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Microphone-Array Signal Processing Jos ´ e A. Apolin ´ ario Jr. and Marcello L. R. de Campos {apolin},{mcampos}@ieee.org IME - Lab. Processamento Digital de Sinais Microphone-Array Signal Processing, c Apolin ´ arioi & Campos – p. 1/19

Microphone-Array Signal Processing

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Page 1: Microphone-Array Signal Processing

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

Outline

Page 3: Microphone-Array Signal Processing

Outline

1. Introduction and Fundamentals

Page 4: Microphone-Array Signal Processing

Outline

1. Introduction and Fundamentals

2. Sensor Arrays and Spatial Filtering

Page 5: Microphone-Array Signal Processing

Outline

1. Introduction and Fundamentals

2. Sensor Arrays and Spatial Filtering

3. Optimal Beamforming

Page 6: Microphone-Array Signal Processing

Outline

1. Introduction and Fundamentals

2. Sensor Arrays and Spatial Filtering

3. Optimal Beamforming

4. Adaptive Beamforming

Page 7: Microphone-Array Signal Processing

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

1. Introduction andFundamentals

Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 3/19

Page 9: Microphone-Array Signal Processing

General concepts

In this course, signals and noise...

Page 10: Microphone-Array Signal Processing

General concepts

In this course, signals and noise...

have spatial dependence

Page 11: Microphone-Array Signal Processing

General concepts

In this course, signals and noise...

have spatial dependence

must be characterized as space-time processes

Page 12: Microphone-Array Signal Processing

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

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

1.2 Signals in Space and Time

Microphone-Array Signal Processing, c©Apolinarioi & Campos – p. 5/19

Page 15: Microphone-Array Signal Processing

Defining operators

Let ∇x(·) and ∇2x(·) be the gradient and Laplacian

operators, i.e.,

Page 16: Microphone-Array Signal Processing

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

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

Wave equation

From Maxwell’s equations,

∇2−→E =1

c2∂2−→E

∂t2

Page 19: Microphone-Array Signal Processing

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

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

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

Monochromatic plane wave

Now assume s(x, t) has a complex exponential form,

Page 23: Microphone-Array Signal Processing

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

Monochromatic plane wave

Substituting the complex exponential form of s(x, t) into thewave equation, we have

Page 25: Microphone-Array Signal Processing

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Monochromatic plane wave

Defining the wavenumber vector k as

k = [kx ky kz]T

Page 40: Microphone-Array Signal Processing

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

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

Monochromatic plane wave

As the plane wave propagates, it advances a distance δxin δt seconds.

Page 43: Microphone-Array Signal Processing

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

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

Monochromatic plane wave

Naturally the plane wave propagates in the direction of thewavenumber vector, i.e.,

Page 46: Microphone-Array Signal Processing

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

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

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

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

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

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

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

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 =

‖k‖

Page 54: Microphone-Array Signal Processing

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 =

‖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

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

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

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

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

Periodic propagating periodic waves

Based on the previous derivations, we observe that:

Page 60: Microphone-Array Signal Processing

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

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

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

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

−∞

S(ω)ejω(t−αTx)dω

Page 64: Microphone-Array Signal Processing

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

−∞

S(ω)ejω(t−αTx)dω

where

S(ω) =

−∞

s(u)e−jωudu

Page 65: Microphone-Array Signal Processing

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

−∞

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