[Paper Review] Diffuse-field coherence of sensors with arbitrary directional responses

Archontis Politis|arXiv (Cornell University)|Aug 27, 2016

Blind Source Separation Techniques13 references3 citations

TL;DR

This paper presents a closed-form analytical formulation for the diffuse-field coherence between sensors with arbitrary directional responses, using spherical harmonic transforms to model angular spectra. It enables accurate coherence estimation for any sensor geometry, orientation, or directivity, including differential beamformers, with applications in beamforming, array calibration, and spatial audio processing.

ABSTRACT

Knowledge of the diffuse-field coherence between array sensors is a basic assumption for a wide range of array processing applications. Explicit relations previously existed only for omnidirectional and first-order directional sensors, or a restricted arrangement of differential patterns. We present a closed-form formulation of the theoretical coherence function between arbitrary directionally band-limited sensors for the general cases that a) the responses of the individual sensors are known or estimated, and the coherence needs to be known for an arbitrary arrangement, and b) that no information on the sensor directionality or on array geometry exists, but calibration measurements around the array are available.

Motivation & Objective

• To address the lack of general analytical solutions for diffuse-field coherence between sensors with arbitrary directional responses, beyond omnidirectional and first-order patterns.
• To enable accurate coherence modeling when sensor directivity is known (analytically or from measurements), regardless of sensor spacing or orientation.
• To extend coherence modeling to beamformers formed by sub-arrays, particularly differential beamformers with arbitrary orientations.
• To provide a framework that works even without prior knowledge of sensor responses or array geometry, using only calibration measurements around the array.

Proposed method

• Utilizes spherical harmonic transforms (SHT) to represent the angular response of sensors and beamformers as finite-bandwidth spectral coefficients.
• Applies the spherical harmonic domain formulation to compute coherence between two sensors based on their angular spectra, leveraging the isotropy of the diffuse field.
• Derives a transformation from differential beamformer weights (e.g., first-order gradient patterns) to spherical harmonic coefficients using Legendre polynomial expansions and normalization.
• Introduces a rotation model to handle arbitrary sensor or beamformer orientations by rotating the angular spectrum coefficients in the spherical harmonic domain.
• Uses the inverse SHT and Parseval’s theorem to compute the coherence function as an inner product of spectral coefficients, accounting for inter-sensor distance via the spherical Bessel function.
• Demonstrates applicability in the absence of sensor models by using calibration data to estimate the angular spectra directly, enabling coherence estimation from measurements alone.

Experimental results

Research questions

• RQ1 What is the general closed-form expression for diffuse-field coherence between sensors with arbitrary directional responses, including non-omnidirectional and differential patterns?
• RQ2 How can the coherence be computed when sensor directivity is known through models, specifications, or measured polar plots?
• RQ3 Can the coherence between beamformers formed by spaced sub-arrays be computed analytically, especially when the beamformers are non-collinear?
• RQ4 How can diffuse-field coherence be estimated without prior knowledge of sensor responses or array geometry, using only calibration measurements?

Key findings

• A closed-form analytical expression for diffuse-field coherence is derived for arbitrary directional sensors, extending beyond the known sinc-function result for omnidirectional sensors.
• The coherence depends solely on the spherical harmonic coefficients of the sensor responses and inter-sensor distance, with the spherical Bessel function of the first kind modeling the spatial decay.
• For differential beamformers, the method enables accurate coherence computation for any orientation, overcoming the prior restriction to collinear patterns in earlier work.
• The transformation from differential weights to spherical harmonic coefficients is derived via Legendre polynomial decomposition and normalization, enabling spectral modeling of common beamformer types.
• When sensor responses and array geometry are unknown, calibration measurements around the array can be used to estimate the angular spectra, enabling coherence estimation without explicit models.
• The method is numerically stable and applicable to band-limited directional responses, making it suitable for practical array signal processing applications.