deg)
Frequency (Hz)
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 20 / 34
Overview
• Introduction & beamforming basics • Data model & definitions
• 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)
11
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 21 / 34
Matched filtering: WNG maximization
• Basic: procedure based on page 13
• 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
ωω
ψωωψωω ωω FF
dFF FF H
H
WNG ==
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 22 / 34
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)( 2MF ψω
ψωω d
dF =
1),().( s.t. ),().(min )( =ψωωωωω dFFFFHH
})().(),().(
arg{max)(2
)(MF
ωω
ψωωω ω FF
dFF F H
H
=
ωωωω
dNnMmf nm ∫ −−==
2MF1..0,..1, )()(min
,FF
12
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 23 / 34
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 ψ :
Hence (for ideal omni-dir. micr.’s) this is matched filter solution
1),( =ψωH
d
ψcos)1( dm −
Σ d
2Δ mΔ
1Δ
M1
ψ
MeF
mj
m
Δ−
=ω
ω)(
∑=
Δ+=M
mmm ky
Mkz
1
][.1][
d(ω,ψ) = 1 e− jωτ 2 (ψ ) ... e− jωτM (ψ )"#$
%&'
T
)(ψτmm =Δ M),()( ψω
ωdF =
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 24 / 34
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),(
maxθψψθω
θω
θωψωθω
===
≤
=
HH
MH H dd
MWNG H
H
==== ..)().(),().(
),(2
ωω
ψωωψθω
FFdF
ideal omni-dir. micr.’s
13
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 25 / 34
02000
40006000
8000 045 90 135
180
0.2
0.4
0.6
0.8
1
Angle (deg)Frequency (Hz)
• Array directivity pattern H(ω,θ) for uniform linear array:
H(ω,θ) has sinc-like shape and is frequency-dependent
Matched filtering example: Delay-and-sum
)2/sin()2/sin(
),(
2/
2/1
)cos(cos)1(
γγ
θω
γ
γ
ψθω
j
jM
M
m
fc
dmj
eMe
eH s
−
−=
−−−
=
= ∑
-20
-10
0
90
270
180 0
Spatial directivity pattern for f=5000 Hz
M=5 microphones d=3 cm inter-microphone distance ψ=60° steering angle fs=16 kHz sampling frequency
=endfire
γ
1),( ==ψθωH
ψ=60° wavelength=4cm
ideal omni-dir. micr.’s
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 26 / 34
0
2000
4000
6000
8000 050
100150
0.20.40.60.8
1
Angle (deg)
Frequency (Hz)
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 ψ) if
M=5, ψ=60°, fs=16 kHz, d=8 cm
)cos1.(
ψ+=d
cf
2.2min
max
λ==≤
fc
fcds
)cos1.(
.. and 0for occurs 2 then2
if 3)
)cos1.(.. and for occurs 2 then
2 if 2)
) all(for for 0 )1integer for 2 1),(
Details...
ψθγ
πψ
ψπθγ
πψ
ωψθγ
θω
−====≥
+====≤
==
==
dcfπ
dcfπ
pπ.pγiffH
ideal omni-dir. micr.’s
)cos1.(
ψ+≥d
cf
14
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 27 / 34
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 ω, large d for small ω) – 2D- (planar) / 3D-arrays
with e.g. ν=1/sqrt(2) (-3 dB)
d
2d
4d
ψω
νsec)1(96
dMcBW
−≈
ideal omni-dir. micr.’s
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 28 / 34
Overview
• Introduction & beamforming basics • Data model & definitions
• 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)
15
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 29 / 34
Super-directive beamforming : DI maximization
• Basic: procedure based on page 13
• 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
ωω
ψωωψωω ωω FF
dFF FF diffuse
noiseH
H
DIΓ
==
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 30 / 34
• 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
ωω
ψωωω ω FF
dFF F diffuse
noiseH
H
Γ=
1),().( s.t. ),().().(min )( =Γ ψωωωωωω dFFFFHdiffuse
noiseH
),(.)}(.{1)( 1
),(.1)}(.{),(
SD ψωωωψωωψω
dFdd
−−Γ
Γ= diffusenoisediffuse
noiseH
ωωωω
dNnMmf nm ∫ −−==
2SD1..0,..1, )()(min
,FF
Super-directive beamforming : DI maximization
16
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 31 / 34
• 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 cm fs=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)
Direc
tivity
(line
ar)
SuperdirectiveDelay-and-sum
0 2000 4000 6000 8000-60
-50
-40
-30
-20
-10
0
10
Frequency (Hz)
Whit
e nois
e gain
(dB)
SuperdirectiveDelay-and-sum
WNG=M= 5
PS: diffuse noise ≈ white noise for high frequencies (cfr. ωèΠ 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
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 32 / 34
Overview
• Introduction & beamforming basics • Data model & definitions
• 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)
17
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 33 / 34
• First-order differential microphone = directional microphone 2 closely spaced microphones, where one microphone signal is delayed (=hardware) and then subtracted from the other micropone signal
• Array directivity pattern:
– First-order high-pass frequency dependence – P(θ) = freq.independent (!) directional response – 0 ≤ α1 ≤ 1 : P(θ) is scaled cosine, shifted up with α1 such that θmax = 0o (=end-fire) and P(θmax )=1
d
Σ τ
θ
+
_
)cos(1),( c
djeH
θτω
θω+−
−=
ωd/c <<π, ωτ <<π
cd /1 +=τ
τα
θααθ cos)1()( 11 −+=P
H (ω,θ ) ≈ jω.(τ + dc
cosθ ) = jωhigh-pass!
.(τ + dc
). P(θ )angle dependence!
Differential microphones : Delay-and-subtract
Digital Audio Signal Processing Version 2017-2018 Lecture-3: Fixed Beamforming 34 / 34
Differential microphones : Delay-and-subtract
• Types: dipole, cardioid, hypercardioid, supercardioid (HJ84)
=endfire
=broadside
Dipole:
α1= 0 (τ=0) zero at 90o
DI=4.8dB
Cardioid:
α1= 0.5 zero at 180o
DI=4.8dB
Supercardioid:
α1= zero at 125o, DI=5.7 dB highest front-to-back ratio
35.02/)13( ≈−
Hypercardioid:
α1= 0.25 zero at 109o highest DI=6.0dB