Applying Automatic Differentiation to Optimize Differential Microphone Array Designs

TL;DR

The paper tackles constrained adaptive non-uniform LDMA design by casting the beamforming optimization as a differentiable problem solvable via automatic differentiation. It directly optimizes array weights and inter-microphone spacings under a distortionless constraint in the target direction and a spacing bound to prevent spatial aliasing, aiming to achieve a prescribed wideband directivity factor. Using a differentiable convex formulation, the method minimizes the mismatch between the actual beampattern and a desired cosine-series beampattern while enforcing $d(w,\theta_d)^H h(w)=1$ and $\delta_{ ext{min}}\le\delta_m\le\delta_{ ext{max}}$, with $\delta_{ ext{max}}<\frac{\lambda}{2}$. Numerical results demonstrate close alignment to target beampatterns and DF/MSE performance across frequencies, with findings suggesting that $M=N+1$ can offer a cost-effective design choice. The approach promises efficient, scalable LDMA design suitable for wideband applications.

Abstract

This paper introduces a novel methodology leveraging differentiable programming to design efficient, constrained adaptive non-uniform Linear Differential Microphone Arrays (LDMAs) with reduced implementation costs. Utilizing an automatic differentiation framework, we propose a differentiable convex approach that enables the adaptive design of a filter with a distortionless constraint in the desired sound direction, while also imposing constraints on microphone positioning to ensure consistent performance. This approach achieves the desired Directivity Factor (DF) over a wide frequency range and facilitates effective recovery of wide-band speech signals at lower implementation costs.

Figures (9)

• Figure 1: An adaptive non-uniform linear microphone array.
• Figure 2: Desired beam-pattern and the beam-pattern of the proposed method for the second order LDMA is compared versus incident angle when the angle of arrival is $\theta_{d} = 0$.
• Figure 3: Desired beam-pattern and the beam-pattern of the proposed method for the second order LDMA is compared versus the incident angle when the angle of arrival is $\theta_{d} = \pi/3$.
• Figure 4: Desired beam-pattern and the beam-pattern of the proposed method for the second order LDMA is compared versus the incident angle when the angle of arrival is $\theta_{d} = \pi$.
• Figure 5: Minimum mean square error of the proposed method for the second order LDMA versus frequency when the angle of arrival is $\theta_{d} = \pi$.