• The derived analytic expression gives the exact frequency(s) where the LCMV fails to provide a stable solution.

• The analytic expression also establishes a relation between the sound source locations and the numerical stability of the LCMV beamformer.

• The provided solutions mitigate the instability problem, however they fail to satisfy the original directional constraints.

E-Mail:

soumitro.

E-Mail:

soumitro.

IEEE SPL Vol. 23 No. 2 (February 2016)

5. Simulation Experiment

6. Conclusions

Fallback solutions

• Modify the directional constraints: For our experiment, the modified main beam direction was set to be while keeping the null in the original direction. Problem: All directional signal components for the specific frequency are suppressed. • Diagonal loading

where

M = 4 ✓1 = 135

d = 3 cm ✓2 = 25

fa = 5.72 kHz f = 7.09 kHz

✓1 = 120

wLCMV(f) = 1v A(f)

⇥AH

(f)1v A(f) + ↵I

⇤1q(f)

↵ = λ1

Ltr

�AH

(f)1v A(f)

4. Power Patternsθ1θ2 θ1θ2θ3

θ1θ2

✓1 = 105

✓2 = 35

f = 10.6 kHz

c = 343 m/s

d = 3 cm

fa = 5.72 kHz

✓1 = 135

✓2 = 35

✓2 = 10

f12 = 7.49 kHz

f12 = 6.76 kHz

• Given a pair of DOAs, and ,the lowest frequency where the columns of the constraint matrix are linearly dependent is given by

• In terms of the spatial aliasing frequency

where

• Two important observations: 1) The frequency of instability lies above the spatial aliasing frequency. 2) The instability also occurs at integer multiples of the lowest frequency of instability.

3. Non-existence of solution

fa = c/(2d)

f =

c

d |cos ✓t cos ✓s|

f =

fa∣∣sin

(✓s+✓t

2

)sin

(✓s✓t

2

)∣∣

✓s ✓t

✓1 = 135

✓2 = 25

f = 7.09 kHz

2. Problem Formulation

• Consider a uniform linear array (ULA) of M microphones located at . The microphone signals are given by

• Assuming signal components are mutually uncorrelated

• Desired signal

• Estimate of the desired signal

y(f) =

LX

l=1

xl(f)

| {z }x(f)

+xv(f)

y(f) = E{y(f)yH(f)}

= x(f) +v(f)

ˆZ(f) = wH(f)y(f)

• Microphone signal of l-th plane wave • Sound pressure of l-th plane wave

• m-th element of steering vector

• Noise PSD matrix

• Direction dependent gain

• Filter weights

• Constraint matrix

Definitions

xl(f) = [Xl(f,d1) . . . Xl(f,dM )]

T

xl(f) = a(✓l, f)Xl(f,d1)

am(✓l, f) = exp{j (m 1)d cos ✓l}

v = 2v(f)I

w(f)

• Formulation: s.t.

• Solution:

• Analyze the columns of the constraint matrix, , to derive an analytic expression for determining the frequency where the columns are linearly dependent, leading to instability in the LCMV solution.

wLCMV(f) = arg min

wwHv(f)w wH

(f)A(f) = qH(f)

wLCMV(f) = 1v A(f)

⇥AH

(f)1v A(f)

⇤1q(f)

LCMV filter

A(f)

Z(f) =

LX

l=1

Q⇤(✓l, f)Xl(f,d1)

d1...M

A(f) = [a(✓1, f), . . . ,a(✓L, f)]

Q(✓l, f)

1. Introduction

• Study the solution of an LCMV beamformer for a ULA, with directional constraints given by steering vectors. • Analyse the columns of the constraint matrix and present an analytic expression to determine the frequency where the LCMV fails to satisfy the constraints. • Provide insight into the behaviour and applicability of the LCMV beamformer beyond the spatial aliasing frequency.

Aim

On the Numerical Instability of an LCMV Beamformer for a Uniform Linear Array

Soumitro Chakrabarty, Emanuel. A. P. Habets

International Audio Laboratories Erlangen

E-Mail: soumitro.chakrabarty@audiolabs-erlangen.de

INTERNATIONAL AUDIO LABORATORIES ERLANGENA joint institution of Fraunhofer IIS and Universität Erlangen-Nürnberg

• Two well known conditions for the existence of the inverse term in LCMV solution: 1) Noise PSD matrix should be full rank. 2) Columns of the constraint matrix should be linearly independent. • The conditions are generic for mathematical operations involving matrix inversions. • No previous study provides any insight into the influence of the geometric setup of the sound scene on the stability of the LCMV solution.

Motivation

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Mixture Signal LCMV Output with modified constraints

LCMV Output LCMV Output with diagonal loading

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