Digital Audio Signal Processing
Lecture-2:
Microphone Array Processing
- Fixed Beamforming Marc Moonen
Dept. E.E./ESAT-STADIUS, KU Leuven
marc.moonen@esat.kuleuven.be
homes.esat.kuleuven.be/~moonen/
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)
•  Lecture-3: Adaptive Beamforming
Digital Audio Signal Processing
Version 2014-2015
Lecture-2: Microphone Array Processing
p. 2
1
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 also a function of frequency (ω)
|H(ω,θ)| for 1 frequency
–  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
Digital Audio Signal Processing
Version 2014-2015
p. 3
Lecture-2: Microphone Array Processing
Introduction
•  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
F1 (ω )
z[k]
+
F2 (ω )
y1[k]
1
y2 [k]
0.5
:
0
0
FM (ω )
yM [k]
N−1
Fm (ω ) = ∑ fm,n .e− jnω
n=0
Fr e 1000
que
ncy 2000
( Hz 3000
)
0
45
90
135
180
Angle (deg)
M
H virtual (ω ,θ ) = ∑ Fm (ω ).H m (ω ,θ )
m =1
This assumes all microphones receive the same signals (so are all in the same position).
However…
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
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2
Introduction
•  However, an additional aspect is that in a microphone array different
microphones are in different positions/locations, hence also receive
different signals
θ
•  Example : uniform linear array
i.e. microphones placed on a line
& uniform inter-micr. distances (d)
& ideal micr. characteristics (p.8)
z[k]
Version 2014-2015
+
F2 (ω )
:
dM
FM (ω )
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
Digital Audio Signal Processing
y1[k]
F1 (ω )
τ m (θ ) =
dM cosθ
d m cosθ
fs
c
ym [k ] = y1[k + τ m ]
d m = (m − 1)d
M
H virtual (ω, θ ) = ∑ Fm (ω ).e− jωτ m (θ )
m=1
p. 5
Lecture-2: Microphone Array Processing
Introduction
•  Beamforming
= `spatial filtering’ based on
microphone characteristics
(directivity patterns)
AND
microphone array configuration
(`spatial sampling’)
z[k]
+
θ
F1 (ω )
y1[k]
F2 (ω )
:
dM
FM (ω )
Digital Audio Signal Processing
Version 2014-2015
Lecture-2: Microphone Array Processing
dM cosθ
p. 6
3
Introduction
•  Background/history: ideas borrowed from antenna array design/
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
•  Classification:
–  fixed beamforming: data-independent, fixed filters Fm
e.g. delay-and-sum, filter-and-sum
–  adaptive beamforming: data-dependent adaptive filters Fm
e.g. LCMV-beamformer, Generalized Sidelobe canceler
•  Applications: voice controlled systems (e.g. Xbox Kinect), speech
communication systems, hearing aids,…
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
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Beamforming basics
Data model: source signal in far-field (see p.12 for near-field)
•  Microphone signals are filtered versions of source signal S(ω) at angle θ
pos.-dep. phase shift
dir. pattern
!#!
"
$
!
#!
" $
Ym (ω , θ ) = H m (ω ,θ ). e − jωτ m (θ ) . S (ω )
•  Stack all microphone signals in a vector
Y(ω ,θ ) = d(ω ,θ ).S (ω )
[
d(ω ,θ ) = H1 (ω ,θ ).e − jωτ1 (θ ) ... H M (ω ,θ ).e − jωτ M (θ )
]
T
d is `steering vector’
•  Output signal after `filter-and-sum’ is H instead of T for convenience (**)
M
Z (ω ,θ ) = ∑ Fm* (ω ).Ym (ω ,θ ) = F H (ω ).Y(ω ,θ ) = {F H (ω ).d(ω ,θ )}.S (ω )
m =1
Digital Audio Signal Processing
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Beamforming basics
Data Model: source signal in far-field
Y(ω ,θ ) = d(ω ,θ ).S (ω )
•  If all microphones have the same directivity pattern Ho(ω,θ), steering
vector can be factored as…
[
]
T
d(ω ,θ ) = H 0 (ω ,θ ). 1 e − jωτ 2 (θ ) ... e − jωτ M (θ )
!#!!!!!
"
$
!#!
" $!!!!
dir. pattern
spatial positions
microphone-1 is used as a reference (=arbitrary)
•  Will often consider arrays with
ideal omni-directional microphones : Ho(ω,θ)=1
Example : uniform linear array, see p.5
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
p. 9
Beamforming basics
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 )
H (ω,θ ) =
Z (ω,θ )
= F H (ω ).d(ω ,θ )
S (ω )
•  Steering direction
= angle θ with maximum amplification (for 1 freq.)
θ max (ω ) = arg maxθ H (ω,θ )
•  Beamwidth (BW)
= region around θmax with (e.g.) amplification > -3dB (for 1 freq.)
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Beamforming basics
Data model: source signal + noise
•  Microphone signals are corrupted by additive noise
Y(ω ,θ ) = d(ω ,θ ).S (ω ) + N(ω )
T
N(ω ) = [N1 (ω ) N 2 (ω ) ... N M (ω )]
•  Define noise correlation matrix as
Φ noise (ω ) = E{N(ω ).N(ω ) H }
•  Will assume noise field is homogeneous, i.e. all diagonal elements of
noise correlation matrix are equal :
Φii (ω ) = Φnoise (ω ) ,
•  Then noise coherence matrix is
Digital Audio Signal Processing
Version 2014-2015
∀i
⎡1 .. ..⎤
1
Γ noise (ω ) =
.Φ noise (ω ) = ⎢⎢.. ! ..⎥⎥
φnoise (ω )
⎢⎣.. ..p. 11 1⎥⎦
Lecture-2: Microphone Array Processing
Beamforming basics
Definitions (2) :
•  Array Gain =improvement in SNR for source at angle θ ( -π<θ< π )
G(ω ,θ ) =
SNRoutput
SNRinput
=
F H (ω ).d(ω ,θ )
2
F H (ω ).Γ noise (ω ).F(ω )
|signal transfer function|^2
|noise transfer function|^2
•  White Noise Gain =array gain for spatially uncorrelated noise (=white)
WNG (ω ,θ ) =
F H (ω ).d(ω , θ )
2
F H (ω ).F(ω )
white
Γnoise
=I
(e.g. sensor noise)
ps: often used as a measure for robustness
•  Directivity =array gain for diffuse noise (=coming from all directions)
DI (ω , θ ) =
F H (ω ).d(ω ,θ )
H
diffuse
noise
F (ω ).Γ
2
.F (ω )
⎛ ω f s (d j − di ) ⎞
⎟⎟
Γijdiffuse (ω ) = sinc⎜⎜
c
⎝
⎠
(ignore this formula)
DI and WNG evaluated at θmax is often used as a performance criterion
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
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6
PS: Near-field beamforming
•  Far-field assumptions not valid for sources close to microphone array
–  spherical wavefronts instead of planar waveforms
–  include attenuation of signals
–  2 coordinates θ,r (=position q) instead of 1 coordinate θ (in 2D case)
•  Different steering vector
d(ω ,θ )
[
d(ω , q) = a1e − jωτ1 (q )
e
e=1 (3D)…2 (2D)
(e.g. with Hm(ω,θ)=1 m=1..M)
am =
q − p ref
q − pm
τ m (q) =
:
a2e − jωτ 2 (q ) … aM e − jωτ M (q )
q − p ref − q − p m
c
]
T
fs
with q position of source
pref position of reference microphone
pm position of mth microphone
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
p. 13
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 practical data model is:
Y(ω , q) = d(ω , q).S (ω ) + N(ω )
T
d(ω , q) = [H1 (ω , q) H 2 (ω , q) ... H M (ω , q)]
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’)
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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)
•  Lecture-3: Adaptive beamforming
Digital Audio Signal Processing
Version 2014-2015
Lecture-2: Microphone Array Processing
p. 15
Filter-and-sum beamformer design
•  Basic: procedure based on
H (ω , θ ) =
page 9
Z (ω ,θ )
= F H (ω ).d(ω , θ )
S (ω )
N −1
T
F(ω ) = [F1 (ω ) ... FM (ω )] , Fm (ω ) = ∑ f m,n .e − jnω
n =0
Array directivity pattern to be matched to given (desired) pattern H d (ω,θ )
over frequency/angle range of interest
•  Non-linear optimization for FIR filter design
(=ignore phase response)
2
min f m ,n ,m =1.. M ,n =0.. N −1 ∫ ∫ ( H (ω , θ ) − H d (ω , θ ) ) dω dθ
θ ω
•  Quadratic optimization for FIR filter design
(=co-design phase response)
2
min f m ,n ,m=1.. M ,n =0.. N −1 ∫ ∫ H (ω , θ ) − H d (ω ,θ ) dω dθ
θ ω
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
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8
Filter-and-sum beamformer design
•  Quadratic optimization for FIR filter design (continued)
Kronecker product
2
min f m ,n ,m=1.. M ,n=0.. N −1 ∫ ∫ H (ω,θ ) − H d (ω,θ ) dω dθ
θ ω
f m = [ f m,0 ...
f m, N −1 ]
T
[
, f = f1T
... f MT
]
T
NMxM
'!!
!
&!!!
%
⎡ e 0. jω ⎤
⎢
⎥
H (With
ω ,θ ) = F H (ω ).d(ω ,θ ) = F1H (ω ) ... FMH (ω ) .d(ω ,θ ) = f T .( I MxM ⊗ ⎢ ( ⎥ ).d(ω ,θ )
⎢e ( N −1). jω ⎥
⎣ !#!!
⎦ !!"
$!!!
[
]
d (ω ,θ )
optimal solution is
foptimal = Q −1. p , Q = ∫
∫ d(ω,θ ).d
θ ω
Digital Audio Signal Processing
H
(ω,θ ) dω dθ , p = ∫
∫ d(ω,θ ).H
θ ω
Version 2014-2015
*
d
(ω,θ ) dω dθ
Lecture-2: Microphone Array Processing
p. 17
Filter-and-sum beamformer design
•  Design example
M=8
Logarithmic array
N=50
fs=8 kHz
1
0.5
0
0
Fre 1000
que 2000
ncy
(H 3000
z)
Digital Audio Signal Processing
0
Version 2014-2015
45
90
135
180
Angle (deg)
Lecture-2: Microphone Array Processing
p. 18
9
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)
•  Lecture-3: Adaptive beamforming
Digital Audio Signal Processing
Version 2014-2015
p. 19
Lecture-2: Microphone Array Processing
Matched filtering : WNG maximization
•  Basic: procedure based on
page 11
•  Maximize White Noise Gain (WNG) for given steering angle ψ
MF
F (ω ) = arg{maxF (ω ) WNG(ω ,ψ )} = arg{maxF (ω )
F H (ω ).d(ω ,ψ )
F H (ω ).F(ω )
2
}
•  A priori knowledge/assumptions:
–  angle-of-arrival ψ of desired signal + corresponding steering vector
–  noise scenario = white
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
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10
Matched filtering : WNG maximization
•  Maximization in
F MF (ω ) = arg{maxF (ω )
F H (ω ).d(ω ,ψ )
F H (ω ).F(ω )
2
}
is equivalent to minimization of noise output power (under
white input noise), subject to unit response for steering angle (**)
minF (ω ) F H (ω).F(ω), s.t. F H (ω).d(ω,ψ ) = 1
•  Optimal solution (`matched filter’) is
F MF (ω ) =
1
d(ω ,ψ )
2
.d(ω ,ψ )
•  [FIR approximation]
2
min f m ,n ,m=1.. M ,n=0.. N −1 ∫ F MF (ω ) − F(ω ) dω
ω
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
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Matched filtering example: Delay-and-sum
•  Basic: Microphone signals are delayed and then summed together
1 M
z[kH ](ω=,ψ ) = 1 .∑ ym [k + Δ m ]
M m =1
ψ
Δ1
1
M
Σ
Δ2
d
Fm (ω ) =
d
Δm
e − jωΔ m
M
( m − 1) d cosψ
•  Fractional delays implemented with truncated interpolation filters (=FIR)
•  Consider array with ideal omni-directional micr’s
Then array can be steered to angle ψ :
d(ω ,ψ )
T
F(ω ) =
Δ m = τ m (ψ )
d(ω ,ψ ) = 1 e − jωτ 2 (ψ ) ... e − jωτ M (ψ )
M
[
]
Hence (for ideal omni-dir. micr.’s) this is matched filter solution
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
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11
Matched filtering example: Delay-and-sum
ideal omni-dir. micr.’s
•  Array directivity pattern H(ω,θ):
1 H
d (ω ,ψ ).d(ω , θ )
M
H (ω , θ ) ≤ 1
H (ω , θ ) =
H (ω , θ = ψ ) = 1
=destructive interference
=constructive interference
(ψ = θ max )
•  White noise gain :
WNG (ω ,θ = ψ ) =
F H (ω ).d(ω ,ψ )
F H (ω ).F(ω )
2
= .. = M
(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).
Digital Audio Signal Processing
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p. 23
Lecture-2: Microphone Array Processing
Matched filtering example: Delay-and-sum
ideal omni-dir. micr.’s
•  Array directivity pattern H(ω,θ) for uniform linear array:
H (ω ,θ ) =
M
∑e
− j ( m −1)ω
m =1
− jMγ / 2
=
e
M=5 microphones
d=3 cm inter-microphone distance
ψ=60° steering angle
fs=16 kHz sampling frequency
d (cos θ − cosψ )
fs
c
sin( Mγ / 2)
e − jγ / 2 sin(γ / 2)
γ
H(ω,θ) has sinc-like shape and is frequency-dependent
H (ω ,θ = ψ ) = 1
Spatial directivity pattern for f=5000 Hz
90
0
1
-10
0.8
0.6
-20
180
0.4
0
=endfire
0.2
0
2000
4000
Frequency (Hz)
6000
8000 0
Digital Audio Signal Processing
wavelength=4cm
45
135
90
Angle (deg)
180
ψ=60°
Version 2014-2015
270
Lecture-2: Microphone Array Processing
p. 24
12
Matched filtering example: Delay-and-sum
ideal omni-dir. micr.’s
f ≥
For
c
d .(1 + cosψ )
an ambiguity, called spatial aliasing, occurs.
This is analogous to time-domain aliasing where now the spatial
sampling (=d) is too large.
c
c
λ
d≤
=
= min
Aliasing does not occur (for any ψ) if
f s 2. f max
2
M=5, ψ=60°, fs=16 kHz, d=8 cm
1
0.8
0.6
0.4
0.2
f =
c
d .(1 + cosψ )
Details...
H (ω ,θ ) = 1 iff γ = 2π.p for integer p
1) γ = 0 for θ = ψ (for all ω )
0
y
enc
qu
Fre
2000
6000
z)
(H
c
d .(1 + cosψ )
π
c
3) if ψ ≥ then γ = 2π occurs for θ = 0 and f = .. =
2
d .(1 − cosψ )
2) if ψ ≤
4000
50
8000 0
Digital Audio Signal Processing
100
150
π
2
then γ = 2π occurs for θ = π and f = .. =
Angle (deg)
Version 2014-2015
Lecture-2: Microphone Array Processing
p. 25
Matched filtering example: Delay-and-sum
ideal omni-dir. micr.’s
•  Beamwidth for a uniform linear array:
96(1 −ν )
secψ
ωdM
BW ≈ c
with e.g. ν=1/sqrt(2) (-3 dB)
hence large dependence on # microphones, distance (compare p.22 & 23)
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
4d
d
2d
Digital Audio Signal Processing
Version 2014-2015
Lecture-2: Microphone Array Processing
p. 26
13
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)
•  Lecture-3: Adaptive beamforming
Digital Audio Signal Processing
Version 2014-2015
p. 27
Lecture-2: Microphone Array Processing
Super-directive beamforming : DI maximization
•  Basic: procedure based on
page 11
•  Maximize Directivity (DI) for given steering angle ψ
SD
F (ω ) = arg{max F (ω ) DI (ω ,ψ )} = arg{max F (ω )
F H (ω ).d(ω ,ψ )
2
diffuse
F H (ω ).Γnoise
.F(ω )
}
•  A priori knowledge/assumptions:
–  angle-of-arrival ψ of desired signal + corresponding steering vector
–  noise scenario = diffuse
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
p. 28
14
Super-directive beamforming : DI maximization
•  Maximization in
SD
F (ω ) = arg{max F (ω )
2
F H (ω ).d(ω ,ψ )
diffuse
F H (ω ).Γnoise
.F(ω )
}
is equivalent to minimization of noise output power (under
diffuse input noise), subject to unit response for steering angle (**)
diffuse
minF (ω ) F H (ω).Γnoise
(ω).F(ω), s.t. F H (ω).d(ω,ψ ) = 1
•  Optimal solution is
F SD (ω ) =
1
diffuse (ω )}−1 .d (ω ,ψ )
d (ω ,ψ ) H .{ Γnoise
diffuse
.{Γnoise
(ω )}−1.d(ω ,ψ )
•  [FIR approximation]
2
min f m ,n ,m=1.. M ,n =0.. N −1 ∫ FSD (ω ) − F(ω ) dω
ω
Digital Audio Signal Processing
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p. 29
Super-directive beamforming : DI maximization
ideal omni-dir. micr.’s
•  Directivity patterns for end-fire steering (ψ=0):
Delay-and-sum beamformer (f=3000 Hz)
Superdirective beamformer (f=3000 Hz)
90
90
0
0
-10
-10
-20
-20
180
0
180
0
270
M=5
d=3 cm
fs=16 kHz
270
Superdirective beamformer has highest DI, but very poor WNG
(at low frequencies, where diffuse noise coherence matrix becomes ill-conditioned)
25
Directivity (linear)
20
DI=25
15
10
DI=5
5
0
0
WNG=5
10
Superdirective
Delay-and-sum
0
White noise gain (dB)
M
hence problems with robustness (e.g. sensor noise) !
2
-10
-20
-30
-40
-50
2000
4000
Frequency (Hz)
6000
8000
-60
0
Superdirective
Delay-and-sum
2000
4000
Frequency (Hz)
6000
PS:
diffuse noise =
white noise for
high frequencies
8000
Digital
Audio Signal
Processing obtained
Version
Microphone Array
Maximum
directivity=M.M
for2014-2015
end-fire steering Lecture-2:
and for frequency->0
(noProcessing
proof)
p. 30
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)
•  Lecture-3: Adaptive beamforming
Digital Audio Signal Processing
Version 2014-2015
Lecture-2: Microphone Array Processing
p. 31
Differential microphones : Delay-and-subtract
•  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
θ
Σ
+
_
Digital Audio Signal Processing
d
H (ω ,θ ) = 1 − e
− jω (τ +
d cosθ
)
c
τ
Version 2014-2015
Lecture-2: Microphone Array Processing
p. 32
16
Differential microphones : Delay-and-subtract
•  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
θ
Σ
+
_
H (ω ,θ ) = 1 − e
d
τ
− jω (τ +
d cosθ
)
c
ωd/c <<π, ωτ <<π
•  Array directivity pattern:
H (ω ,θ ) ≈ ω .(τ +
high -pass
angle dependence
!
!
d
d
cosθ ) = ω .(τ + ). P(θ )
c
c
τ
α1 =
–  First-order high-pass frequency dependence
τ +d /c
–  P(θ) = freq.independent (!) directional response
P(θ ) = α1 + (1 − α1 ) cosθ
–  0 ≤ α1 ≤ 1 : P(θ) is scaled cosine, shifted up with α1
such that θmax = 0o (=end-fire) and P(θmax )=1
Digital Audio Signal Processing
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Lecture-2: Microphone Array Processing
Differential microphones : Delay-and-subtract
•  Types: dipole, cardioid, hypercardioid, supercardioid (HJ84)
Cardioid: α1= 0.5
zero at 180o
DI = 4.8 dB
Dipole: α1= 0 (τ=0)
zero at 90o
DI = 4.8 dB
=broadside
=broadside
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Digital Audio Signal Processing
Version 2014-2015
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Lecture-2: Microphone Array Processing
p. 34
17
Differential microphones : Delay-and-subtract
Hypercardioid: α1= 0.25
Supercardioid: α1= ( 3 − 1) / 2 ≈ 0.35
zero at 109o
zero at 125o, DI=5.7 dB
highest DI=6.0 dB
highest front-to-back ratio
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Digital Audio Signal Processing
Version 2014-2015
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Lecture-2: Microphone Array Processing
p. 35
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)
•  Lecture-3: Adaptive beamforming
Digital Audio Signal Processing
Version 2014-2015
Lecture-2: Microphone Array Processing
p. 36
18