microphone-array signal processing
TRANSCRIPT
![Page 1: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/1.jpg)
Microphone-Array Signal Processing
Jose A. Apolinario Jr. and Marcello L. R. de Campos
{apolin},{mcampos}@ieee.org
IME − Lab. Processamento Digital de Sinais
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 1/19
![Page 2: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/2.jpg)
Outline
![Page 3: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/3.jpg)
Outline
1. Introduction and Fundamentals
![Page 4: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/4.jpg)
Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
![Page 5: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/5.jpg)
Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
3. Optimal Beamforming
![Page 6: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/6.jpg)
Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
3. Optimal Beamforming
4. Adaptive Beamforming
![Page 7: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/7.jpg)
Outline
1. Introduction and Fundamentals
2. Sensor Arrays and Spatial Filtering
3. Optimal Beamforming
4. Adaptive Beamforming
5. DoA Estimation with Microphone Arrays
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 2/19
![Page 8: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/8.jpg)
1. Introduction andFundamentals
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 3/19
![Page 9: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/9.jpg)
General concepts
In this course, signals and noise...
![Page 10: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/10.jpg)
General concepts
In this course, signals and noise...
have spatial dependence
![Page 11: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/11.jpg)
General concepts
In this course, signals and noise...
have spatial dependence
must be characterized as space-time processes
![Page 12: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/12.jpg)
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.
![Page 13: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/13.jpg)
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
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 4/19
![Page 14: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/14.jpg)
1.2 Signals in Space and Time
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 5/19
![Page 15: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/15.jpg)
Defining operators
Let ∇x(·) and ∇2x(·) be the gradient and Laplacian
operators, i.e.,
![Page 16: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/16.jpg)
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
![Page 17: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/17.jpg)
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.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 6/19
![Page 18: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/18.jpg)
Wave equation
From Maxwell’s equations,
∇2−→E =1
c2∂2−→E
∂t2
![Page 19: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/19.jpg)
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.
![Page 20: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/20.jpg)
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
![Page 21: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/21.jpg)
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.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 7/19
![Page 22: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/22.jpg)
Monochromatic plane wave
Now assume s(x, t) has a complex exponential form,
![Page 23: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/23.jpg)
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.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 8/19
![Page 24: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/24.jpg)
Monochromatic plane wave
Substituting the complex exponential form of s(x, t) into thewave equation, we have
![Page 25: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/25.jpg)
Monochromatic plane wave
Substituting the complex exponential form of s(x, t) into thewave equation, we have
k2xs(x, t) + k2
ys(x, t) + k2zs(x, t) =
1
c2ω2s(x, t)
![Page 26: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/26.jpg)
Monochromatic plane wave
Substituting the complex exponential form of s(x, t) into thewave equation, we have
k2xs(x, t) + k2
ys(x, t) + k2zs(x, t) =
1
c2ω2s(x, t)
or, after canceling s(x, t),
![Page 27: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/27.jpg)
Monochromatic plane wave
Substituting the complex exponential form of s(x, t) into thewave equation, we have
k2xs(x, t) + k2
ys(x, t) + k2zs(x, t) =
1
c2ω2s(x, t)
or, after canceling s(x, t),
k2x + k2
y + k2z =
1
c2ω2
![Page 28: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/28.jpg)
Monochromatic plane wave
Substituting the complex exponential form of s(x, t) into thewave equation, we have
k2xs(x, t) + k2
ys(x, t) + k2zs(x, t) =
1
c2ω2s(x, t)
or, after canceling s(x, t),
k2x + k2
y + k2z =
1
c2ω2
constraints to be satisfiedby the parametersof the scalar field
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 9/19
![Page 29: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/29.jpg)
Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
![Page 30: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/30.jpg)
Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
![Page 31: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/31.jpg)
Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
plane
![Page 32: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/32.jpg)
Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
plane
![Page 33: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/33.jpg)
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:
![Page 34: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/34.jpg)
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
![Page 35: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/35.jpg)
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
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 10/19
![Page 36: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/36.jpg)
Monochromatic plane wave
From the constraints imposed by the complex exponentialform, s(x, t) = Aej(ωt−kxx−kyy−kzz) is
monochromatic
plane
![Page 37: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/37.jpg)
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
![Page 38: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/38.jpg)
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.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 11/19
![Page 39: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/39.jpg)
Monochromatic plane wave
Defining the wavenumber vector k as
k = [kx ky kz]T
![Page 40: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/40.jpg)
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)
![Page 41: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/41.jpg)
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
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 12/19
![Page 42: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/42.jpg)
Monochromatic plane wave
As the plane wave propagates, it advances a distance δxin δt seconds.
![Page 43: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/43.jpg)
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)]
![Page 44: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/44.jpg)
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
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 13/19
![Page 45: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/45.jpg)
Monochromatic plane wave
Naturally the plane wave propagates in the direction of thewavenumber vector, i.e.,
![Page 46: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/46.jpg)
Monochromatic plane wave
Naturally the plane wave propagates in the direction of thewavenumber vector, i.e.,
k and δx point in the same direction.
![Page 47: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/47.jpg)
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‖
![Page 48: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/48.jpg)
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‖
![Page 49: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/49.jpg)
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,
![Page 50: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/50.jpg)
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‖
![Page 51: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/51.jpg)
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
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 14/19
![Page 52: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/52.jpg)
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.
![Page 53: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/53.jpg)
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‖
![Page 54: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/54.jpg)
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.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 15/19
![Page 55: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/55.jpg)
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.
![Page 56: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/56.jpg)
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.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 16/19
![Page 57: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/57.jpg)
Periodic propagating periodic waves
Any arbitrary periodic waveform s(x, t) = s(t−αTx) with
fundamental period ω0 can be represented as a sum:
s(x, t) = s(t−αTx) =
∞∑
n=−∞
Snejnω0(t−α
Tx)
![Page 58: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/58.jpg)
Periodic propagating periodic waves
Any arbitrary periodic waveform s(x, t) = s(t−αTx) with
fundamental period ω0 can be represented as a sum:
s(x, t) = s(t−αTx) =
∞∑
n=−∞
Snejnω0(t−α
Tx)
The coefficients are given by
Sn =1
T
∫ T
0
s(u)e−jnω0udu
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 17/19
![Page 59: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/59.jpg)
Periodic propagating periodic waves
Based on the previous derivations, we observe that:
![Page 60: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/60.jpg)
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.
![Page 61: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/61.jpg)
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.
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 18/19
![Page 62: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/62.jpg)
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
![Page 63: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/63.jpg)
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
![Page 64: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/64.jpg)
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
![Page 65: Microphone-Array Signal Processing](https://reader031.vdocument.in/reader031/viewer/2022021207/62061d5a8c2f7b173004b65a/html5/thumbnails/65.jpg)
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...
Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 19/19