Frequency-Invariant Beamforming in Elevation and Azimuth via Autograd and Concentric Circular Microphone Arrays

TL;DR

This work addresses frequency-invariant beamforming with dual-axis (azimuth and elevation) control using concentric circular microphone arrays (CCMAs). It proposes a differentiable optimization framework powered by autograd to jointly optimize ring weights and intra-ring contributions under beampattern constraints, balancing beamwidth, directivity, and invariance across frequencies. The approach introduces flexible loss functions that incorporate directivity and white-noise gain along with invariance terms, and demonstrates superior elevation resolution at low frequencies compared with standard delay-and-sum and related methods. The results indicate that the CCMA with gradient-based optimization yields sharper mainlobes and robust dual-axis performance, improving spatial audio sensing and acoustic localization across broad frequency ranges. The framework offers a versatile, differentiable pathway for frequency-invariant beamforming in 3D acoustic sensing applications.

Abstract

The use of planar and concentric circular microphone arrays in beamforming has gained attention due to their ability to optimize both azimuth and elevation angles, making them ideal for spatial audio tasks like sound source localization and noise suppression. Unlike linear arrays, which restrict steering to a single axis, 2D arrays offer dual-axis optimization, although elevation control remains challenging. This study explores the integration of autograd, an automatic differentiation tool, with concentric circular arrays to impose beamwidth and frequency invariance constraints. This enables continuous optimization over both angles while maintaining performance across a wide frequency range. We evaluate our method through simulations of beamwidth, white noise gain, and directivity across multiple frequencies. A comparative analysis is presented against standard and advanced beamformers, including delay-and-sum, modified delay-and-sum, a Jacobi-Anger expansion-based method, and a Gaussian window-based gradient descent approach. Our method achieves superior spatial selectivity and narrower mainlobes, particularly in the elevation axis at lower frequencies. These results underscore the effectiveness of our approach in enhancing beamforming performance for acoustic sensing and spatial audio applications requiring precise dual-axis control.

Figures (6)

• Figure 1: Problem geometry.
• Figure 2: Fitted parabola to the beampattern.
• Figure 3: Beampattern comparison by correcting beamwidths smaller than the objective by explicitly increasing the estimated mainlobe beamwidth ($\mathcal{L}_1$) vs reducing the directivity factor ($\mathcal{L}_2$).
• Figure 4: Performance evolution with the iterations of the base array optimized with Eq. \ref{['eq:loss_1']} ($\mathcal{L}_1$) as objective.
• Figure 5: Results obtained for different parametrization of $\mathcal{L}_3$ in terms of metrics s frequency. Top row shows $\alpha=\{0, 0.25, 0.5, 0.75, 1\}$, $\lambda_1 = \lambda_2, \lambda_3 = 0$. Second row shows the effect of varying $\lambda_1 = \{0, 0.1, 0.5, 0.75, 1, 1.5, 2\}$ with $\alpha=1, \lambda_2=1$ and $\lambda_3=0$. The third row sweeps $\lambda_2 = \{0, 0.5, 0.75, 1, 1.5, 2, 5\}$ for $\alpha = 0, \lambda_1 = 0$ and $\lambda_3 = 0$. Finally the last row evaluates $\lambda_3 = \{0.001, 0.005, 0.01, 0.05\}$ while fixing $\alpha=1$, $\lambda_1$ and $\lambda_2=0$.