Planar Concentric Circular Microphone Array

• Planar Concentric Circular Microphone Arrays are sensor configurations with multiple rings of microphones arranged in a common plane, each with a distinct radius for enhanced spatial sampling.
• They enable robust sound field reconstruction, frequency-invariant beamforming, and precise source localization by mitigating modal nulls and spatial aliasing through circular harmonic expansion and regularized inversion.
• Recent advancements incorporate virtual microphone methods and neural network augmentation to reduce hardware costs while maintaining high resolution and noise resilience in diverse acoustic applications.

A planar concentric circular microphone array (CCMA) consists of multiple circular rings of microphones arranged in the same plane with a common center, each ring having a distinct radius. This sensor geometry provides enhanced spatial sampling characteristics for multidimensional sound field reconstruction, frequency-invariant beamforming, and robust source localization. Theoretical, algorithmic, and practical innovations in CCMA design eliminate modal nulls due to Bessel function zeros and mitigate spatial aliasing, which hampers traditional single-ring circular arrays. Recent work in both inverse sound field reconstruction and real-time, robust spatial audio systems demonstrates that CCMAs—whether implemented physically or augmented via virtual microphones—constitute the reference architecture for spatially resolved acoustic sensing and beamforming, especially under physical and cost constraints.

1. Array Geometry and Sampling Principles

A planar CCMA comprises $N_r$ concentric rings, each at radius $R_n$ ($n=1,\ldots,N_r$), with $M_n$ microphones uniformly distributed in azimuth:

$$\theta_{n,m} = 2\pi (m-1)/M_n, \quad m=1,\ldots,M_n$$

Total sensors: $M = \sum_{n=1}^{N_r} M_n$. The innermost radius $R_1$ is chosen just outside any active source region for exterior field problems; the outermost radius $R_{N_r}$ is set inside the reconstructable domain of interest (Nguyen et al., 2023).

To represent fields up to modal order $L$ without angular aliasing, each ring must satisfy $M_n \geq 2L+1$ microphones, and the modal order is dictated by $L \approx k R_1$, with $k = 2\pi f / c$ the wavenumber at target frequency $f$ (Nguyen et al., 2023, Zhang et al., 2019).

Optimal microphone distribution on each ring is also constrained by spatial aliasing: the chord between adjacent mics on radius $\rho_r$ must not exceed $\lambda_{\min}/2$ (where $\lambda_{\min}=c/f_{\max}$), leading to a minimum $$M_r = \lfloor \pi / \arcsin(\lambda_{\min} / (4\rho_r))\rfloor$$ (Ortigoso-Narro et al., 24 Nov 2025). For broadband operation and modal decomposition across frequencies, microphone count increases with ring radius.

2. Circular Harmonic Expansion and Signal Modeling

The primary mathematical tool for CCMA-based sound field representation is circular harmonic expansion (CHE). The measured complex pressures $p_{n,m}=p(R_n, \theta_{n,m}) + \text{noise}$ on the rings are related to the sound field in polar coordinates $(r,\theta)$ by:

$$p(r,\theta) \approx \sum_{\ell=-L}^{L} a^+_\ell H^{(2)}_\ell(k r) e^{j \ell \theta} + \sum_{\ell=-L}^{L} a^-_\ell H^{(1)}_\ell(k r) e^{j \ell \theta}$$

where $H^{(1,2)}_\ell$ are Hankel functions and $a^\pm_\ell$ are outgoing/incoming modal coefficients (Nguyen et al., 2023). For purely radiating exterior fields, $a^-_\ell = 0$.

Measurement stacking and matrix formulation yield $p = H a + \eta$, with $H\in\mathbb{C}^{M\times(2L+1)}$ encoding modal responses per sensor and $\eta$ representing noise. Overdetermined systems (robust when $M \gg 2L+1$) are solved via Tikhonov-regularized least squares:

$\hat{a} = (H^H H + \lambda I)^{-1} H^H p$

Continuous sound field values are then reconstructed anywhere outside $R_1$ via the expansion using the estimated $\hat{a}^+_\ell$ coefficients.

3. Beamforming, Source Localization, and Frequency-Invariant Modal Processing

The concentric structure enables beamforming and source localization with enhanced robustness. In broadband settings, circular harmonic SRP (steered-response power) and frequency-invariant beamformers decompose the array output into modal coefficients up to $|l|\leq L$. UCCA (two-ring) and general CCMA arrays suppress the deep spatial nulls aligned with zeros of Bessel functions $J_l(kR)$, which afflict single-ring UCAs, by shifting modal zeros across radii and combining their contributions (Zhang et al., 2019, Zhao et al., 24 Feb 2024):

$$p(\theta, \omega) = \sum_{m=1}^M \sum_{n=-N}^N w_{m,n} J_n(k R_m) e^{j n \theta}$$

Compensating inverse filters (computed per ring and modal order) solve for robust modal outputs even in the proximity of Bessel zeros, thereby maintaining wide usable frequency bands for SRP and other metrics (Zhang et al., 2019).

Designing ring radii such that their corresponding Bessel function zeros interleave maximizes the frequency interval in which no order vanishes across all rings, contributing to higher modal rank and angular resolution.

Advanced beamforming with CMCAs enables dual-axis (azimuth, elevation) spatial selectivity, leveraging continuous optimization via autograd frameworks. The objective function imposes beamwidth and frequency invariance constraints directly on the beampattern:

$\text{Minimize} \; \mathcal{L}_3 = P + I + \Delta$

where $$P = -\alpha \log_{10}[\text{DF}] - (1-\alpha)\log_{10}[\text{WNG}]$$, $I$ penalizes deviations in directivity and noise gain, and $\Delta$ enforces mainlobe symmetry (Ortigoso-Narro et al., 24 Nov 2025). In practice, ring-specific weights and Gaussian intra-ring windows are learned to optimize spatial selectivity, beamwidth, and noise robustness across wide frequency bands.

4. Practical Implementation, Performance, and Trade-offs

Experimental implementations underscore the advantages of CMCAs over single-ring arrays. In controlled room settings, a two-ring UCCA (9 mics at 6 cm, 7 mics at 4 cm) achieved sub-degree mean angular error and nearly 100% success in real-time source localization at both near (1 m) and far (3 m) fields for white noise and speech, demonstrating resilience to reverberation and noise (Zhang et al., 2019).

Performance comparison across array types shows:

Array	Success (Near)	Success (Far)	Mean Error ($\sigma$)
UCCA	100% (noise)	99.5% (noise)	$\sim$0°, $\sigma\sim$0.5°
UCA (single)	76.5–97% (noise)	55.5–73% (noise)	$\sigma>2°$ near Bessel zeros

Processing pipelines run in real time (32 ms frame, 352 ms total latency), and the modal domain processing decouples complexity from microphone count, scaling instead with modal order and scan resolution (Zhang et al., 2019).

Key trade-offs include increased hardware and calibration complexity for multiple rings versus sharp improvements in bandwidth, modal stability, and aliasing suppression. More rings and wider apertures yield narrower beams but higher microphone counts and greater computational cost. Mainlobe width versus white-­noise gain (WNG) is explicitly managed via weight regularization and constraints in optimization (Ortigoso-Narro et al., 24 Nov 2025).

5. Virtual Microphone Methods and Neural Network-Aided Arrays

A principal limitation of physical CCMAs is hardware cost: a Q-ring CCMA with $Q>1$ typically doubles or triples the number of channels, increasing synchronization and calibration overhead. Virtual microphone methods, using acoustics-informed neural networks (AINNs), have emerged as an alternative to circumvent modal nulls and aliasing without physically increasing the array size (Zhao et al., 24 Feb 2024).

The AINN is trained on physical mic pressure data (e.g., from the outer ring at $R_1$), with inputs as physical and virtual 2D positions and outputs as complex pressure:

• The network minimizes a composite loss: $L_\text{data}$ (fit to mic data) plus $L_\text{phys}$ (Helmholtz PDE constraint over an auxiliary sample grid).
• After training, virtual mic pressures on ring(s) $R_2$ are predicted, augmenting the measurement vector used for covariance estimation and MVDR beamforming.

Experimental results show that a CCMA with virtual microphones matches the performance of a physical dual-ring array in removing Bessel-zero nulls and further, if enough virtual sensors are added, entirely suppresses spatial aliasing up to 22 kHz. For a 30-mic ring at $R=0.12$ m, directivity index and WNG remain above +10 dB at all frequencies. A plausible implication is that, for cost-sensitive or deployable arrays, AINN-augmented designs offer the modal robustness of multi-ring architectures without attendant hardware, at the expense of network inference accuracy and training (Zhao et al., 24 Feb 2024).

6. Design Guidelines and Best Practices

Best-practice CCMA design is characterized by:

• Selecting $N_r\geq2$ rings with radii spanning the reconstruction or localization region; maximizing interleaving of Bessel zeros while ensuring each ring independently satisfies angular sampling requirements ($M_n\geq 2L+1$).
• Ensuring total measurement count $M$ exceeds $(2L+1)$ (number of modal coefficients) by a factor of 2–3 for regularized inversion robustness.
• Setting truncation order $L$ at $L \approx kR_1$ for desired highest frequency $f$ (Nguyen et al., 2023).
• Choosing spatial regularization and processing parameters (Tikhonov $\lambda$, frequency windowing, frame length) based on application noise floor and update latency.
• Deploying frequency-invariant beamformers by learning ring-level and intra-ring weights using automatic differentiation to achieve dual-axis beamwidth and mainlobe stability (Ortigoso-Narro et al., 24 Nov 2025).
• When feasible, using virtual microphone expansion via AINNs to reduce hardware overhead without sacrificing performance (Zhao et al., 24 Feb 2024).

Distinct from single-ring or linear arrays, CCMAs deliver high-fidelity spatial sampling, wideband field capture, precise source localization, and beamforming with minimized spatial nulls and aliasing vulnerabilities.

7. Applications and Impact

Planar CCMAs are foundational in diverse settings:

• Acoustic field reconstruction: Superior performance in 2D exterior field estimation via CHE, with normalized mean squared errors (NMSE) better than –20 dB achievable at SNR ≥ 20 dB (Nguyen et al., 2023).
• Real-time source localization: Achieving sub-degree angular precision and near-perfect success rates in reverberant environments, e.g., robust steered response power scanning (Zhang et al., 2019).
• Frequency-invariant beamforming and spatial audio: Enabling dual-axis mainlobe shaping with high directivity and white-noise gain across wide frequency bands for 3D acoustic capture, noise suppression, and immersive rendering (Ortigoso-Narro et al., 24 Nov 2025).
• Deployable and cost-sensitive systems: Combination with AINN-based virtual sensing allows high modal fidelity while minimizing physical sensor resources, facilitating scalable spatial audio and field capture architectures (Zhao et al., 24 Feb 2024).

CCMAs are thus the architecture of choice for high-resolution, robust, and flexible spatial acoustic sensing across scientific, industrial, and emerging audio applications.