microphone-array signal processingaquarius.ime.eb.br/~apolin/micarray/sbrt_11_minicurso_all.pdf ·...
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Microphone-Array Signal Processing
Jose A. Apolinario Jr. and Marcello L. R. de Campos
apolin , [email protected]
IME − Lab. Processamento Digital de Sinais
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 1/115
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Outline
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Outline
1. Introduction and Fundamentals
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Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
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Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
3. Optimal Beamforming
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Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
3. Optimal Beamforming
4. Adaptive Beamforming
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Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
3. Optimal Beamforming
4. Adaptive Beamforming
5. DoA Estimation with Microphone Arrays
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1. Introduction andFundamentals
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 3/115
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General concepts
In this course, signals and noise...
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General concepts
In this course, signals and noise...
have spatial dependence
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General concepts
In this course, signals and noise...
have spatial dependence
must be characterized as space-time processes
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General concepts
In this course, signals and noise...
have spatial dependence
must be characterized as space-time processes
An array of sensors is represented in the figure below.
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General concepts
In this course, signals and noise...
have spatial dependence
must be characterized as space-time processes
An array of sensors is represented in the figure below.
SENSORS
ARRAY
PROCESSOR
interference
signal #1signal #2
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1.2 Signals in Space and Time
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 5/115
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Defining operators
Let ∇x(·) and ∇2x(·) be the gradient and Laplacian
operators, i.e.,
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Defining operators
Let ∇x(·) and ∇2x(·) be the gradient and Laplacian
operators, i.e.,
∇x(·) =∂(·)∂x
−→ı x +∂(·)∂y
−→ı y +∂(·)∂z
−→ı z
=
[∂(·)∂x
∂(·)∂y
∂(·)∂z
]T
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Defining operators
Let ∇x(·) and ∇2x(·) be the gradient and Laplacian
operators, i.e.,
∇x(·) =∂(·)∂x
−→ı x +∂(·)∂y
−→ı y +∂(·)∂z
−→ı z
=
[∂(·)∂x
∂(·)∂y
∂(·)∂z
]T
and
∇2x(·) =
∂2(·)∂x2
+∂2(·)∂y2
+∂2(·)∂z2
respectively.
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Wave equation
From Maxwell’s equations,
∇2−→E =1
c2∂2−→E
∂t2
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Wave equation
From Maxwell’s equations,
∇2−→E =1
c2∂2−→E
∂t2
or, for s(x, t) a general scalar field,
∂2s
∂x2+∂2s
∂y2+∂2s
∂z2=
1
c2∂2s
∂t2
where: c is the propagation speed,−→E is the electric field
intensity, and x = [x y z]T is a position vector.
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Wave equation
From Maxwell’s equations,
∇2−→E =1
c2∂2−→E
∂t2
or, for s(x, t) a general scalar field,
∂2s
∂x2+∂2s
∂y2+∂2s
∂z2=
1
c2∂2s
∂t2
where: c is the propagation speed,−→E is the electric field
intensity, and x = [x y z]T is a position vector.
“vector” wave equation
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Wave equation
From Maxwell’s equations,
∇2−→E =1
c2∂2−→E
∂t2
or, for s(x, t) a general scalar field,
∂2s
∂x2+∂2s
∂y2+∂2s
∂z2=
1
c2∂2s
∂t2
where: c is the propagation speed,−→E is the electric field
intensity, and x = [x y z]T is a position vector.
“vector” wave equation
“scalar” wave equation
Note: From this point onwards the terms wave and field will be used interchangeably.
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Monochromatic plane wave
Now assume s(x, t) has a complex exponential form,
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Monochromatic plane wave
Now assume s(x, t) has a complex exponential form,
s(x, t) = Aej(ωt−kxx−kyy−kzz)
where A is a complex constant and kx, ky, kz, and ω ≥ 0are real constants.
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Monochromatic plane wave
Substituting the complex exponential form of s(x, t) into thewave equation, we have
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Monochromatic plane wave
Substituting the complex exponential form of s(x, t) into thewave equation, we have
k2xs(x, t) + k2ys(x, t) + k2zs(x, t) =1
c2ω2s(x, t)
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Monochromatic plane wave
Substituting the complex exponential form of s(x, t) into thewave equation, we have
k2xs(x, t) + k2ys(x, t) + k2zs(x, t) =1
c2ω2s(x, t)
or, after canceling s(x, t),
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Monochromatic plane wave
Substituting the complex exponential form of s(x, t) into thewave equation, we have
k2xs(x, t) + k2ys(x, t) + k2zs(x, t) =1
c2ω2s(x, t)
or, after canceling s(x, t),
k2x + k2y + k2z =1
c2ω2
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Monochromatic plane wave
Substituting the complex exponential form of s(x, t) into thewave equation, we have
k2xs(x, t) + k2ys(x, t) + k2zs(x, t) =1
c2ω2s(x, t)
or, after canceling s(x, t),
k2x + k2y + k2z =1
c2ω2
constraints to be satisfiedby the parametersof the scalar field
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Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
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Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
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Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
plane
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Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
plane
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Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
planeFor example, take the position at the origin
of the coordinate space:
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Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
planeFor example, take the position at the origin
of the coordinate space:
x = [0 0 0]T
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Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
planeFor example, take the position at the origin
of the coordinate space:
x = [0 0 0]T
s(0, t) = Aejωt
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Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
plane
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Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
planeThe value of s(x, t) is the same for all
points lying on the plane
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Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
planeThe value of s(x, t) is the same for all
points lying on the plane
kxx+ kyy + kzz = C
where C is a constant.
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Monochromatic plane wave
Defining the wavenumber vector k as
k = [kx ky kz]T
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Monochromatic plane wave
Defining the wavenumber vector k as
k = [kx ky kz]T
we can rewrite the equation for the monochromatic planewave as
s(x, t) = Aej(ωt−kTx)
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Monochromatic plane wave
Defining the wavenumber vector k as
k = [kx ky kz]T
we can rewrite the equation for the monochromatic planewave as
s(x, t) = Aej(ωt−kTx) The planes where s(x, t)is constant are perpendicularto the wavenumber vector k
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Monochromatic plane wave
As the plane wave propagates, it advances a distance δxin δt seconds.
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Monochromatic plane wave
As the plane wave propagates, it advances a distance δxin δt seconds.
Therefore,
s(x, t) = s(x+ δx, t+ δt)
⇐⇒ Aej(ωt−kTx) = Aej[ω(t+δt)−kT (x+δx)]
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Monochromatic plane wave
As the plane wave propagates, it advances a distance δxin δt seconds.
Therefore,
s(x, t) = s(x+ δx, t+ δt)
⇐⇒ Aej(ωt−kTx) = Aej[ω(t+δt)−kT (x+δx)]
=⇒ ωδt− kT δx = 0
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Monochromatic plane wave
Naturally the plane wave propagates in the direction of thewavenumber vector, i.e.,
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Monochromatic plane wave
Naturally the plane wave propagates in the direction of thewavenumber vector, i.e.,
k and δx point in the same direction.
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Monochromatic plane wave
Naturally the plane wave propagates in the direction of thewavenumber vector, i.e.,
k and δx point in the same direction.
Therefore,
kT δx = ‖k‖‖δx‖
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Monochromatic plane wave
Naturally the plane wave propagates in the direction of thewavenumber vector, i.e.,
k and δx point in the same direction.
Therefore,
kT δx = ‖k‖‖δx‖
=⇒ ωδt = ‖k‖‖δx‖
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Monochromatic plane wave
Naturally the plane wave propagates in the direction of thewavenumber vector, i.e.,
k and δx point in the same direction.
Therefore,
kT δx = ‖k‖‖δx‖
=⇒ ωδt = ‖k‖‖δx‖or, equivalently,
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Monochromatic plane wave
Naturally the plane wave propagates in the direction of thewavenumber vector, i.e.,
k and δx point in the same direction.
Therefore,
kT δx = ‖k‖‖δx‖
=⇒ ωδt = ‖k‖‖δx‖or, equivalently,
‖δx‖δt
=ω
‖k‖
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Monochromatic plane wave
Naturally the plane wave propagates in the direction of thewavenumber vector, i.e.,
k and δx point in the same direction.
Therefore,
kT δx = ‖k‖‖δx‖
=⇒ ωδt = ‖k‖‖δx‖or, equivalently,
‖δx‖δt
=ω
‖k‖
Remember the constraints?‖k‖2 = ω2/c2
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Monochromatic plane wave
After T = 2π/ω seconds, the plane wave has completedone cycle and it appears as it did before, but its wavefronthas advanced a distance of one wavelength, λ.
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Monochromatic plane wave
After T = 2π/ω seconds, the plane wave has completedone cycle and it appears as it did before, but its wavefronthas advanced a distance of one wavelength, λ.
For ‖δx‖ = λ and δt = T = 2πω
,
T =λ‖k‖ω
=⇒ λ =2π
‖k‖
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Monochromatic plane wave
After T = 2π/ω seconds, the plane wave has completedone cycle and it appears as it did before, but its wavefronthas advanced a distance of one wavelength, λ.
For ‖δx‖ = λ and δt = T = 2πω
,
T =λ‖k‖ω
=⇒ λ =2π
‖k‖
The wavenumber vector, k, may be considered aspatial frequency variable, just as ω is atemporal frequency variable.
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Monochromatic plane wave
We may rewrite the wave equation as
s(x, t) = Aej(ωt−kTx)
= Aejω(t−αTx)
where α = k/ω is the slowness vector.
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Monochromatic plane wave
We may rewrite the wave equation as
s(x, t) = Aej(ωt−kTx)
= Aejω(t−αTx)
where α = k/ω is the slowness vector.
As c = ω/‖k‖, vector α has a magnitude which is thereciprocal of c.
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Periodic propagating periodic waves
Any arbitrary periodic waveform s(x, t) = s(t−αTx) withfundamental period ω0 can be represented as a sum:
s(x, t) = s(t−αTx) =∞∑
n=−∞
Snejnω0(t−αTx)
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Periodic propagating periodic waves
Any arbitrary periodic waveform s(x, t) = s(t−αTx) withfundamental period ω0 can be represented as a sum:
s(x, t) = s(t−αTx) =∞∑
n=−∞
Snejnω0(t−αTx)
The coefficients are given by
Sn =1
T
∫ T
0
s(u)e−jnω0udu
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Periodic propagating periodic waves
Based on the previous derivations, we observe that:
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Periodic propagating periodic waves
Based on the previous derivations, we observe that:
The various components of s(x, t) have differentfrequencies ω = nω0 and different wavenumbervectors, k.
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Periodic propagating periodic waves
Based on the previous derivations, we observe that:
The various components of s(x, t) have differentfrequencies ω = nω0 and different wavenumbervectors, k.
The waveform propagates in the direction of theslowness vector α = k/ω.
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Nonperiodic propagating waves
More generally, any function constructed as the integral ofcomplex exponentials who also have a defined andconverged Fourier transform can represent a waveform
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Nonperiodic propagating waves
More generally, any function constructed as the integral ofcomplex exponentials who also have a defined andconverged Fourier transform can represent a waveform
s(x, t) = s(t−αTx) =1
2π
∫ ∞
−∞
S(ω)ejω(t−αTx)dω
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Nonperiodic propagating waves
More generally, any function constructed as the integral ofcomplex exponentials who also have a defined andconverged Fourier transform can represent a waveform
s(x, t) = s(t−αTx) =1
2π
∫ ∞
−∞
S(ω)ejω(t−αTx)dω
where
S(ω) =
∫ ∞
−∞
s(u)e−jωudu
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Nonperiodic propagating waves
More generally, any function constructed as the integral ofcomplex exponentials who also have a defined andconverged Fourier transform can represent a waveform
s(x, t) = s(t−αTx) =1
2π
∫ ∞
−∞
S(ω)ejω(t−αTx)dω
where
S(ω) =
∫ ∞
−∞
s(u)e−jωudu
We will come back to this later...
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Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
3. Optimal Beamforming
4. Adaptive Beamforming
5. DoA Estimation with Microphone Arrays
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2. Sensor Arrays and SpatialFiltering
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2.1 Wavenumber-Frequency Space
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Space-time Fourier Transform
The four-dimensional Fourier transform of the space-timesignal s(x, t) is given by
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Space-time Fourier Transform
The four-dimensional Fourier transform of the space-timesignal s(x, t) is given by
S(k, ω) =
∫ ∞
−∞
∫ ∞
−∞
s(x, t)e−j(ωt−kTx)dxdt
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Space-time Fourier Transform
The four-dimensional Fourier transform of the space-timesignal s(x, t) is given by
S(k, ω) =
∫ ∞
−∞
∫ ∞
−∞
s(x, t)e−j(ωt−kTx)dxdt
temporal frequency
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Space-time Fourier Transform
The four-dimensional Fourier transform of the space-timesignal s(x, t) is given by
S(k, ω) =
∫ ∞
−∞
∫ ∞
−∞
s(x, t)e−j(ωt−kTx)dxdt
temporal frequency
spatial frequency: wavenumber
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Space-time Fourier Transform
The four-dimensional Fourier transform of the space-timesignal s(x, t) is given by
S(k, ω) =
∫ ∞
−∞
∫ ∞
−∞
s(x, t)e−j(ωt−kTx)dx dt
s(x, t) =1
(2π)4
∫ ∞
−∞
∫ ∞
−∞
S(k, ω)ej(ωt−kTx)dk dω
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Space-time Fourier Transform
We have already concluded that if the space-time signal isa propagating waveform such that s(x, t) = s(t−αT
0 x),then its Fourier transform is equal to
S(k, ω) = S(ω)δ(k− ωα0)
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Space-time Fourier Transform
We have already concluded that if the space-time signal isa propagating waveform such that s(x, t) = s(t−αT
0 x),then its Fourier transform is equal to
S(k, ω) = S(ω)δ(k− ωα0)
Remember the nonperiodic propagating wave Fouriertransform?
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Space-time Fourier Transform
We have already concluded that if the space-time signal isa propagating waveform such that s(x, t) = s(t−αT
0 x),then its Fourier transform is equal to
S(k, ω) = S(ω)δ(k− ωα0)
Remember the nonperiodic propagating wave Fouriertransform?
This means that s(x, t) only has energy along the directionof k = k0 = ωα0 in the wavenumber-frequency space.
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2.2 Frequency-Wavenumber (WN) Responseand Beam patterns (BP)
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Signals at the sensors
An array is a set of N (isotropic) sensors located atpositions pn, n = 0, 1, · · · , N − 1
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Signals at the sensors
An array is a set of N (isotropic) sensors located atpositions pn, n = 0, 1, · · · , N − 1
The sensors spatially sample the signal field atlocations pn
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Signals at the sensors
An array is a set of N (isotropic) sensors located atpositions pn, n = 0, 1, · · · , N − 1
The sensors spatially sample the signal field atlocations pn
At the sensors, the set of N signals are denoted by
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Signals at the sensors
An array is a set of N (isotropic) sensors located atpositions pn, n = 0, 1, · · · , N − 1
The sensors spatially sample the signal field atlocations pn
At the sensors, the set of N signals are denoted by
f(t,p) =
f(t,p0)
f(t,p1)...
f(t,pN−1)
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Array output
N−1
h (t)
h (t)1 Σ
...
f (t, )p
pN−1
f (t, )
f (t, )p1
y(t)
...
0
h (t)
0
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Array output
N−1
h (t)
h (t)1 Σ
...
f (t, )p
pN−1
f (t, )
f (t, )p1
y(t)
...
0
h (t)
0
y(t) =
N−1∑
n=0
∫ ∞
−∞
hn(t− τ)fn(τ,pn)dτ
=
∫ ∞
−∞
hT (t− τ)f(τ,p)dτ
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Array output
N−1
h (t)
h (t)1 Σ
...
f (t, )p
pN−1
f (t, )
f (t, )p1
y(t)
...
0
h (t)
0
y(t) =
N−1∑
n=0
∫ ∞
−∞
hn(t− τ)fn(τ,pn)dτ
=
∫ ∞
−∞
hT (t− τ)f(τ,p)dτ
where h(t) = [ho(t) h1(t) · · · hN−1(t)]T
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 28/115
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In the frequency domain, · · ·
Y (ω) =
∫ ∞
−∞
y(t)e−jωtdt
= HT (ω)F (ω)
![Page 86: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/86.jpg)
In the frequency domain, · · ·
Y (ω) =
∫ ∞
−∞
y(t)e−jωtdt
= HT (ω)F (ω)
where
H(ω) =
∫ ∞
−∞
h(t)e−jωtdt
F (ω) = F (ω,p) =
∫ ∞
−∞
f(t,p)e−jωtdt
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 29/115
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Plane wave propagating · · ·
Consider a plane wave propagating in the direction ofvector a:
a =
−sinθcosφ−sinθsinφ−cosθ
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Plane wave propagating · · ·
Consider a plane wave propagating in the direction ofvector a:
a =
−sinθcosφ−sinθsinφ−cosθ
If f(t) is the signal that would be received at the origin,then:
f(t,p) =
f(t− τ0)
f(t− τ1)...
f(t− τN−1)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 30/115
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Plane wave (assuming φ = 90o)
plane wave
θ
−sinθ
z
y
a
pn
a
u
e
y
z
α
θ 1
−cos
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Plane wave (assuming φ = 90o)
plane wave
θ
−sinθ
z
y
a
pn
a
u
e
y
z
α
θ 1
−cos
e =?
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Plane wave (assuming φ = 90o)
plane wave
θ
−sinθ
z
y
a
pn
a
u
e
y
z
α
θ 1
−cos
e =?
e = cτn
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Plane wave (assuming φ = 90o)
plane wave
θ
−sinθ
z
y
a
pn
a
u
e
y
z
α
θ 1
−cos
e =?
e = cτn
⇒ τn = ec
![Page 93: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/93.jpg)
Plane wave (assuming φ = 90o)
plane wave
θ
−sinθ
z
y
a
pn
a
u
e
y
z
α
θ 1
−cos
e =?
e = cτn
⇒ τn = ec BUT
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Plane wave (assuming φ = 90o)
plane wave
θ
−sinθ
z
y
a
pn
a
u
e
y
z
α
θ 1
−cos
e =?
e = cτn
⇒ τn = ec BUT
e = ‖pn‖cos(α) = ‖u‖︸︷︷︸
=1
‖pn‖cos(α)
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Plane wave (assuming φ = 90o)
plane wave
θ
−sinθ
z
y
a
pn
a
u
e
y
z
α
θ 1
−cos
e =?
e = cτn
⇒ τn = ec BUT
e = ‖pn‖cos(α) = ‖u‖︸︷︷︸
=1
‖pn‖cos(α)
∴ τn = −uTpn
c= aTpn
c
![Page 96: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/96.jpg)
Plane wave (assuming φ = 90o)
plane wave
θ
−sinθ
z
y
a
pn
a
u
e
y
z
α
θ 1
−cos
e =?
e = cτn
⇒ τn = ec BUT
e = ‖pn‖cos(α) = ‖u‖︸︷︷︸
=1
‖pn‖cos(α)
∴ τn = −uTpn
c= aTpn
c
τn is the time since the plane wave hits the sensor at location pn until it reaches point (0, 0).
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 31/115
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Back to the frequency domain
Then, we have:
F (ω) =
∫∞
−∞e−jωtf(t− τ0)dt
∫∞
−∞e−jωtf(t− τ1)dt
...∫∞
−∞e−jωtf(t− τN−1)dt
=
e−jωτ0
e−jωτ1
...e−jωτN−1
F (ω)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 32/115
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Definition of Wavenumber
For plane waves propagating in a locally homogeneousmedium:
k =ω
ca =
2π
c/fa =
2π
λa = −2π
λu
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Definition of Wavenumber
For plane waves propagating in a locally homogeneousmedium:
k =ω
ca =
2π
c/fa =
2π
λa = −2π
λu
Wavenumber Vector ("spatial frequency")
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Definition of Wavenumber
For plane waves propagating in a locally homogeneousmedium:
k =ω
ca =
2π
c/fa =
2π
λa = −2π
λu
Wavenumber Vector ("spatial frequency")
Note that |k| = 2πλ
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Definition of Wavenumber
For plane waves propagating in a locally homogeneousmedium:
k =ω
ca =
2π
c/fa =
2π
λa = −2π
λu
Wavenumber Vector ("spatial frequency")
Note that |k| = 2πλ
Therefore
ωτn =ω
caTpn = kTpn
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 33/115
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Array Manifold Vector
And we have
F (ω) =
e−jkTp0
e−jkTp1
...
e−jkTp
N−1
F (ω) = F (ω)vk(k)
![Page 103: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/103.jpg)
Array Manifold Vector
And we have
F (ω) =
e−jkTp0
e−jkTp1
...
e−jkTp
N−1
F (ω) = F (ω)vk(k)
Array Manifold Vector
![Page 104: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/104.jpg)
Array Manifold Vector
And we have
F (ω) =
e−jkTp0
e−jkTp1
...
e−jkTp
N−1
F (ω) = F (ω)vk(k)
Array Manifold Vector
In this particular example, we can usehn(t) =
1Nδ(t+ τn) such that
y(t) = f(t)
Following, we have the delay-and-sum beamformer.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 34/115
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Delay-and-sum Beamformer
τ
Σ
...
1N
... y(t)
f (t− )0
f (t− )1
f (t− )N−1+τN−1
+τ1
+τ0τ
τ
![Page 106: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/106.jpg)
Delay-and-sum Beamformer
τ
Σ
...
1N
... y(t)
f (t− )0
f (t− )1
f (t− )N−1+τN−1
+τ1
+τ0τ
τ
A common delay is added in each channel to make theoperations physically realizable
![Page 107: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/107.jpg)
Delay-and-sum Beamformer
τ
Σ
...
1N
... y(t)
f (t− )0
f (t− )1
f (t− )N−1+τN−1
+τ1
+τ0τ
τ
A common delay is added in each channel to make theoperations physically realizable
Since F hn(t) = F
1Nδ(t+ τn)
= ejωτn
![Page 108: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/108.jpg)
Delay-and-sum Beamformer
τ
Σ
...
1N
... y(t)
f (t− )0
f (t− )1
f (t− )N−1+τN−1
+τ1
+τ0τ
τ
A common delay is added in each channel to make theoperations physically realizable
Since F hn(t) = F
1Nδ(t+ τn)
= ejωτn
We can write
HT (ω) =1
NvH
k(k)
![Page 109: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/109.jpg)
Delay-and-sum Beamformer
τ
Σ
...
1N
... y(t)
f (t− )0
f (t− )1
f (t− )N−1+τN−1
+τ1
+τ0τ
τ
A common delay is added in each channel to make theoperations physically realizable
Since F hn(t) = F
1Nδ(t+ τn)
= ejωτn
We can write
HT (ω) =1
NvH
k(k)
Array Manifold Vector
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 35/115
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LTI System
ejωt h(t) H(ω)ejωt
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LTI System
ejωt h(t) H(ω)ejωt
Space-time signals (base functions):
fn(t,p) = ejω(t−τn) = ej(ωt−kTp
n)
Note that ωτn = kTpn
![Page 112: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/112.jpg)
LTI System
ejωt h(t) H(ω)ejωt
Space-time signals (base functions):
fn(t,p) = ejω(t−τn) = ej(ωt−kTp
n)
Note that ωτn = kTpn
∴ f(t,p) = ejωtvk(k)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 36/115
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Frequency-Wavenumber Response Function
The response of the array to this plane wave is:
y(t,k) = HT (ω)vk(k)ejωt
![Page 114: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/114.jpg)
Frequency-Wavenumber Response Function
The response of the array to this plane wave is:
y(t,k) = HT (ω)vk(k)ejωt
After taking the Fourier transform, we have:
Y (ω,k) = HT (ω)vk(k)
![Page 115: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/115.jpg)
Frequency-Wavenumber Response Function
The response of the array to this plane wave is:
y(t,k) = HT (ω)vk(k)ejωt
After taking the Fourier transform, we have:
Y (ω,k) = HT (ω)vk(k)
And we define the Frequency-Wavenumber ResponseFunction:
Υ(ω,k) , HT (ω)vk(k)Upsilon
Υ(ω,k) describes the complex gain of an array to aninput plane wave with wavenumber k and temporalfrequency ω.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 37/115
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Beam Pattern and Bandpass Signal
BEAM PATTERN is the Frequency WavenumberResponse Function evaluated versus the direction:
B(ω : θ, φ) = Υ(ω,k)
Note that k = 2πλa(θ, φ), and a is the unit vector with
spherical coordinates angles θ and φ
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Beam Pattern and Bandpass Signal
BEAM PATTERN is the Frequency WavenumberResponse Function evaluated versus the direction:
B(ω : θ, φ) = Υ(ω,k)
Note that k = 2πλa(θ, φ), and a is the unit vector with
spherical coordinates angles θ and φ
Let’s write a bandpass signal:
f(t,pn) =√2Ref(t,pn)e
jωct, n = 0, 1, · · · , N − 1
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Beam Pattern and Bandpass Signal
BEAM PATTERN is the Frequency WavenumberResponse Function evaluated versus the direction:
B(ω : θ, φ) = Υ(ω,k)
Note that k = 2πλa(θ, φ), and a is the unit vector with
spherical coordinates angles θ and φ
Let’s write a bandpass signal:
f(t,pn) =√2Ref(t,pn)e
jωct, n = 0, 1, · · · , N − 1
ωc corresponds to the carrier frequency and thecomplex envelope f(t,pn) is bandlimited to the region
|ω − ωc︸ ︷︷ ︸
ωL
| ≤ 2πBs/2ωc
2πBs
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 38/115
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Bandlimited and Narrowband Signals
Bandlimited plane wave:f(t,pn) =
√2Ref(t− τn)e
jωc(t−τn), n = 0, 1, · · · , N − 1
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Bandlimited and Narrowband Signals
Bandlimited plane wave:f(t,pn) =
√2Ref(t− τn)e
jωc(t−τn), n = 0, 1, · · · , N − 1
Maximum travel time (∆Tmax) across the (linear) array:travel time between the two sensors at the extremities(signal arriving along the end-fire)
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Bandlimited and Narrowband Signals
Bandlimited plane wave:f(t,pn) =
√2Ref(t− τn)e
jωc(t−τn), n = 0, 1, · · · , N − 1
Maximum travel time (∆Tmax) across the (linear) array:travel time between the two sensors at the extremities(signal arriving along the end-fire)
Assuming the origin is at the array’s center of gravity:∑N−1
n=0 pn = 0 ⇒ τn ≤ ∆Tmax
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Bandlimited and Narrowband Signals
Bandlimited plane wave:f(t,pn) =
√2Ref(t− τn)e
jωc(t−τn), n = 0, 1, · · · , N − 1
Maximum travel time (∆Tmax) across the (linear) array:travel time between the two sensors at the extremities(signal arriving along the end-fire)
Assuming the origin is at the array’s center of gravity:∑N−1
n=0 pn = 0 ⇒ τn ≤ ∆Tmax
In Narrowband (NB) signals, Bs∆Tmax ≪ 1
⇒ f(t− τn) ≃ f(t) andf(t,pn) =
√2Ref(t)e−jωcτnejωct
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Phased-Array
For NB signals, the delay is approximated by aphase-shift:⇒ delay&sum beamformer ≡ PHASED ARRAY
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Phased-Array
For NB signals, the delay is approximated by aphase-shift:⇒ delay&sum beamformer ≡ PHASED ARRAY
τ
Σ
...
1N
... y(t)
f(t)
f(t)
f(t)
0 0j
e+ ω τ
0 1j
e+ ω τ
0 N−1j
e+ ω τf (t− )N−1
f (t− )1
0f (t− )τ
τ
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Phased-Array
For NB signals, the delay is approximated by aphase-shift:⇒ delay&sum beamformer ≡ PHASED ARRAY
τ
Σ
...
1N
... y(t)
f(t)
f(t)
f(t)
0 0j
e+ ω τ
0 1j
e+ ω τ
0 N−1j
e+ ω τf (t− )N−1
f (t− )1
0f (t− )τ
τ
The phased array can be implemented adjusting thegain and phase to achieve a desired beam pattern
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NB Beamformers
In narrowband beamformers: y(t,k) = wHvk(k)ejωt
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NB Beamformers
In narrowband beamformers: y(t,k) = wHvk(k)ejωt
y(t)Σ
...
w*0
w*1
w*N−1
...
f (t− )N−1
f (t− )1
0f (t− )τ
τ
τ
y (t)
y (t)
0
1
N−1y (t)
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NB Beamformers
In narrowband beamformers: y(t,k) = wHvk(k)ejωt
y(t)Σ
...
w*0
w*1
w*N−1
...
f (t− )N−1
f (t− )1
0f (t− )τ
τ
τ
y (t)
y (t)
0
1
N−1y (t)
Υ(ω,k) = wH
︸︷︷︸
HT(ω)
vk(k)
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2.3 Uniform Linear Arrays (ULA)
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Uniformly Spaced Linear Arrays
(grazing angle)
_(broadside angle)
r
d
φ
θ
z (array axis = "endfire")
y
x
2θ = π − θ_
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ULAAn ULA along axis z: N−1
d
z
y
x
(azimuth angle)
φ
θ
(polar angle)
0
1
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ULAAn ULA along axis z: N−1
d
z
y
x
(azimuth angle)
φ
θ
(polar angle)
0
1
Location of the elements:pzn = (n− N−1
2)d, for n = 0, 1, · · · , N − 1
pxn = pyn = 0
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ULAAn ULA along axis z: N−1
d
z
y
x
(azimuth angle)
φ
θ
(polar angle)
0
1
Location of the elements:pzn = (n− N−1
2)d, for n = 0, 1, · · · , N − 1
pxn = pyn = 0
Therefore, pn =
0
0
(n− N−12
)d
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ULA
Array manifold vector:
vk(k) =
e−jkTp0
...
e−jkTp
n
...
e−jkTp
N−1
[kx ky kz ]
0
0[
n− N−12
]
d
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ULA
Array manifold vector:
vk(k) =[
e−jkTp0 e−jk
Tp1 · · · e−jk
Tp
N−1
]T
∴ vk(k) = vk(kz) =
e+j(N−1)
2kzd
e+j(N−12
−1)kzd
...e−j(N−1
2)kzd
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Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
3. Optimal Beamforming
4. Adaptive Beamforming
5. DoA Estimation with Microphone Arrays
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3. Optimal Beamforming
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3.1 Introduction
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Introduction
Scope: use the statistical representation of signal andnoise to design array processors that are optimal in astatistical sense.
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Introduction
Scope: use the statistical representation of signal andnoise to design array processors that are optimal in astatistical sense.
We assume that the appropriate statistics are known.
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Introduction
Scope: use the statistical representation of signal andnoise to design array processors that are optimal in astatistical sense.
We assume that the appropriate statistics are known.
Our objective of interest is to estimate the waveform ofa plane-wave impinging on the array in the presence ofnoise and interfering signals.
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Introduction
Scope: use the statistical representation of signal andnoise to design array processors that are optimal in astatistical sense.
We assume that the appropriate statistics are known.
Our objective of interest is to estimate the waveform ofa plane-wave impinging on the array in the presence ofnoise and interfering signals.
Even if a particular beamformer developed in thischapter has good performance, it does not guaranteethat its adaptive version (next chapter) will. However, ifthe performance is poor, it is unlikely that the adaptiveversion will be useful.
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3.2 Optimal Beamformers
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MVDR Beamformer
Snapshot model in the frequency domain:
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MVDR Beamformer
Snapshot model in the frequency domain:
In many applications, we implement a beamforming inthe frequency domain (ωm = ωc +m 2π
∆Tand M varies
from −M−12
to M−12
if odd and from −M2
to M2− 1 if
even).X
DiscreteFourierTransform
∆TX
∆TX
∆T
∆TY
∆TY ω(M−1)/2
( ),k
ω m
∆TY
FourierTransform atM Frequencies
NB Beamformer
NB Beamformer
NB Beamformer
( )
ω ,k−(M−1)/2( )
ω(M−1)/2
ω ,k−(M−1)/2
,k( )
ωm ,k( )
( )
,k y(t)(t)X Inverse
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MVDR Beamformer
Snapshot model in the frequency domain:
In many applications, we implement a beamforming inthe frequency domain (ωm = ωc +m 2π
∆Tand M varies
from −M−12
to M−12
if odd and from −M2
to M2− 1 if
even).X
DiscreteFourierTransform
∆TX
∆TX
∆T
∆TY
∆TY ω(M−1)/2
( ),k
ω m
∆TY
FourierTransform atM Frequencies
NB Beamformer
NB Beamformer
NB Beamformer
( )
ω ,k−(M−1)/2( )
ω(M−1)/2
ω ,k−(M−1)/2
,k( )
ωm ,k( )
( )
,k y(t)(t)X Inverse
In order to generate these vectors, divide theobservation interval T in K disjoint intervals of duration∆T : (k − 1)∆T ≤ t < k∆T, k = 1, · · · ,K.
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MVDR Beamformer
∆T must be significantly greater than the propagationtime across the array.
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MVDR Beamformer
∆T must be significantly greater than the propagationtime across the array.
∆T also depends on the bandwidth of the input signal.
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MVDR Beamformer
∆T must be significantly greater than the propagationtime across the array.
∆T also depends on the bandwidth of the input signal.
Assume an input signal with BW Bs centered in fc
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MVDR Beamformer
∆T must be significantly greater than the propagationtime across the array.
∆T also depends on the bandwidth of the input signal.
Assume an input signal with BW Bs centered in fc
In order to develop the frequency-domain snapshotmodel for the case in which the desired signals and theinterfering signals can de modeled as plane waves, wehave two cases: desired signals are deterministic orsamples of a random process.
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MVDR Beamformer
Let’s assume the case where the signal is nonrandombut unknown; we initially consider the case of singleplane-wave signal.
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MVDR Beamformer
Let’s assume the case where the signal is nonrandombut unknown; we initially consider the case of singleplane-wave signal.
Frequency-domain snapshot consists of signal plusnoise: X(ω) = Xs(ω) +N (ω)
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MVDR Beamformer
Let’s assume the case where the signal is nonrandombut unknown; we initially consider the case of singleplane-wave signal.
Frequency-domain snapshot consists of signal plusnoise: X(ω) = Xs(ω) +N (ω)
The signal vector can be written asXs(ω) = F (ω)v(ω : ks) where F (ω) is thefrequency-domain snapshot of the source signal andv(ω : ks) is the array manifold vector for a plane-wavewith wavenumber ks.
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MVDR Beamformer
Let’s assume the case where the signal is nonrandombut unknown; we initially consider the case of singleplane-wave signal.
Frequency-domain snapshot consists of signal plusnoise: X(ω) = Xs(ω) +N (ω)
The signal vector can be written asXs(ω) = F (ω)v(ω : ks) where F (ω) is thefrequency-domain snapshot of the source signal andv(ω : ks) is the array manifold vector for a plane-wavewith wavenumber ks.
The noise snapshot is a zero-mean random vectorN (ω) with spectral matrix given bySn(ω) = Sc(ω) + σ2
ωI
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MVDR Beamformer
We process X(ω) with the 1×N operator WH(ω):
Yω ( )ω( )ω
XW
H( )
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MVDR Beamformer
We process X(ω) with the 1×N operator WH(ω):
Yω ( )ω( )ω
XW
H( )
Distortionless criterion (in the absence of noise):
Y (ω) = F (ω)
= WH(ω)Xs(ω) = F (ω)WH(ω)v(ω : ks)
=⇒WH(ω)v(ω : ks) = 1
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MVDR Beamformer
In the presence of noise, we have:
Y (ω) = F (ω) + Yn(ω)
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MVDR Beamformer
In the presence of noise, we have:
Y (ω) = F (ω) + Yn(ω)
The mean square of the output noise is:
E[|Yn(ω)|2] = WH(ω)Sn(ω)W (ω)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 55/115
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MVDR BeamformerIn the MVDR beamformer, we want to minimize
E[|Yn(ω)|2] subject to WH(ω)v(ω : ks) = 1
![Page 160: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/160.jpg)
MVDR BeamformerIn the MVDR beamformer, we want to minimize
E[|Yn(ω)|2] subject to WH(ω)v(ω : ks) = 1
Using the method of Lagrange multipliers, we definethe following cost function to be minimized
F = WH(ω)Sn(ω)Wω
+ λ[WH(ω)v(ω : ks)− 1
]+ λ∗
[vH(ω : ks)W (ω)− 1
]
![Page 161: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/161.jpg)
MVDR BeamformerIn the MVDR beamformer, we want to minimize
E[|Yn(ω)|2] subject to WH(ω)v(ω : ks) = 1
Using the method of Lagrange multipliers, we definethe following cost function to be minimized
F = WH(ω)Sn(ω)Wω
+ λ[WH(ω)v(ω : ks)− 1
]+ λ∗
[vH(ω : ks)W (ω)− 1
]
...and the result (suppressing ω and ks) is
WHmvdr = Λsv
HS−1n where Λs =
[vHS−1
n v]−1
![Page 162: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/162.jpg)
MVDR BeamformerIn the MVDR beamformer, we want to minimize
E[|Yn(ω)|2] subject to WH(ω)v(ω : ks) = 1
Using the method of Lagrange multipliers, we definethe following cost function to be minimized
F = WH(ω)Sn(ω)Wω
+ λ[WH(ω)v(ω : ks)− 1
]+ λ∗
[vH(ω : ks)W (ω)− 1
]
...and the result (suppressing ω and ks) is
WHmvdr = Λsv
HS−1n where Λs =
[vHS−1
n v]−1
This result is referred to as MVDR or CaponBeamformer.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 56/115
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Constrained Optimal Filtering
The gradient of ξ with respect to w (real case):
∇wξ =
∂ξ
∂w0∂ξ
∂w1...∂ξ
∂wN−1
![Page 164: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/164.jpg)
Constrained Optimal Filtering
The gradient of ξ with respect to w (real case):
∇wξ =
∂ξ
∂w0∂ξ
∂w1...∂ξ
∂wN−1
From the definition above, it is easy to show that:∇w(bTw) = ∇w(wTb) = b
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Constrained Optimal Filtering
The gradient of ξ with respect to w (real case):
∇wξ =
∂ξ
∂w0∂ξ
∂w1...∂ξ
∂wN−1
From the definition above, it is easy to show that:∇w(bTw) = ∇w(wTb) = b
Also ∇w(wTRw) = RTw +Rw
![Page 166: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/166.jpg)
Constrained Optimal Filtering
The gradient of ξ with respect to w (real case):
∇wξ =
∂ξ
∂w0∂ξ
∂w1...∂ξ
∂wN−1
From the definition above, it is easy to show that:∇w(bTw) = ∇w(wTb) = b
Also ∇w(wTRw) = RTw +Rw
which, when R is symmetric, leads to∇w(wTRw) = 2Rw
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 57/115
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Constrained Optimal Filtering
We now assume the complex case w = a+ jb.
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Constrained Optimal Filtering
We now assume the complex case w = a+ jb.
The gradient becomes ∇wξ =
∂ξ
∂a0+ j ∂ξ
∂b0∂ξ
∂a1+ j ∂ξ
∂b1...
∂ξ
∂aN−1+ j ∂ξ
∂bN−1
![Page 169: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/169.jpg)
Constrained Optimal Filtering
We now assume the complex case w = a+ jb.
The gradient becomes ∇wξ =
∂ξ
∂a0+ j ∂ξ
∂b0∂ξ
∂a1+ j ∂ξ
∂b1...
∂ξ
∂aN−1+ j ∂ξ
∂bN−1
· · · which corresponds to ∇wξ = ∇aξ + j∇bξ
![Page 170: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/170.jpg)
Constrained Optimal Filtering
We now assume the complex case w = a+ jb.
The gradient becomes ∇wξ =
∂ξ
∂a0+ j ∂ξ
∂b0∂ξ
∂a1+ j ∂ξ
∂b1...
∂ξ
∂aN−1+ j ∂ξ
∂bN−1
· · · which corresponds to ∇wξ = ∇aξ + j∇bξ
Let us define the derivative ∂∂w
(with respect to w):
∂∂w
= 12
∂∂a0
− j ∂∂b0
∂∂a1
− j ∂∂b1
...∂
∂aN−1− j ∂
∂bN−1
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 58/115
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Constrained Optimal Filtering
The conjugate derivative with respect to w is
∂∂w∗
= 12
∂∂a0
+ j ∂∂b0
∂∂a1
+ j ∂∂b1
...∂
∂aN−1+ j ∂
∂bN−1
![Page 172: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/172.jpg)
Constrained Optimal Filtering
The conjugate derivative with respect to w is
∂∂w∗
= 12
∂∂a0
+ j ∂∂b0
∂∂a1
+ j ∂∂b1
...∂
∂aN−1+ j ∂
∂bN−1
Therefore, ∇wξ = ∇aξ + j∇bξ is equivalent to 2 ∂ξ
∂w∗.
![Page 173: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/173.jpg)
Constrained Optimal Filtering
The conjugate derivative with respect to w is
∂∂w∗
= 12
∂∂a0
+ j ∂∂b0
∂∂a1
+ j ∂∂b1
...∂
∂aN−1+ j ∂
∂bN−1
Therefore, ∇wξ = ∇aξ + j∇bξ is equivalent to 2 ∂ξ
∂w∗.
The complex gradient may be slightly tricky ifcompared to the simple real gradient. For this reason,we exemplify the use of the complex gradient bycalculating ∇wE[|e(k)|2].
![Page 174: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/174.jpg)
Constrained Optimal Filtering
The conjugate derivative with respect to w is
∂∂w∗
= 12
∂∂a0
+ j ∂∂b0
∂∂a1
+ j ∂∂b1
...∂
∂aN−1+ j ∂
∂bN−1
Therefore, ∇wξ = ∇aξ + j∇bξ is equivalent to 2 ∂ξ
∂w∗.
The complex gradient may be slightly tricky ifcompared to the simple real gradient. For this reason,we exemplify the use of the complex gradient bycalculating ∇wE[|e(k)|2].∇wE[e(k)e∗(k)] = Ee∗(k)[∇we(k)] + e(k)[∇we
∗(k)]
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 59/115
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Constrained Optimal Filtering
We compute each gradient ...
∇we(k) =∇a[d(k)−wHx(k)] + j∇b[d(k)−wHx(k)]
=− x(k)− x(k) = −2x(k)
![Page 176: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/176.jpg)
Constrained Optimal Filtering
We compute each gradient ...
∇we(k) =∇a[d(k)−wHx(k)] + j∇b[d(k)−wHx(k)]
=− x(k)− x(k) = −2x(k)and
∇we∗(k) =∇a[d
∗(k)−wTx∗(k)] + j∇b[d∗(k)−wTx∗(k)]
=− x∗(k) + x∗(k) = 0
![Page 177: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/177.jpg)
Constrained Optimal Filtering
We compute each gradient ...
∇we(k) =∇a[d(k)−wHx(k)] + j∇b[d(k)−wHx(k)]
=− x(k)− x(k) = −2x(k)and
∇we∗(k) =∇a[d
∗(k)−wTx∗(k)] + j∇b[d∗(k)−wTx∗(k)]
=− x∗(k) + x∗(k) = 0
such that the final result is
∇wE[e(k)e∗(k)] =− 2E[e∗(k)x(k)]
=− 2E[x(k)[d(k)−wHx(k)]∗=− 2E[x(k)d∗(k)]
︸ ︷︷ ︸
p
+2E[x(k)xH(k)]︸ ︷︷ ︸
R
w
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 60/115
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Constrained Optimal Filtering
Which results in the Wiener solution w = R−1p.
![Page 179: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/179.jpg)
Constrained Optimal Filtering
Which results in the Wiener solution w = R−1p.
When a set of linear constraints involving thecoefficient vector of an adaptive filter is imposed, theresulting problem (LCAF)—admitting the MSE as theobjective function—can be stated as minimizingE[|e(k)|2] subject to CHw = f .
![Page 180: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/180.jpg)
Constrained Optimal Filtering
Which results in the Wiener solution w = R−1p.
When a set of linear constraints involving thecoefficient vector of an adaptive filter is imposed, theresulting problem (LCAF)—admitting the MSE as theobjective function—can be stated as minimizingE[|e(k)|2] subject to CHw = f .
The output of the processor is y(k) = wHx(k).
![Page 181: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/181.jpg)
Constrained Optimal Filtering
Which results in the Wiener solution w = R−1p.
When a set of linear constraints involving thecoefficient vector of an adaptive filter is imposed, theresulting problem (LCAF)—admitting the MSE as theobjective function—can be stated as minimizingE[|e(k)|2] subject to CHw = f .
The output of the processor is y(k) = wHx(k).
It is worth mentioning that the most general casecorresponds to having a reference signal, d(k). It is,however, usual to have no reference signal as inLinearly-Constrained Minimum-Variance (LCMV)applications. In LCMV, if f = 1, the system is oftenreferred to as Minimum-Variance DistortionlessResponse (MVDR).
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 61/115
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Constrained Optimal Filtering
Using Lagrange multipliers, we form
ξ(k) = E[e(k)e∗(k)] +LTRRe[CHw − f ] +L
TI Im[CHw − f ]
![Page 183: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/183.jpg)
Constrained Optimal Filtering
Using Lagrange multipliers, we form
ξ(k) = E[e(k)e∗(k)] +LTRRe[CHw − f ] +L
TI Im[CHw − f ]
We can also represent the above expression with a complex
L given by LR + jLI such that
ξ(k) = E[e(k)e∗(k)] +Re[LH(CHw − f)]
= E[e(k)e∗(k)] +1
2L
H(CHw − f) +1
2L
T (CTw∗ − f∗)
![Page 184: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/184.jpg)
Constrained Optimal Filtering
Using Lagrange multipliers, we form
ξ(k) = E[e(k)e∗(k)] +LTRRe[CHw − f ] +L
TI Im[CHw − f ]
We can also represent the above expression with a complex
L given by LR + jLI such that
ξ(k) = E[e(k)e∗(k)] +Re[LH(CHw − f)]
= E[e(k)e∗(k)] +1
2L
H(CHw − f) +1
2L
T (CTw∗ − f∗)
Noting that e(k) = d(k)−wHx(k), we compute:
∇wξ(k) = ∇w
E[e(k)e∗(k)] +1
2L
H(CHw − f) +1
2L
T (CTw∗ − f∗)
= E[−2x(k)e∗(k)] + 0+CL
= −2E[x(k)d∗(k)] + 2E[x(k)xH(k)]w +CL
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 62/115
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Constrained Optimal Filtering
By using R = E[x(k)xH(k)] and p = E[d∗(k)x(k)], thegradient is equated to zero and the results can bewritten as (note that stationarity was assumed for theinput and reference signals): −2p+ 2Rw +CL = 0
![Page 186: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/186.jpg)
Constrained Optimal Filtering
By using R = E[x(k)xH(k)] and p = E[d∗(k)x(k)], thegradient is equated to zero and the results can bewritten as (note that stationarity was assumed for theinput and reference signals): −2p+ 2Rw +CL = 0
Which leads to w = 12R−1(2p−CL)
![Page 187: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/187.jpg)
Constrained Optimal Filtering
By using R = E[x(k)xH(k)] and p = E[d∗(k)x(k)], thegradient is equated to zero and the results can bewritten as (note that stationarity was assumed for theinput and reference signals): −2p+ 2Rw +CL = 0
Which leads to w = 12R−1(2p−CL)
If we pre-multiply the previous expression by CH anduse CHw = f , we find L:L = 2(CHR−1C)−1(CHR−1p− f)
![Page 188: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/188.jpg)
Constrained Optimal Filtering
By using R = E[x(k)xH(k)] and p = E[d∗(k)x(k)], thegradient is equated to zero and the results can bewritten as (note that stationarity was assumed for theinput and reference signals): −2p+ 2Rw +CL = 0
Which leads to w = 12R−1(2p−CL)
If we pre-multiply the previous expression by CH anduse CHw = f , we find L:L = 2(CHR−1C)−1(CHR−1p− f)
By replacing L, we obtain the Wiener solution for thelinearly constrained adaptive filter:wopt = R−1p+R−1C(CHR−1C)−1(f −CHR−1p)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 63/115
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Constrained Optimal Filtering
The optimal solution for LCAF:wopt = R−1p+R−1C(CHR−1C)−1(f −CHR−1p)
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Constrained Optimal Filtering
The optimal solution for LCAF:wopt = R−1p+R−1C(CHR−1C)−1(f −CHR−1p)
Note that if d(k) = 0, then p = 0, and we have (LCMV):wopt = R−1C(CHR−1C)−1f
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Constrained Optimal Filtering
The optimal solution for LCAF:wopt = R−1p+R−1C(CHR−1C)−1(f −CHR−1p)
Note that if d(k) = 0, then p = 0, and we have (LCMV):wopt = R−1C(CHR−1C)−1f
Yet with d(k) = 0 but f = 1 (MVDR)wopt = R−1C(CHR−1C)−1
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Constrained Optimal Filtering
The optimal solution for LCAF:wopt = R−1p+R−1C(CHR−1C)−1(f −CHR−1p)
Note that if d(k) = 0, then p = 0, and we have (LCMV):wopt = R−1C(CHR−1C)−1f
Yet with d(k) = 0 but f = 1 (MVDR)wopt = R−1C(CHR−1C)−1
For this case, d(k) = 0, the cost function is termedminimum output energy (MOE) and is given byE[|e(k)|2] = wHRw
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Constrained Optimal Filtering
The optimal solution for LCAF:wopt = R−1p+R−1C(CHR−1C)−1(f −CHR−1p)
Note that if d(k) = 0, then p = 0, and we have (LCMV):wopt = R−1C(CHR−1C)−1f
Yet with d(k) = 0 but f = 1 (MVDR)wopt = R−1C(CHR−1C)−1
For this case, d(k) = 0, the cost function is termedminimum output energy (MOE) and is given byE[|e(k)|2] = wHRw
Also note that in case we do not have constraints (Cand f are nulls), the optimal solution above becomesthe unconstrained Wiener solution R−1p.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 64/115
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The GSC
We start by doing a transformation in the coefficient vector.
Let T = [C B] such that
w = Tw = [C B]
[wU
−wL
]
= CwU −BwL
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The GSC
We start by doing a transformation in the coefficient vector.
Let T = [C B] such that
w = Tw = [C B]
[wU
−wL
]
= CwU −BwL
Matrix B is usually called the Blocking Matrix and werecall that CHw = g such thatCHw = CHCwU −CHBwL = g.
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The GSC
We start by doing a transformation in the coefficient vector.
Let T = [C B] such that
w = Tw = [C B]
[wU
−wL
]
= CwU −BwL
Matrix B is usually called the Blocking Matrix and werecall that CHw = g such thatCHw = CHCwU −CHBwL = g.
If we impose the condition BHC = 0 or, equivalently,CHB = 0, we will have wU = (CHC)−1g.
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The GSC
We start by doing a transformation in the coefficient vector.
Let T = [C B] such that
w = Tw = [C B]
[wU
−wL
]
= CwU −BwL
Matrix B is usually called the Blocking Matrix and werecall that CHw = g such thatCHw = CHCwU −CHBwL = g.
If we impose the condition BHC = 0 or, equivalently,CHB = 0, we will have wU = (CHC)−1g.
wU is fixed and termed the quiescent weight vector;the minimization process will be carried out only in thelower part, also designated wGSC = wL.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 65/115
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The GSC
It is shown below how to split the transformation matrixinto two parts: a fixed path and an adaptive path.
reference signal
input
reference signal
signal
inputsignal
Σ−
+
−
MN
p MN−p
MN−p
p
B−
C
+
−
F
B
Σ
Σ−
1
+1
MNΣ+
− ≡MN
MN
MN
1
T+
− Σw(k)
B
MNp
MN
p
MN−p
1
1
C
Σ+
− Σ− +
(k)GSC
w
(c)
(a) (b)
(d)MN−p
w(k)
wU(k)
w (k)L
wU(k)
w (k)L
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 66/115
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The GSC
This structure (detailed below) was named theGeneralized Sidelobe Canceller (GSC).
+
−
F
B
Σ
HF
Σ
(k)x
− +
y(k)(k)
(k)
=
(k)xH
B x(k)
GSC
GSCx
ye(k)
d(k)
(k−1)GSC
w
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The GSC
This structure (detailed below) was named theGeneralized Sidelobe Canceller (GSC).
+
−
F
B
Σ
HF
Σ
(k)x
− +
y(k)(k)
(k)
=
(k)xH
B x(k)
GSC
GSCx
ye(k)
d(k)
(k−1)GSC
w
It is always possible to have the overall equivalentcoefficient vector which is given by w = F−BwGSC .
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The GSC
This structure (detailed below) was named theGeneralized Sidelobe Canceller (GSC).
+
−
F
B
Σ
HF
Σ
(k)x
− +
y(k)(k)
(k)
=
(k)xH
B x(k)
GSC
GSCx
ye(k)
d(k)
(k−1)GSC
w
It is always possible to have the overall equivalentcoefficient vector which is given by w = F−BwGSC .
If we pre-multiply last equation by BH and isolatewGSC , we find wGSC = −(BHB)−1BHw.
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The GSC
This structure (detailed below) was named theGeneralized Sidelobe Canceller (GSC).
+
−
F
B
Σ
HF
Σ
(k)x
− +
y(k)(k)
(k)
=
(k)xH
B x(k)
GSC
GSCx
ye(k)
d(k)
(k−1)GSC
w
It is always possible to have the overall equivalentcoefficient vector which is given by w = F−BwGSC .
If we pre-multiply last equation by BH and isolatewGSC , we find wGSC = −(BHB)−1BHw.
Knowing that T = [C B] and that THT = I, it followsthat P = I−C(CHC)−1CH = B(BHB)BH .
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 67/115
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The GSC
A simple procedure to find the optimal GSC solutioncomes from the unconstrained Wiener solution appliedto the unconstrained filter: wGSC−OPT = R−1
GSCpGSC
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The GSC
A simple procedure to find the optimal GSC solutioncomes from the unconstrained Wiener solution appliedto the unconstrained filter: wGSC−OPT = R−1
GSCpGSC
From the figure, it is clear that:RGSC = E[xGSCx
HGSC ] = E[BHxxHB] = BHRB
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The GSC
A simple procedure to find the optimal GSC solutioncomes from the unconstrained Wiener solution appliedto the unconstrained filter: wGSC−OPT = R−1
GSCpGSC
From the figure, it is clear that:RGSC = E[xGSCx
HGSC ] = E[BHxxHB] = BHRB
The cross-correlation vector is given as:
pGSC = E[d∗GSCxGSC ]
= E[FHx− d]∗[BHx]= E[−BHd∗x+BHxxHF]
= −BHp+BHRF
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The GSC
A simple procedure to find the optimal GSC solutioncomes from the unconstrained Wiener solution appliedto the unconstrained filter: wGSC−OPT = R−1
GSCpGSC
From the figure, it is clear that:RGSC = E[xGSCx
HGSC ] = E[BHxxHB] = BHRB
The cross-correlation vector is given as:
pGSC = E[d∗GSCxGSC ]
= E[FHx− d]∗[BHx]= E[−BHd∗x+BHxxHF]
= −BHp+BHRF
· · · and wGSC−OPT = (BHRB)−1(−BHp+BHRF)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 68/115
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The GSCA common case is when d(k) = 0:
+
−Σ
HF (k)x
(k) =
(k)xH
B x(k)GSCx
y (k)GSC
GSCe (k)
B
Fd (k) = GSC
(k−1)GSC
w
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The GSCA common case is when d(k) = 0:
+
−Σ
HF (k)x
(k) =
(k)xH
B x(k)GSCx
y (k)GSC
GSCe (k)
B
Fd (k) = GSC
(k−1)GSC
w
We have dropped the negative sign that should existaccording to the notation used. Although we defineeGSC(k) = −e(k), the inversion of the sign in the errorsignal actually results in the same results because theerror function is always based on the absolute value.
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The GSCA common case is when d(k) = 0:
+
−Σ
HF (k)x
(k) =
(k)xH
B x(k)GSCx
y (k)GSC
GSCe (k)
B
Fd (k) = GSC
(k−1)GSC
w
We have dropped the negative sign that should existaccording to the notation used. Although we defineeGSC(k) = −e(k), the inversion of the sign in the errorsignal actually results in the same results because theerror function is always based on the absolute value.
In this case, the optimum filter wOPT is:F−BwGSC−OPT = F−B(BHRB)−1BHRF =
R−1C(CHR−1C)−1f (LCMV solution)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 69/115
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The GSC
Choosing the blocking matrix: B plays an importantrole since its choice determines computationalcomplexity and even robustness against numericalinstability.
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The GSC
Choosing the blocking matrix: B plays an importantrole since its choice determines computationalcomplexity and even robustness against numericalinstability.
Since the only need for B is having its columns forminga basis orthogonal to the constraints, BHC = 0, amyriad of options are possible.
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The GSC
Choosing the blocking matrix: B plays an importantrole since its choice determines computationalcomplexity and even robustness against numericalinstability.
Since the only need for B is having its columns forminga basis orthogonal to the constraints, BHC = 0, amyriad of options are possible.
Let us recall the paper by Griffiths and Jim where theterm GSC was coined; let
CT =
1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 1
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The GSC
Choosing the blocking matrix: B plays an importantrole since its choice determines computationalcomplexity and even robustness against numericalinstability.
Since the only need for B is having its columns forminga basis orthogonal to the constraints, BHC = 0, amyriad of options are possible.
Let us recall the paper by Griffiths and Jim where theterm GSC was coined; let
CT =
1 1 1 1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 1 1 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 1
With simple constraint matrices, simple blockingmatrices satisfying BTC = 0 are possible.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 70/115
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The GSC
For this particular example, the paper presents twopossibilities. The first one (orthogonal) is:
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The GSC
For this particular example, the paper presents twopossibilities. The first one (orthogonal) is:
BT1 =
1 1 −1 −1 0 0 0 0 0 0 0 0
1 −1 −1 1 0 0 0 0 0 0 0 0
1 −1 1 −1 0 0 0 0 0 0 0 0
0 0 0 0 1 1 −1 −1 0 0 0 0
0 0 0 0 1 −1 −1 1 0 0 0 0
0 0 0 0 1 −1 1 −1 0 0 0 0
0 0 0 0 0 0 0 0 1 1 −1 −1
0 0 0 0 0 0 0 0 1 −1 −1 1
0 0 0 0 0 0 0 0 1 −1 1 −1
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 71/115
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The GSC
And the second possibility (non-orthogonal) is:
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The GSC
And the second possibility (non-orthogonal) is:
BT2 =
1 −1 0 0 0 0 0 0 0 0 0 0
0 1 −1 0 0 0 0 0 0 0 0 0
0 0 1 −1 0 0 0 0 0 0 0 0
0 0 0 0 1 −1 0 0 0 0 0 0
0 0 0 0 0 1 −1 0 0 0 0 0
0 0 0 0 0 0 1 −1 0 0 0 0
0 0 0 0 0 0 0 0 1 −1 0 0
0 0 0 0 0 0 0 0 0 1 −1 0
0 0 0 0 0 0 0 0 0 0 1 −1
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 72/115
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The GSC
SVD: the blocking matrix can be produced with thefollowing Matlab command lines,
[U,S,V]=svd(C);B3=U(:,p+1:M * N); % p=N in this case
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The GSC
SVD: the blocking matrix can be produced with thefollowing Matlab command lines,
[U,S,V]=svd(C);B3=U(:,p+1:M * N); % p=N in this case
BT3 is given by:
−0.50 −0.17 −0.17 0.83 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.25 −0.42 0.08 0.08 0.75 −0.25 −0.25 −0.25 0.00 0.00 0.00 0.00
0.25 −0.42 0.08 0.08 −0.25 0.75 −0.25 −0.25 0.00 0.00 0.00 0.00
0.25 −0.42 0.08 0.08 −0.25 −0.25 0.75 −0.20 0.00 0.00 0.00 0.00
0.25 −0.42 0.08 0.08 −0.25 −0.25 −0.25 0.75 0.00 0.00 0.00 0.00
0.25 0.08 −0.42 0.08 0.00 0.00 0.00 0.00 0.75 −0.25 −0.25 −0.25
0.25 0.08 −0.42 0.08 0.00 0.00 0.00 0.00 −0.25 0.75 −0.25 −0.25
0.25 0.08 −0.42 0.08 0.00 0.00 0.00 0.00 −0.25 −0.25 0.75 −0.25
0.25 0.08 −0.42 0.08 0.00 0.00 0.00 0.00 −0.25 −0.25 −0.25 0.75
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 73/115
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The GSC
QRD: the blocking matrix can be produced with thefollowing Matlab command lines,
[Q,R]=qr(C);B4=Q(:,p+1:M * N);
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The GSC
QRD: the blocking matrix can be produced with thefollowing Matlab command lines,
[Q,R]=qr(C);B4=Q(:,p+1:M * N);
B4 was identical to B3 (SVD).
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The GSC
QRD: the blocking matrix can be produced with thefollowing Matlab command lines,
[Q,R]=qr(C);B4=Q(:,p+1:M * N);
B4 was identical to B3 (SVD).
Two other possibilities are: the one presented in[Tseng Griffiths 88] where a decomposition procedureis introduced in order to offer an effectiveimplementation structure and the other one concernedto a narrowband BF implemented with GSC where B iscombined with a wavelet transform [Chu Fang 99].
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The GSC
QRD: the blocking matrix can be produced with thefollowing Matlab command lines,
[Q,R]=qr(C);B4=Q(:,p+1:M * N);
B4 was identical to B3 (SVD).
Two other possibilities are: the one presented in[Tseng Griffiths 88] where a decomposition procedureis introduced in order to offer an effectiveimplementation structure and the other one concernedto a narrowband BF implemented with GSC where B iscombined with a wavelet transform [Chu Fang 99].
Finally, a new efficient linearly constrained adaptivescheme which can also be visualized as a GSCstructure can be found in [Campos&Werner&ApolinárioIEEE-TSP Sept. 2002].
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Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
3. Optimal Beamforming
4. Adaptive Beamforming
5. DoA Estimation with Microphone Arrays
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4. Adaptive Beamforming
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4.1 Introduction
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Introduction
Scope: instead of assuming knowledge about thestatistical properties of the signals, beamformers aredesigned based on statistics gathered online.
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Introduction
Scope: instead of assuming knowledge about thestatistical properties of the signals, beamformers aredesigned based on statistics gathered online.
Different algorithms may be employed for iterativelyapproximating the desired solution.
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Introduction
Scope: instead of assuming knowledge about thestatistical properties of the signals, beamformers aredesigned based on statistics gathered online.
Different algorithms may be employed for iterativelyapproximating the desired solution.
We will briefly cover a small subset of algorithms forconstrained adaptive filters.
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Introduction
Linearly constrained adaptive filters (LCAF) have foundapplication in numerous areas, such as spectrumanalysis, spatial-temporal processing, antenna arrays,interference suppression, among others.
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Introduction
Linearly constrained adaptive filters (LCAF) have foundapplication in numerous areas, such as spectrumanalysis, spatial-temporal processing, antenna arrays,interference suppression, among others.
LCAF algorithms incorporate into the solutionapplication-specific requirements translated into a setof linear equations to be satisfied by the coefficients.
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Introduction
Linearly constrained adaptive filters (LCAF) have foundapplication in numerous areas, such as spectrumanalysis, spatial-temporal processing, antenna arrays,interference suppression, among others.
LCAF algorithms incorporate into the solutionapplication-specific requirements translated into a setof linear equations to be satisfied by the coefficients.
For example, if direction of arrival of the signal ofinterest is known, jammer suppression can takeplace through spatial filtering without the need oftraining signal, or
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Introduction
Linearly constrained adaptive filters (LCAF) have foundapplication in numerous areas, such as spectrumanalysis, spatial-temporal processing, antenna arrays,interference suppression, among others.
LCAF algorithms incorporate into the solutionapplication-specific requirements translated into a setof linear equations to be satisfied by the coefficients.
For example, if direction of arrival of the signal ofinterest is known, jammer suppression can takeplace through spatial filtering without the need oftraining signal, or in systems withconstant-envelope modulation (e.g., M-PSK), aconstant-modulus constraint can mitigate multipathpropagation effects.
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4.2 Constrained FIR Filters
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Broadband Array Beamformer
1
M
2 Σ
ΣAlgorithm:
wmin ξ(k)s.t. C w=fH
1x (k)w (k)
w (k)
w (k)
d(k)
y(k)
+e(k)
-
x (k)2
x (k)M
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Optimal Constrained MSE Filter
We look for
minw
ξ(k) s.t. CHw = f ,
where
ξ(k) = E [|e(k)|2]C is the MN × p constraint matrix
f is the p× 1 gain vector
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Optimal Constrained MSE Filter
The optimal beamformer is
w(k) = R−1p+R−1C(CHR−1C
)−1 (f −CHR−1p
)
where:
R = E[x(k)xH(k)
]and p = E [d∗(k)x(k)]
w(k) =[wT
1 (k) wT2 (k) · · · wT
M(k)]T
x(k) =[xT1 (k) x
T2 (k) · · · xT
M(k)]T
xTi (k) = [xi(k) xi(k − 1) · · · xi(k −N + 1)]
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The Constrained LS Beamformer
In the absence of statistical information, we may choose
minw
[
ξ(k) =k∑
i=0
λk−i|d(i)−wHx(i)|2]
s.t. CHw = f
with λ ∈ (0, 1],
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The Constrained LS Beamformer
In the absence of statistical information, we may choose
minw
[
ξ(k) =k∑
i=0
λk−i|d(i)−wHx(i)|2]
s.t. CHw = f
with λ ∈ (0, 1], which gives, as solution,
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The Constrained LS Beamformer
In the absence of statistical information, we may choose
minw
[
ξ(k) =k∑
i=0
λk−i|d(i)−wHx(i)|2]
s.t. CHw = f
with λ ∈ (0, 1], which gives, as solution,
w(k) = R−1(k)p(k)
+R−1(k)C(CHR−1(k)C
)−1 [f −CHR−1(k)p(k)
],
where
R(k) =∑k
i=0 λk−ix(i)xH(i), and p(k) =
∑k
i=0 λk−id∗(i)x(i).
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The Constrained LMS Algorithm
A (cheaper) alternative cost function is
minw
[ξ(k) = ‖w(k)−w(k − 1)‖2 + µ|e(k)|2
]s.t. CHw(k) = f ,
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The Constrained LMS Algorithm
A (cheaper) alternative cost function is
minw
[ξ(k) = ‖w(k)−w(k − 1)‖2 + µ|e(k)|2
]s.t. CHw(k) = f ,
which gives, as solution,
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The Constrained LMS Algorithm
A (cheaper) alternative cost function is
minw
[ξ(k) = ‖w(k)−w(k − 1)‖2 + µ|e(k)|2
]s.t. CHw(k) = f ,
which gives, as solution,
w(k) = w(k − 1) + µe∗(k)[
I−C(CHC
)−1CH
]
x(k),
where e(k) = d(k)−wH(k − 1)x(k), µ is a positive small
constant called step size.
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The Constrained LMS Algorithm
A (cheaper) alternative cost function is
minw
[ξ(k) = ‖w(k)−w(k − 1)‖2 + µ|e(k)|2
]s.t. CHw(k) = f ,
which gives, as solution,
w(k) = P [w(k − 1) + µe∗(k)x(k)] + F,
where e(k) = d(k)−wH(k − 1)x(k), µ is a positive small
constant called step size, P = C(CHC
)−1CH , and
F = C(CHC
)−1f .
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The Constrained AP Algorithm
We may wish to trade complexity for speed of convergence:
minw
[ξ(k) = ‖w(k)−w(k − 1)‖2
]s.t.
XT (k)w∗(k) = d(k)
CHw(k) = f ,
where
d(k) = [d(k) d(k − 1) · · · d(k − L+ 1)]T
X(k) = [x(k) x(k − 1) · · · x(k − L+ 1)]T
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The Constrained AP Algorithm
We may wish to trade complexity for speed of convergence:
minw
[ξ(k) = ‖w(k)−w(k − 1)‖2
]s.t.
XT (k)w∗(k) = d(k)
CHw(k) = f ,
which gives, as solution,
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The Constrained AP Algorithm
We may wish to trade complexity for speed of convergence:
minw
[ξ(k) = ‖w(k)−w(k − 1)‖2
]s.t.
XT (k)w∗(k) = d(k)
CHw(k) = f ,
which gives, as solution,
w(k) = P [w(k − 1) + µX(k)t(k)] + F
where
e(k) = d(k)−XT (k)w∗(k − 1)
t(k) =[XH(k)PX(k)
]−1e∗(k)
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Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
3. Optimal Beamforming
4. Adaptive Beamforming
5. DoA Estimation with Microphone Arrays
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5. DOA Estimation withMicrophone Arrays
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5.0 Signal Preparation
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It is usual to find a delayed signal represented by amultiplication of the signal with exponential ejωoτ
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It is usual to find a delayed signal represented by amultiplication of the signal with exponential ejωoτ
First thing to note: when this is the case, the signal isnarrow band with a center frequency in ω0 (in thecontinuous-time domain, it corresponds to a carrierfrequency Ω0 = fsω0)
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It is usual to find a delayed signal represented by amultiplication of the signal with exponential ejωoτ
First thing to note: when this is the case, the signal isnarrow band with a center frequency in ω0 (in thecontinuous-time domain, it corresponds to a carrierfrequency Ω0 = fsω0)
But, most importantly, the delay is well representedonly if the signal is also analytic, i. e., having onlynon-negative frequency components.
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It is usual to find a delayed signal represented by amultiplication of the signal with exponential ejωoτ
First thing to note: when this is the case, the signal isnarrow band with a center frequency in ω0 (in thecontinuous-time domain, it corresponds to a carrierfrequency Ω0 = fsω0)
But, most importantly, the delay is well representedonly if the signal is also analytic, i. e., having onlynon-negative frequency components.
An analytic signal, mathematically, can be obtained bymultiplying its Fourier transform by the continuousHeaviside step function:
Xa(ejω) = 2X(ejω)u(ω), u(ω) =
0, ω < 0
1, ω = 0
1, ω > 0
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Let x(n) = s(n) cos(ω0n), s(n) having a maximumfrequency component (ωm) much lower than ω0:
0 0.5 1 1.5 2 2.5 3 3.5 4−0.5
0
0.5
1
1.5Bandbase signal
t=nT in s−4000 −3000 −2000 −1000 0 1000 2000 3000 4000
0
0.2
0.4
0.6
0.8
1
f in Hz
S(f
)0 0.5 1 1.5 2 2.5 3 3.5 4
−1.5
−1
−0.5
0
0.5
1
1.5Carrier signal
t=nT in s−4000 −3000 −2000 −1000 0 1000 2000 3000 4000
0
0.2
0.4
0.6
0.8
1
f in Hz
0 0.5 1 1.5 2 2.5 3 3.5 4−1.5
−1
−0.5
0
0.5
1
1.5Modulated signal
t=nT in s−4000 −3000 −2000 −1000 0 1000 2000 3000 4000
0
0.2
0.4
0.6
0.8
1
f in Hz
X(f
)
s(n
)co
s(ω0n)
x(n
)
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If x(n) = s(n)ejω0n, thenx(n)e−jω0τ = s(n)ejω0(n−τ) ≈ x(n− τ) if τ ≪ 1/ωm
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If x(n) = s(n)ejω0n, thenx(n)e−jω0τ = s(n)ejω0(n−τ) ≈ x(n− τ) if τ ≪ 1/ωm
But if x(n) = s(n)cos(ω0n), then x(n)e−jω0τ 6= x(n− τ)
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If x(n) = s(n)ejω0n, thenx(n)e−jω0τ = s(n)ejω0(n−τ) ≈ x(n− τ) if τ ≪ 1/ωm
But if x(n) = s(n)cos(ω0n), then x(n)e−jω0τ 6= x(n− τ)
We can make
x(n) = s(n)cos(ω0n) =s(n)
2ejω0n
︸ ︷︷ ︸
x+(n)
+s(n)
2e−jω0n
︸ ︷︷ ︸
x−(n)
such that
x(n− τ) ≈ x+e−jω0τ + x−(n)e
+jω0τ = s(n)cos(ω0(n− τ))
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If x(n) = s(n)ejω0n, thenx(n)e−jω0τ = s(n)ejω0(n−τ) ≈ x(n− τ) if τ ≪ 1/ωm
But if x(n) = s(n)cos(ω0n), then x(n)e−jω0τ 6= x(n− τ)
We can make
x(n) = s(n)cos(ω0n) =s(n)
2ejω0n
︸ ︷︷ ︸
x+(n)
+s(n)
2e−jω0n
︸ ︷︷ ︸
x−(n)
such that
x(n− τ) ≈ x+e−jω0τ + x−(n)e
+jω0τ = s(n)cos(ω0(n− τ))
· · · but, how to obtain x+(n) or a scaled copy? Usingthe Hilbert Transform xH(n) = HT x(n) where
XH(ejω) =
jX(ejω),−π < ω < 0
X(ejω), ω = 0
−jX(ejω), 0 < ω < π
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 92/115
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Knowing thatx(n) = x−(n) + x+(n) = F−1 X−(e
jω) +X+(ejω), we
compute y(n) = x(n) + jxH(n)
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Knowing thatx(n) = x−(n) + x+(n) = F−1 X−(e
jω) +X+(ejω), we
compute y(n) = x(n) + jxH(n)
y(n) =
F−1X−(ejω) +X+(e
jω) + j[jX−(ejω)− jX+(e
jω)︸ ︷︷ ︸
XH (ejω)
]
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Knowing thatx(n) = x−(n) + x+(n) = F−1 X−(e
jω) +X+(ejω), we
compute y(n) = x(n) + jxH(n)
y(n) =
F−1X−(ejω) +X+(e
jω) + j[jX−(ejω)− jX+(e
jω)︸ ︷︷ ︸
XH (ejω)
]
= F−1X−(ejω) +X+(e
jω)−X−(ejω) +X+(e
jω)
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Knowing thatx(n) = x−(n) + x+(n) = F−1 X−(e
jω) +X+(ejω), we
compute y(n) = x(n) + jxH(n)
y(n) =
F−1X−(ejω) +X+(e
jω) + j[jX−(ejω)− jX+(e
jω)︸ ︷︷ ︸
XH (ejω)
]
= F−1X−(ejω) +X+(e
jω)−X−(ejω) +X+(e
jω)
Therefore y(n) = 2F−1X+(ejω) = s(n)ejω0n which is
analytic!
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 93/115
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Signal Model
Consider xm(t) the signal from the m-th microphone(prior to the A/D converter) corresponding to audiofrom D sources (directions θ1 to θD) plus noise:xm(t) = s1(t− τm(θ1)) + · · · + sD(t− τm(θD)) + nm(t)
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Signal Model
Consider xm(t) the signal from the m-th microphone(prior to the A/D converter) corresponding to audiofrom D sources (directions θ1 to θD) plus noise:xm(t) = s1(t− τm(θ1)) + · · · + sD(t− τm(θD)) + nm(t)
Assuming τm(θd) = Tτm(θd) in s (τm(θd) in number ofsamples), after the A/D converter and + jHT tomake it an analytic signal, we could writexm(n) = s1(n)e
−jω0τm(θ1)+ · · ·+ sD(n)e−jω0τm(θD)+nm(n)
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Signal Model
Consider xm(t) the signal from the m-th microphone(prior to the A/D converter) corresponding to audiofrom D sources (directions θ1 to θD) plus noise:xm(t) = s1(t− τm(θ1)) + · · · + sD(t− τm(θD)) + nm(t)
Assuming τm(θd) = Tτm(θd) in s (τm(θd) in number ofsamples), after the A/D converter and + jHT tomake it an analytic signal, we could writexm(n) = s1(n)e
−jω0τm(θ1)+ · · ·+ sD(n)e−jω0τm(θD)+nm(n)
For an array with M microphones, we would have:
x(n)︸︷︷︸
M×1
= A︸︷︷︸
M×D
s(n)︸︷︷︸
D×1
+n(n)︸︷︷︸
M×1
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 94/115
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5.1 Signal model
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Assume, initially, we have D narrowband signalscoming from unknown directions:
+
+ X
X
+
x1(t)
xM (t)
x1(n)
xM (n)
h∗
1
h∗
M
y(n) = hHx(n)
j
j
HT
HT
A/D
A/Ds1(t)
sD(t)
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Assume, initially, we have D narrowband signalscoming from unknown directions:
+
+ X
X
+
x1(t)
xM (t)
x1(n)
xM (n)
h∗
1
h∗
M
y(n) = hHx(n)
j
j
HT
HT
A/D
A/Ds1(t)
sD(t)
x(n) =
e−jω0τ1(θ1)s1(n) + · · ·+ e−jω0τ1(θD)sD(n) + n1(n)
...
e−jω0τM (θ1)s1(n) + · · ·+ e−jω0τM (θD)sD(n) + nM (n)
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Assume, initially, we have D narrowband signalscoming from unknown directions:
+
+ X
X
+
x1(t)
xM (t)
x1(n)
xM (n)
h∗
1
h∗
M
y(n) = hHx(n)
j
j
HT
HT
A/D
A/Ds1(t)
sD(t)
x(n) =
e−jω0τ1(θ1)s1(n) + · · ·+ e−jω0τ1(θD)sD(n) + n1(n)
...
e−jω0τM (θ1)s1(n) + · · ·+ e−jω0τM (θD)sD(n) + nM (n)
Such that the output signal can be written as
y(n) = hHx(n) = hH [As(n) + n(n)]
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 96/115
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If we now assume one single signal, s(n), coming fromdirection θ, thenx(n) = s(n)a(θ) + n(n)
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If we now assume one single signal, s(n), coming fromdirection θ, thenx(n) = s(n)a(θ) + n(n)
And the output signal becomesy(n) = hHa(θ)s(n) + hHn(n)
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If we now assume one single signal, s(n), coming fromdirection θ, thenx(n) = s(n)a(θ) + n(n)
And the output signal becomesy(n) = hHa(θ)s(n) + hHn(n)
If we make hHa(θ) = 1, the output signal wouldcorrespond to y(n) = s(n) + hHn(n)
︸ ︷︷ ︸
noise
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If we now assume one single signal, s(n), coming fromdirection θ, thenx(n) = s(n)a(θ) + n(n)
And the output signal becomesy(n) = hHa(θ)s(n) + hHn(n)
If we make hHa(θ) = 1, the output signal wouldcorrespond to y(n) = s(n) + hHn(n)
︸ ︷︷ ︸
noise
Also note that E[|y(n)|2] = hHRxh, Rx = E[x(n)xH(n)]
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 97/115
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5.2 Non-parametric methods: BF (beamforminga.k.a. Delay & Sum) and Capon
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DS DoA
If x(n) were spatially white, i.e. Rx = I, we wouldobtain E[|y(n)|2] = hHh
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DS DoA
If x(n) were spatially white, i.e. Rx = I, we wouldobtain E[|y(n)|2] = hHh
Minimizing E[|y(n)|2] = hHh s.t. hHa(θ) = 1, the result,
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DS DoA
If x(n) were spatially white, i.e. Rx = I, we wouldobtain E[|y(n)|2] = hHh
Minimizing E[|y(n)|2] = hHh s.t. hHa(θ) = 1, the result,after using Lagrange multiplier, taking the gradient,
and equating to zero, is h = a(θ)/M which leads to
E[|y(n)|2] = aH(θ)Rxa(θ)M2
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DS DoA
If x(n) were spatially white, i.e. Rx = I, we wouldobtain E[|y(n)|2] = hHh
Minimizing E[|y(n)|2] = hHh s.t. hHa(θ) = 1, the result,after using Lagrange multiplier, taking the gradient,
and equating to zero, is h = a(θ)/M which leads to
E[|y(n)|2] = aH(θ)Rxa(θ)M2
Omitting factor 1M2 , we estimate the autocorrelation
matrix as Rx = 1N
∑N
n=1 x(n)xH(n) and find the
direction of interest by varying θ and obtaining the
peak in PDS(θ) = aH(θ)Rxa(θ)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 99/115
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CaponIn the method known as Capon, we minimizeE[|y(n)|2] = hHRxh subject to hHa(θ) = 1
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CaponIn the method known as Capon, we minimizeE[|y(n)|2] = hHRxh subject to hHa(θ) = 1
Using Lagrange multiplier, we writeξ = hHRxh+ λ(hHa(θ)− 1), and make ∇hξ = 0 such
that h = R−1x a(θ)
aH(θ)R−1x a(θ)
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CaponIn the method known as Capon, we minimizeE[|y(n)|2] = hHRxh subject to hHa(θ) = 1
Using Lagrange multiplier, we writeξ = hHRxh+ λ(hHa(θ)− 1), and make ∇hξ = 0 such
that h = R−1x a(θ)
aH(θ)R−1x a(θ)
Replacing the above coefficient vector in E[|y(n)|2], weobtain E[|y(n)|2] = 1
aH(θ)R−1x a(θ)
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CaponIn the method known as Capon, we minimizeE[|y(n)|2] = hHRxh subject to hHa(θ) = 1
Using Lagrange multiplier, we writeξ = hHRxh+ λ(hHa(θ)− 1), and make ∇hξ = 0 such
that h = R−1x a(θ)
aH(θ)R−1x a(θ)
Replacing the above coefficient vector in E[|y(n)|2], weobtain E[|y(n)|2] = 1
aH(θ)R−1x a(θ)
Therefore, in the Capon DoA, we estimateRx = 1
N
∑N
n=1 x(n)xH(n) and find the direction of
interest by varying θ and obtaining the peak in
PCAPON (θ) =1
aH(θ)R−1x a(θ)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 100/115
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5.3 Eigenvalue-Based DoA
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MUSIC
Coming back to the previous model of D sources, wewrite x(n) = As(n) + n(n)
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MUSIC
Coming back to the previous model of D sources, wewrite x(n) = As(n) + n(n)
We assume D < M (number of signals lower than thenumber of sensors); this method is known asparametric for we make this assumption
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MUSIC
Coming back to the previous model of D sources, wewrite x(n) = As(n) + n(n)
We assume D < M (number of signals lower than thenumber of sensors); this method is known asparametric for we make this assumption
Also note that A is M ×D, s is D × 1, and n(n) isM × 1
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MUSIC
Coming back to the previous model of D sources, wewrite x(n) = As(n) + n(n)
We assume D < M (number of signals lower than thenumber of sensors); this method is known asparametric for we make this assumption
Also note that A is M ×D, s is D × 1, and n(n) isM × 1
We then write Rx = E[x(n)xH(n)
]= ARsA
H +Rn,this last matrix becoming Rn = σ2
nI when assumingspatially white noise; Rs is the D ×D autocorrelationmatrix of the signal vector, i.e., E
[s(n)sH(n)
]
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 102/115
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MUSICRx = ARsA
H +Rn with D < M implies thatARsA
H is singular (rank D), its determinant is equal tozero and, therefore, det [Rx − σ2
nI] = 0 and σ2n is a
(minimum) eigenvalue with multiplicity M −D
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MUSICRx = ARsA
H +Rn with D < M implies thatARsA
H is singular (rank D), its determinant is equal tozero and, therefore, det [Rx − σ2
nI] = 0 and σ2n is a
(minimum) eigenvalue with multiplicity M −D
Spectral decomposition of matrix Rx: vector em beingan eigenvector of Rx means that Rxem = λmem.Collecting all eigenvectors in matrix E, we may writeRxE = EΛ = [e1 · · · eM ]diag [λ1 · · · λM ]⇒ Rx = EΛEH
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MUSICRx = ARsA
H +Rn with D < M implies thatARsA
H is singular (rank D), its determinant is equal tozero and, therefore, det [Rx − σ2
nI] = 0 and σ2n is a
(minimum) eigenvalue with multiplicity M −D
Spectral decomposition of matrix Rx: vector em beingan eigenvector of Rx means that Rxem = λmem.Collecting all eigenvectors in matrix E, we may writeRxE = EΛ = [e1 · · · eM ]diag [λ1 · · · λM ]⇒ Rx = EΛEH
Dividing matrix E in two parts, the first D columns andthe last N =M −D columns, we have:E = [e1 · · · eD
︸ ︷︷ ︸
ES
eD+1 · · · eM︸ ︷︷ ︸
EN
] = [ES EN ]
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 103/115
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MUSIC
Noting that EEH = I, we can write ESEHS +ENE
HN = I
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MUSIC
Noting that EEH = I, we can write ESEHS +ENE
HN = I
The columns of ES span the D-dimensional signalsubspace while the columns of EN span theN -dimensional noise subspace
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MUSIC
Noting that EEH = I, we can write ESEHS +ENE
HN = I
The columns of ES span the D-dimensional signalsubspace while the columns of EN span theN -dimensional noise subspace
A vector in the signal subspace is a linear combinationof the columns of ES. An example:∑D
d=1 xded = ESx,x = [x1 · · · xD]T
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MUSIC
Noting that EEH = I, we can write ESEHS +ENE
HN = I
The columns of ES span the D-dimensional signalsubspace while the columns of EN span theN -dimensional noise subspace
A vector in the signal subspace is a linear combinationof the columns of ES. An example:∑D
d=1 xded = ESx,x = [x1 · · · xD]T
We can find the distance d from a vector v to the signalsubspace ES by obtaining x that minimizesd = |v −ESx|; the result is d2 = vHENE
HNv
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 104/115
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MUSIC
The squared distance from vector a(θ) to the signalsubspace (spanned by ES) is d2 = aH(θ)ENE
HNa(θ)
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MUSIC
The squared distance from vector a(θ) to the signalsubspace (spanned by ES) is d2 = aH(θ)ENE
HNa(θ)
When θ belongs to θ1 · · · θD, this distance should beclose to zero
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MUSIC
The squared distance from vector a(θ) to the signalsubspace (spanned by ES) is d2 = aH(θ)ENE
HNa(θ)
When θ belongs to θ1 · · · θD, this distance should beclose to zero
Its inverse will present peaks. In algorithm MUSIC, weestimate D from the eigenvalues of Rx; from itseigenvectors, we form ES and EN , and by varying θ,we shall find peaks in the directions of θ1 to θD in
PMUSIC(θ) =1
d2a(θ)
=1
aH(θ)ENEHNa(θ)
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MUSIC
The squared distance from vector a(θ) to the signalsubspace (spanned by ES) is d2 = aH(θ)ENE
HNa(θ)
When θ belongs to θ1 · · · θD, this distance should beclose to zero
Its inverse will present peaks. In algorithm MUSIC, weestimate D from the eigenvalues of Rx; from itseigenvectors, we form ES and EN , and by varying θ,we shall find peaks in the directions of θ1 to θD in
PMUSIC(θ) =1
d2a(θ)
=1
aH(θ)ENEHNa(θ)
If RS is required, we computeRS =
(AHA
)−1AH (Rx − σ2
nI)A(AHA
)−1
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 105/115
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5.4 GCC-Based DoA
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 106/115
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GCCM microphones of an array are inpositions p1 to pM :
ECM 800
x
y
zMic 1 positioned at p1
2
3
4
5
6
M = 7
−u
θ
φ
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GCCM microphones of an array are inpositions p1 to pM :
ECM 800
x
y
zMic 1 positioned at p1
2
3
4
5
6
M = 7
−u
θ
φ
−u: unit vector in thedirection of propagation
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GCCM microphones of an array are inpositions p1 to pM :
ECM 800
x
y
zMic 1 positioned at p1
2
3
4
5
6
M = 7
−u
θ
φ
−u: unit vector in thedirection of propagation
θ: grazing angle(π2
- elevation angle)
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GCCM microphones of an array are inpositions p1 to pM :
ECM 800
x
y
zMic 1 positioned at p1
2
3
4
5
6
M = 7
−u
θ
φ
−u: unit vector in thedirection of propagation
θ: grazing angle(π2
- elevation angle)
φ: horizontal angle(azimuth)
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GCCM microphones of an array are inpositions p1 to pM :
ECM 800
x
y
zMic 1 positioned at p1
2
3
4
5
6
M = 7
−u
θ
φ
−u: unit vector in thedirection of propagation
θ: grazing angle(π2
- elevation angle)
φ: horizontal angle(azimuth)
u =
sin θ cosφ
sin θ sinφ
cos θ
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 107/115
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GCCWe are interested in the TDoAbetween mics m and l
WAVEFRONT
pm
plu dml
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GCCWe are interested in the TDoAbetween mics m and l
WAVEFRONT
pm
plu dml
Note that dml = uT (pm − pl︸ ︷︷ ︸
∆pml
)
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GCCWe are interested in the TDoAbetween mics m and l
WAVEFRONT
pm
plu dml
Note that dml = uT (pm − pl︸ ︷︷ ︸
∆pml
)
TDoA:τml =
dml
vsound= τmlT = τml
fs
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GCCWe are interested in the TDoAbetween mics m and l
WAVEFRONT
pm
plu dml
Note that dml = uT (pm − pl︸ ︷︷ ︸
∆pml
)
TDoA:τml =
dml
vsound= τmlT = τml
fs
τml (in number of samples)is to be obtained fromthe peak of rxmxl
(τ)
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GCCWe are interested in the TDoAbetween mics m and l
WAVEFRONT
pm
plu dml
Note that dml = uT (pm − pl︸ ︷︷ ︸
∆pml
)
TDoA:τml =
dml
vsound= τmlT = τml
fs
τml (in number of samples)is to be obtained fromthe peak of rxmxl
(τ)
rxmxl(τ) = E[xm(n)xl(n− τ)]
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 108/115
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GCCWhen the sound frontwave firsthits microphone m (τml < 0):
0 20 40 60 80 100−5
0
5
0 20 40 60 80 100−5
0
5
−50 0 50−50
0
50
n
n
τ
xm(n
)xl(n
)rxm
xl(τ
)
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GCCWhen the sound frontwave firsthits microphone m (τml < 0):
0 20 40 60 80 100−5
0
5
0 20 40 60 80 100−5
0
5
−50 0 50−50
0
50
n
n
τ
xm(n
)xl(n
)rxm
xl(τ
)
When it first hitsmic l (τml > 0):
0 20 40 60 80 100−5
0
5
0 20 40 60 80 100−5
0
5
−50 0 50−20
0
20
n
n
τ
xm(n
)xl(n
)rxm
xl(τ
)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 109/115
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GCCAn estimate for the correlation can be given as:rxmxl
(τ) =∑∞
−∞ xm(n)xl(n− τ) = xm(τ) ∗ xl(−τ)
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GCCAn estimate for the correlation can be given as:rxmxl
(τ) =∑∞
−∞ xm(n)xl(n− τ) = xm(τ) ∗ xl(−τ)
The cross-power spectrum density (CPSD):Rxmxl
(ejω) = Fxm(τ) ∗ xl(−τ) = Xm(ejω)Xl(e
−jω)
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GCCAn estimate for the correlation can be given as:rxmxl
(τ) =∑∞
−∞ xm(n)xl(n− τ) = xm(τ) ∗ xl(−τ)
The cross-power spectrum density (CPSD):Rxmxl
(ejω) = Fxm(τ) ∗ xl(−τ) = Xm(ejω)Xl(e
−jω)
We may assume the modelxm(n) = s(n) ∗ hm(n) + nm(n) and similarly for xl(n)
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GCCAn estimate for the correlation can be given as:rxmxl
(τ) =∑∞
−∞ xm(n)xl(n− τ) = xm(τ) ∗ xl(−τ)
The cross-power spectrum density (CPSD):Rxmxl
(ejω) = Fxm(τ) ∗ xl(−τ) = Xm(ejω)Xl(e
−jω)
We may assume the modelxm(n) = s(n) ∗ hm(n) + nm(n) and similarly for xl(n)
Hence, considering very small additive error and realsequences, we findRxmxl
(ejω) ≈ |S(ejω)|2Hm(ejω)H∗
l (ejω) and
rxmxl(τ) ≈ 1
2π
∫ π
−πHm(e
jω)H∗l (e
jω)Rs(ejω)ejωτdω
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GCCAn estimate for the correlation can be given as:rxmxl
(τ) =∑∞
−∞ xm(n)xl(n− τ) = xm(τ) ∗ xl(−τ)
The cross-power spectrum density (CPSD):Rxmxl
(ejω) = Fxm(τ) ∗ xl(−τ) = Xm(ejω)Xl(e
−jω)
We may assume the modelxm(n) = s(n) ∗ hm(n) + nm(n) and similarly for xl(n)
Hence, considering very small additive error and realsequences, we findRxmxl
(ejω) ≈ |S(ejω)|2Hm(ejω)H∗
l (ejω) and
rxmxl(τ) ≈ 1
2π
∫ π
−πHm(e
jω)H∗l (e
jω)Rs(ejω)ejωτdω
Which motivates the GCC:
rGxmxl(τ) =
1
2π
∫ π
−π
ψ(ω)Rxmxl(ejω)ejωτdω
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 110/115
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GCCTypes of ψ(ω)
Classical cross-correlation:ψ(ω) = 1
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GCCTypes of ψ(ω)
Classical cross-correlation:ψ(ω) = 1
Maximum Likelihood (ML):
ψ(ω) = |Xm(ejω)|||Xl(ejω)|
Rnn(ejω)Rxm (ejω)+Rnl
(ejω)Rxl(ejω)
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GCCTypes of ψ(ω)
Classical cross-correlation:ψ(ω) = 1
Maximum Likelihood (ML):
ψ(ω) = |Xm(ejω)|||Xl(ejω)|
Rnn(ejω)Rxm (ejω)+Rnl
(ejω)Rxl(ejω)
Rxm(ejω) = |Xm(e
jω)|2
Rxl(ejω) = |Xl(e
jω)|2
Rnm(ejω) = |Nm(e
jω)|2 (estimated during silence interval)
Rnl(ejω) = |Nl(e
jω)|2 (estimated during silence interval)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 111/115
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GCCPHAT (Phase Transform):ψ(ω) = 1
|Rxmxl(ejω)
|
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GCCPHAT (Phase Transform):ψ(ω) = 1
|Rxmxl(ejω)
|
Replacing this function in the expression of rGxmxl(τ):
rPHATxmxl
(τ) = 12π
∫ π
−π
Rxmxl(ejω)
|Rxmxl(ejω)|
ejωτdω in which,
after making Rxmxl(ejω) = |S(ejω)|2Hm(e
jω)H∗l (e
jω),we have rPHAT
xmxl(τ) = 1
2π
∫ π
−πej(∢Hm−∢Hl+ωπ)dω
![Page 323: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/323.jpg)
GCCPHAT (Phase Transform):ψ(ω) = 1
|Rxmxl(ejω)
|
Replacing this function in the expression of rGxmxl(τ):
rPHATxmxl
(τ) = 12π
∫ π
−π
Rxmxl(ejω)
|Rxmxl(ejω)|
ejωτdω in which,
after making Rxmxl(ejω) = |S(ejω)|2Hm(e
jω)H∗l (e
jω),we have rPHAT
xmxl(τ) = 1
2π
∫ π
−πej(∢Hm−∢Hl+ωπ)dω
For the PHAT, in case of havinghm(n) = αmδ(n) and hl(n) = αlδ(n−∆τ),the cross-correlation would berPHATxmxl
(τ) = δ(τ +∆τ) ⇒ peak in τml = −∆τ
(a perfect indication of a temporal delay!)
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 112/115
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LS solutionAssuming we have all possible(M(M − 1)/2) delays τml, we want angles φ and θ
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LS solutionAssuming we have all possible(M(M − 1)/2) delays τml, we want angles φ and θ
We define a cost function:
ξ =(τ12 −∆pT
12u)2
+ · · · +(
τ(M−1)M −∆pT(M−1)Mu
)2
with τml = τml/fs and ∆pml = (pm − pl)/vsound
![Page 326: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/326.jpg)
LS solutionAssuming we have all possible(M(M − 1)/2) delays τml, we want angles φ and θ
We define a cost function:
ξ =(τ12 −∆pT
12u)2
+ · · · +(
τ(M−1)M −∆pT(M−1)Mu
)2
with τml = τml/fs and ∆pml = (pm − pl)/vsound
We then find u that minimizes ξ by making ∇uξ = 0:
Au = b
where A = ∆p12∆pT12 + · · · +∆p(M−1)M∆pT
(M−1)M
and b = τ12∆p12 + · · · + τ(M−1)M∆p(M−1)M
![Page 327: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/327.jpg)
LS solutionAssuming we have all possible(M(M − 1)/2) delays τml, we want angles φ and θ
We define a cost function:
ξ =(τ12 −∆pT
12u)2
+ · · · +(
τ(M−1)M −∆pT(M−1)Mu
)2
with τml = τml/fs and ∆pml = (pm − pl)/vsound
We then find u that minimizes ξ by making ∇uξ = 0:
Au = b
where A = ∆p12∆pT12 + · · · +∆p(M−1)M∆pT
(M−1)M
and b = τ12∆p12 + · · · + τ(M−1)M∆p(M−1)M
And this unit vector is given as u =
uxuyuz
= A−1b
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 113/115
![Page 328: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/328.jpg)
Azimuth and elevation
Knowing u and also the fact that it corresponds to
sin θ cosφ
sin θ sinφ
cos θ
, · · ·
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Azimuth and elevation
Knowing u and also the fact that it corresponds to
sin θ cosφ
sin θ sinφ
cos θ
, · · ·
· · · we compute the azimuth:
φ = arctanuyux
![Page 330: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/330.jpg)
Azimuth and elevation
Knowing u and also the fact that it corresponds to
sin θ cosφ
sin θ sinφ
cos θ
, · · ·
· · · we compute the azimuth:
φ = arctanuyux
And the elevation:
elevation = 90 − θ = 90 − arccosuz
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 114/115
![Page 331: Microphone-Array Signal Processingaquarius.ime.eb.br/~apolin/MicArray/sbrt_11_minicurso_ALL.pdf · Microphone-Array Signal Processing Jose A. Apolin´ ario Jr. and Marcello L. R](https://reader036.vdocuments.us/reader036/viewer/2022062413/5b15206c7f8b9a4e2c8da4ea/html5/thumbnails/331.jpg)
Last slide ,
Thank you!
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 115/115