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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
A Microphone Array Tutorial
Iain McCowan
August 20, 2004
Iain McCowan Microphone Array Tutorial 1
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
What is a Microphone Array?
What is a Microphone Array?
A microphone array is an array of microphones:
Multiple microphones combined to act as a single device.
A microphone array can be used to:Discriminate between sounds based on direction.
e.g. Input, Speaker 1, Speaker 2.Locate sound sources.
A microphone array provides hands-free/distant acquisition.Less constraining on users.Can be used for surveillance.
Iain McCowan Microphone Array Tutorial 2
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/ -
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
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
+ 2fy2
+ 2fz2
)
c is the speed of propagation, which depends on the type andtemperature of the fluid.
r is the position vector, r =
x y zT
.
s is the amplitude of the wave (e.g. sound pressure level).
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
Our ModelSound PropagationContinuous AperturesAperture DirectivityDiscrete Apertures
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
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
Our ModelSound PropagationContinuous AperturesAperture DirectivityDiscrete Apertures
Continuous Apertures
An aperture is 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 aperture sensitivity 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
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
Our ModelSound PropagationContinuous AperturesAperture DirectivityDiscrete Apertures
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 the directivityfunction, as it gives the response as a function of the directionof arrival of the plane wave (recalling that k = g(f, ,)).
Iain McCowan Microphone Array Tutorial 6
Our Model
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
Our ModelSound PropagationContinuous AperturesAperture DirectivityDiscrete Apertures
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
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Our Model
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
Our ModelSound PropagationContinuous AperturesAperture DirectivityDiscrete Apertures
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)}.
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Our Model
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
Our ModelSound PropagationContinuous AperturesAperture DirectivityDiscrete Apertures
Directivity of a Linear Aperture
For a linear aperture r = x of 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
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I d iOur Model
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
Our ModelSound PropagationContinuous AperturesAperture DirectivityDiscrete Apertures
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
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I t d tiMicrophone Array Directivity Pattern
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
p y yVarying the Number of MicrophonesVarying the Length of the ArrayVariation with FrequencyAliasing and Symmetry
Microphone Array Directivity Pattern
A microphone array is a discrete receiving aperture.
A plot of the directivity function over different angles ofarrival is known as the directivity pattern.
Directivity of Continuous Aperture
D(f, k) =
V
w(f, r)ejkrdr
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
p y yVarying the Number of MicrophonesVarying the Length of the ArrayVariation with FrequencyAliasing and Symmetry
Microphone Array Directivity Pattern
A microphone array is a discrete receiving aperture.
A plot of the directivity function over different angles ofarrival is known as the directivity pattern.
Directivity of Linear Array
D(f, k) =N1n=0
wn(f)ejkrn ,
where thereare N microphones and rn is the location of the n
th microphone.
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IntroductionMicrophone Array Directivity Pattern
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
Varying the Number of MicrophonesVarying the Length of the ArrayVariation with FrequencyAliasing and Symmetry
Microphone Array Directivity Pattern
A microphone array is a discrete receiving aperture.
A plot of the directivity function over different angles ofarrival is known as the directivity pattern.
Horizontal Directivity of Uniform Linear Array
D(f,) =N1n=0
wn(f)ej2f
cndcos,
where there are N microphones, d is the uniforminter-element spacing, and = /2.
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
Varying the Number of MicrophonesVarying the Length of the ArrayVariation with FrequencyAliasing and Symmetry
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,)|
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Introduction Microphone Array Directivity PatternV i th N b f Mi h
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IntroductionFundamentals
Analysis of Directivity PatternBeamforming
Varying the Number of MicrophonesVarying the Length of the ArrayVariation with FrequencyAliasing and Symmetry
Varying the Length of the Array
Varying the array length, L = Nd, for fixed N decreases 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,)|
Iain McCowan Microphone Array Tutorial 12
Introduction Microphone Array Directivity PatternVar ing the N mber of Micro hones
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FundamentalsAnalysis of Directivity Pattern
Beamforming
Varying the Number of MicrophonesVarying the Length of the ArrayVariation with FrequencyAliasing and Symmetry
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,)|
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Introduction Microphone Array Directivity PatternVarying the Number of Microphones
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FundamentalsAnalysis of Directivity Pattern
Beamforming
Varying the Number of MicrophonesVarying the Length of the ArrayVariation with FrequencyAliasing and Symmetry
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
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IntroductionDefinition
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FundamentalsAnalysis of Directivity Pattern
Beamforming
DefinitionDelay-Sum BeamformerBeyond the Delay-Sum Beamformer
Beamforming
Horizontal Directivity of Uniform Linear Array
D(f,) =
N1n=0
wn(f)ej2f
cndcos,
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 desired steering andshaping of the directivity pattern.
This is referred to as microphone array beamforming.
Iain McCowan Microphone Array Tutorial 15
IntroductionF d l
Definition
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FundamentalsAnalysis of Directivity Pattern
Beamforming
DefinitionDelay-Sum BeamformerBeyond the Delay-Sum Beamformer
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 desired steering andshaping of the directivity pattern.
This is referred to as microphone array beamforming.
Iain McCowan Microphone Array Tutorial 15
IntroductionF d t l
Definition
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FundamentalsAnalysis of Directivity Pattern
Beamforming
DefinitionDelay-Sum BeamformerBeyond the Delay-Sum Beamformer
Delay-Sum Beamformer
For the uniform linear array, using a simple time-delay filter:
wn(f) =1
Nej
2fc
ndcoss
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-known delay-sum beamformer.
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IntroductionFundamentals
Definition
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FundamentalsAnalysis of Directivity Pattern
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:
the superdirective beamformer, which maximises the gain inthe desired direction while minimising average gain over all
other directions.adaptive beamformers, which dynamically update filters tominimise the power from localised noise sources.
Iain McCowan Microphone Array Tutorial 17
IntroductionFundamentals
Definition
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FundamentalsAnalysis of Directivity Pattern
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
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Superdirective Beamformer
0.5
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Iain McCowan Microphone Array Tutorial 18
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