Superdirective beamforming Superdirective beamforming robust against microphone robust against microphone mismatchmismatch

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Superdirective beamforming Superdirective beamforming

robust against microphone robust against microphone

mismatch mismatch

Simon Doclo, Marc Moonen

Dept. of Electrical Engineering (ESAT-SCD), KU Leuven, Belgium

ICASSP-2006, May 17 2006

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Overview Overview

• Fixed superdirective beamforming:

o Optimal suppression of diffuse noise field

o Sensitive to uncorrelated noise and microphone mismatch

• Robust design procedures:

o Limit white noise gain

o Take into account statistics of microphone characteristics:

– Mean and worst-case directivity factor – Mean noise and distortion energy

– Mean deviation from desired directivity pattern

• Simulation results:

o Mean/worst-case directivity factor is preferred procedure

o Suitable parameter range for other design procedures

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3

Fixed beamforming Fixed beamforming

• Speech and noise sources with overlapping spectrum at different positions

Exploit spatial diversity by using multiple microphones Spatial focus on speech source + suppress noise and reverberation from certain directions

• Fixed beamformers:

o Direction of speech source and microphone configuration assumed to be known

o Applications: hearing aids, teleconferencing, pre-processing stage in adaptive beamformers (GSC)

o Different types: delay-and-sum beamformer, differential microphone array, frequency-invariant beamformer, superdirective beamformer

Fixed beamforming

Superdirective beamforming

Robust

superdirective beamforming

Simulation results

Conclusions

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Design of fixed beamformer (1) Design of fixed beamformer (1)

• Configuration:

o Linear microphone array (N microphones, distance dn) o Far-field assumption

o Speech source (_s,_s) + noise field

• Steering vector:

2( , , )

   

v

   

s

     S     

Y g V g

n

    ^{, ,}   A

n

    ^{, ,}  e

^^j^{  }ⁿ^ ^, ^

microphone characteristics

(gain, phase)

delay (position)

Robust

Conclusions

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5

Design of fixed beamformer (2) Design of fixed beamformer (2)

• Output signal:

• Directivity pattern: transfer function between source and output

 

^H

   

Z   W  Y 

 ^{, ,} 

^H

   ^{, ,} 

H

    W 

g

  

• Array Gain: SNR improvement between input and output signal

with normalised noise correlation matrix (i.e. noise coherence matrix for homogeneous noise field)

• Directivity factor (DF): ability of array to suppress diffuse noise

     

H 2

s H

VV

G

 

 ^   

W g WΦ



W

VV

   Φ 

     

2

VV

H

s

H diff

DF

 

 ^   

W g WΦ



W

     

   

H 2

s

WNG H

 

 ^  

W g

W W

• White noise gain (WNG): ability of array to suppress spatially uncorrelated noise (e.g. sensor noise)  measure for robustness

Robust

Conclusions

Depend on

mic char

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Superdirective beamforming (1) Superdirective beamforming (1)

• Optimization criteria: maximize array gain for diffuse noise

• Solution:

min , s.t. 1

VV

H diff H

s



W WΦ W



W g

 

1

VV

diff

s

sd H diff

s s

W





Φ g

gΦ g



Sensitive to uncorrelated noise, i.e. small WNG, especially for small-size microphone arrays at low frequencies

• WNG constraint: limit amplification of uncorrelated noise

 needs to be chosen in function of the amount of sensor noise

• Solution:

min , s.t. 1,

VV

diff

s

H

H H

  

W WΦ W



W g W W

 

1

, 1

VV

diff

s

sd H diff

s s

W _









Φ



g

Φ I g g



I



2

max , s.t. 1

VV

H

s H

H diff s



W

W g W g WΦ W



Unity constraint in speech direction

Robust

Conclusions

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7

Superdirective beamforming (2) Superdirective beamforming (2)

• Sensitivity to microphone mismatch:

o N=3, [0 0.01 0.025]m, _s=0^o, fs=16 kHz

o Deviation: [0 2 0]dB, [-5 10 5]^o, [0.001 –0.001 0.001]m

o Determine  such that requirements are met for this mismatch

0 1000 2000 3000 4000 5000 6000 7000 8000

0 1 2 3 4 5 6 7 8 9 10

Frequency [Hz]

Decrease of Directivity factor [dB]

=0

=0.0001

=0.001

=0.01

=0.1

=1

=10

Robust

Conclusions

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Robust superdirective beamforming Robust superdirective beamforming

• In practice microphone characteristics are never exactly known

• Instead of measuring/calibrating or limiting WNG, take all feasible microphone characteristics into account and optimise a mean performance criterion using probability as weight:

1. Mean (or worst-case) directivity factor

2. Weighted sum of mean noise and distortion energy

3. Mean deviation from desired superdirective directivity pattern

• Related to earlier proposed design procedures for robust beamformers with an arbitrary directivity pattern

[S. Doclo, M. Moonen, “Design of broadband beamformers robust against gain and phase errors in the microphone array characteristics,” IEEE Trans. Signal Processing, vol. 51, no. 10, pp. 2511-2526, Oct. 2003]

Take into account stochastic deviations in design

 ^{, ,}   ^{, ,} 

^j ⁿ^ ^{, ,} ^ ^j ⁿ^cos^c ^f^s

n n

A a e e

 

    

      

^ ^

Knowledge of probability density function required ^f



  ^A

Robust

Conclusions

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9

Robust superdirective beamforming Robust superdirective beamforming

1. Mean directivity factor

o filter W cannot be extracted from integrals  discrete sum

o Iterative optimization techniques (e.g. quasi-Newton method)

   

0 1

, ( )

0

(

1

)

0 1

m A AN N N

DF DF f A f A dA dA

    

  

W



W A

 

   

0 1

0 1 0 1

, ( ) ( )

N

m N N

A A

DF DF f A f A A A



   

    

W



W A

 

2. Worst-case directivity factor

o finite grid of microphone characteristics

o Minimax optimization problem (e.g. sequential quadratic program)

 

1

 

2

 

_tot

 

T

DF DF DFK

 

  

F W W W



W

min

max min

_k

 

DF



k DF

W W

Robust

Conclusions

Preferred design procedures

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Robust superdirective beamforming Robust superdirective beamforming

3. Weighted sum of mean noise and distortion energy

 ^,     

tm vm dm

J W

 

J W

 

J W

   

0 1

( )

0

(

1

)

0 1

N VV

H diff

vm A A N N

J f A f A dA dA

    

  

W



WΦ



A W

 

   

0 1

2

0 1 0 1

1 ( ) ( )

N

H

dm A A s N N

J f A f A dA dA

    

   

W  W g A  

4. Mean deviation from desired superdirective directivity pattern

   

0 1

,

, ( )

0

(

1

)

0 1

LS m A AN LS N N

J J f A f A dA dA

    

  

W



W A

 

 ^, 

₀² ₀

 ^{, ,}   ^{, ,}   ^{, ,} 

²

JLS ^{W A}

  

^{ } F

  

H

   

D

  

d d

 

directivity pattern of superdirective BF when no microphone

mismatch occurs directivity pattern for

specific microphone characteristic A

Robust

Conclusions

Quadratic cost functions  closed-from expression

Not optimising directivity factors

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11

Simulation results (1) Simulation results (1)

• Set-up and microphone characteristics:

o N=3, [0 0.01 0.025]m, _s=0^o, fs=16 kHz, design frequency 1 kHz o Nominal microphone characteristics: A_n()=1, same pdf

o Only gain deviations: uniform gain pdf (a=1, sa=0.3)

o Grid spacing for mean/worst-case directivity factor: a=0.02 o Measures: DF without deviation + mean/worst-case DF

 

f

_

A

Robust

Conclusions

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Simulation results (2) Simulation results (2)

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹ 10²

0 2 4 6 8



DF [dB]

Directivity factor no deviation

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹ 10²

0 2 4



DFm [dB]

Mean directivity factor - max = 4.88dB

10^-4 10^-3 10^-2 10^-1 10⁰ 10¹ 10²

-40 -20 0



DFmin [dB]

Worst-case directivity factor - max = 2.43dB

=0.01

=0.07

• Limit WNG: parameter  needs to be tuned

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Simulation results (3) Simulation results (3)

• Directivity patterns: with/without gain deviation [0.7 1.3 1.2]

Robust

Conclusions

-20 -10 0

30

210

60

240

90

270 120

300

330

180 0

Mean directivity factor W_m

-20 -10 0

30

210

60

240

90

270 120

300

330

180 0

Worst-case directivity factor W_min

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-20 -10 0 10 20 30

30

210

60

240

90

270 120

300 150

330

180 0

Superdirective (^{=0), J}_vm^{=17dB, J}_dm^=17dB

-20 -10 0

30

210

60

240

90

270 120

300 150

330

180 0

^{=0.03, J}_vm^{=-4.1dB, J}_dm^=-9.5dB

-20 -10 0

30

210

60 90

120 150

330

180 0

^{=0.1, J}_vm^{=-3.6dB, J}_dm^=-12dB

-20 -10 0

30

210

60 90

120 150

330

180 0

Delay-and-sum (⁼^{), J}_vm^{=-0.2dB, J}_dm^=-20dB

Simulation results (4) Simulation results (4)

Robust

Conclusions

scale: 30dB

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15

Conclusions Conclusions

• Superdirective beamforming:

o Commonly used fixed beamforming technique (hearing aids) o Maximises array gain for diffuse noise field

o Sensitive to uncorrelated noise and microphone mismatch, especially for small-size arrays at low frequencies

Robust

Conclusions

• Robustness improvement:

o Limit white noise gain  parameter  needs to be tuned

o Take into account statistics of microphone characteristics and optimize mean performance criterion:

– Mean/worst-case directivity factor: preferred designed procedure – Weighted sum of mean noise and distortion energy  parameter

 needs to be tuned

– Mean deviation from desired directivity pattern  lowest performance

Superdirective beamforming Superdirective beamforming robust against microphone robust against microphone mismatchmismatch