• 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: [email protected]
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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