A Microphone Array Tutorial

5/25/2018 A Microphone Array Tutorial

1/22

IntroductionFundamentals

Analysis of Directivity PatternBeamforming

A Microphone Array Tutorial

Iain McCowan

August 20, 2004

Iain McCowan Microphone Array Tutorial 1
http://find/http://goback/


2/22



What is a Microphone Array?

What is a Microphone Array?

A microphone array is anarray of microphones:

Multiple microphones combined to act as a single device.

A microphone array can be used to:Discriminatebetween sounds based on direction.

e.g. Input,Speaker 1,Speaker 2.Locatesound sources.

A microphone array provideshands-free/distantacquisition.Less constraining on users.Can be used for surveillance.

http://localhost/Users/imccowan/Downloads/s2_in.wavhttp://localhost/Users/imccowan/Downloads/s2_sp1.wavhttp://localhost/Users/imccowan/Downloads/s2_sp2.wavhttp://localhost/Users/imccowan/Downloads/s2_sp2.wavhttp://localhost/Users/imccowan/Downloads/s2_sp1.wavhttp://localhost/Users/imccowan/Downloads/s2_in.wavhttp://find/http://goback/


3/22



Our ModelSound PropagationContinuous AperturesAperture DirectivityDiscrete Apertures

Our Model

The Wave Equation

2s(t, r) =

1

c22

t2s(t, r)

where:

2 is the Laplacian operator(For Cartesian coordinates, 2f =

2fx2

+ 2f

y2+

2fz2

)

cis the speed of propagation, which depends on the type andtemperature of the fluid.

r is the position vector, r=

x y zT

.

sis the amplitude of the wave (e.g. sound pressure level).

http://find/


4/22




Sound Propagation

Sound propagates through a fluid (e.g. air) as a longitudinalpressure wave, with speed c 340ms1 in air at 20oC.

x

y

z

k

Plane Waves

For a plane wave, the solution to the waveequation takes the form:

s(f, r) =s(f)ejkr

where:

k= 2fc

sin cos sin sin cos

(,) is the direction of propagation, and ris a position vector relative to the sound

source location.Iain McCowan Microphone Array Tutorial 4

O M d l
http://find/


5/22




Continuous Apertures

Anapertureis a spatial region that transmits or receivespropagating waves,

e.g. an antenna for EM waves, a hole in an opaque screenfor optics, a microphone for acoustics.

The aperturesensitivity function w(f, r) gives the response asa function of position on the aperture.

Sources(f)

s(f)e-jk r

w(f,r)

.

r

k

Aperture


O M d l
http://find/


6/22




Aperture Directivity

The received signal at a given point on the aperture is:

s(f)w(f, r)ejkr

Aperture ResponseThe total response of the aperture to signal s(f) is thus:

D(f, k) =

V

w(f, r)ejkrdr

The aperture response is also known as thedirectivityfunction, as it gives the response as a function of the directionof arrivalof the plane wave (recalling that k= g(f, ,)).


Our Model
http://goforward/http://find/


7/22




Fourier Transform Relationship

From Time to Frequency Domain

The Fourier Transform operationtransforms domain from t :

X() =

x(t)ejtdt,

commonly denoted as:

X() =Ft{x(t)}.

note = 2f


Our Model
http://find/


8/22




Fourier Transform Relationship

From Time to Frequency Domain

The Fourier Transform operationtransforms domain from t :

X() =

x(t)ejtdt,

commonly denoted as:

X() =Ft{x(t)}.

note = 2f

From Sensitivity to Directivity

At a fixed frequency, thedirectivity is:

D(k) =V

w(r)ejkrdr

which can be denoted:

D(k) =Fr{w(r)}.


Our Model
http://find/


9/22




Directivity of a Linear Aperture

For a linear aperture r=xof length L, having uniformsensitivity,w(f, r) = rect(x/L) D(f, r) =L sinc(kxL).

s(f)

w(f,r)

L

0

1

02/L3/L /L/L 2/L 3/L

|D(f,k)|

-L/2 L/2

w(f,r)

L

x

xk


I d iOur Model
http://find/


10/22




Directivity of a Discrete Linear Aperture

This directivity can be approximated by a discrete aperture,which spatially samples the continuous aperture.

s(f)

w (f,r )

Ln n

0

1

02/L3/L /L/L 2/L 3/L

|D(f,k)|

-L/2 L/2

w (f,r )

L

x

xk

n n

n


I t d tiMicrophone Array Directivity Pattern
http://find/


11/22



p y yVarying the Number of MicrophonesVarying the Length of the ArrayVariation with FrequencyAliasing and Symmetry

Microphone Array Directivity Pattern

A microphone array is adiscrete receiving aperture.

A plot of the directivity function over different angles ofarrival is known as thedirectivity pattern.

Directivity of Continuous Aperture

D(f, k) =

V

w(f, r)ejkrdr


IntroductionMicrophone Array Directivity Pattern
http://find/


12/22



p y yVarying the Number of MicrophonesVarying the Length of the ArrayVariation with FrequencyAliasing and Symmetry




Directivity of Linear Array

D(f, k) =N1n=0

wn(f)ejkrn ,

where thereareNmicrophones andrnis the location of then

th microphone.


http://find/


13/22



Varying the Number of MicrophonesVarying the Length of the ArrayVariation with FrequencyAliasing and Symmetry




Horizontal Directivity of Uniform Linear Array

D(f,) =N1n=0

wn(f)ej2f

c ndcos ,

where there are N microphones, d is the uniforminter-element spacing, and =/2.


http://find/


14/22




Varying the Number of Microphones

Varying the number of sensors, N, for a given array lengthdecreases the sidelobe level (f=1 kHz,L=0.5 m,wn(f) =

1N

):

Directivity Pattern for Varying N

0 20 40 60 80 100 120 140 160 180

N = 3

N = 5

N = 10

|D(f,)|


Introduction Microphone Array Directivity PatternV i h N b f Mi h


15/22




Varying the Length of the Array

Varying the array length, L=Nd, for fixed Ndecreases themain lobe width (f=1 kHz,N=5,wn(f) =

1N

):

Directivity Pattern for Varying L

0 20 40 60 80 100 120 140 160 180

d = 0.1 m

d = 0.2 m

d = 0.15 m

|D(f,)|


Introduction Microphone Array Directivity PatternV i th N b f Mi h
http://find/


16/22

FundamentalsAnalysis of Directivity Pattern

Beamforming


Variation with Frequency

Varying the frequency of interest 400Hz f 3000Hz(N=5,d=0.1 m,wn(f) =

1N

):

Directivity Pattern for Varying f

0 20 40 60 80 100 120 140 160 180 0

500

1000

1500

2000

2500

3000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f

|D(f,)|


Introduction Microphone Array Directivity PatternVarying the Number of Microphones
http://find/


17/22


Beamforming


Aliasing and Symmetry

Spatial Aliasing: Analogous to the Nyquist frequency in temporalsampling, we have a restriction on minimum spatialsampling rate. Linear arrays require inter-element spacingd< min2 to avoid copies of the main lobe appearing in thedirectivity pattern, where min is the smallest wavelength

of interest (corresponding to highest frequency).Symmetry of Directivity Pattern: For a linear array, the directivity

pattern is symmetrical about the array axis.

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0


IntroductionDefinition
http://find/


18/22


Beamforming

DefinitionDelay-Sum BeamformerBeyond the Delay-Sum Beamformer

Beamforming

Horizontal Directivity of Uniform Linear Array

D(f,) =

N1n=0

wn(f)ej2f

c ndcos ,

The term wn(f) represents a filter applied to microphone n.

The analysis so far has assumed uniform, frequency invariant,

microphone filters wn(f) = 1N.

In general, we can design filters to give a desiredsteeringandshapingof the directivity pattern.

This is referred to as microphone array beamforming.


IntroductionF d l

Definition
http://find/


19/22


Beamforming


Beamforming

Microphone Array Beamformer Output

y(f) =

N1n=0

wn(f)sn(f)

The term wn(f) represents a filter applied to microphone n.

The analysis so far has assumed uniform, frequency invariant,

microphone filters wn(f) = 1N.

In general, we can design filters to give a desiredsteeringandshapingof the directivity pattern.

This is referred to as microphone array beamforming.


IntroductionF d t l

Definition
http://find/


20/22


Beamforming


Delay-Sum Beamformer

For the uniform linear array, using a simple time-delay filter:

wn(f) = 1

Nej

2fc ndcos s

will lead to a horizontal directivity function:

D(f,) = 1

N

N1n=0

ej2fc nd(coscoss),

steering the main lobe of the directivity pattern to the sourcedirection s.

This is the well-knowndelay-sum beamformer.


IntroductionF ndamentals

Definition


21/22


Beamforming

Delay-Sum BeamformerBeyond the Delay-Sum Beamformer

Beyond the Delay-Sum Beamformer

Delay-sum is the simplest beamformer: it just ensures that theresponse maximum occurs for a given direction.

Many more sophisticated beamforming techniques exist, whichdiffer in the design criteria for the beamforming filters wn(f).

Examples include:

thesuperdirectivebeamformer, which maximises the gain inthe desired direction while minimising average gain over all

other directions.adaptivebeamformers, which dynamically update filters tominimise the power from localised noise sources.



Definition


22/22


Beamforming

Delay-Sum BeamformerBeyond the Delay-Sum Beamformer

Delay-Sum vs Superdirective Beamforming

For a circular array (N=8, radius = 10 cm)

Delay-sum Beamformer

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

250 Hz

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

500 Hz

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

1000 Hz

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

2000 Hz

Superdirective Beamformer

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

250 Hz

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

500 Hz

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

1000 Hz

0.5

1

30

210

60

240

90

270

120

300

150

330

180 0

2000 Hz


Documents

A Microphone Array Tutorial