An approach to optimal design of circular harmonics based modal beamformers for circular arrays is presented. Theoretical analysis shows the equivalence between the element-space and circular-harmonics-domain beamformers. By deriving the expression for the signal and noise covariance matrices, array manifold, and the array response in terms of the circular-harmonics-domain beamforming weights, the weights design problem is formulated as one of minimum variance distortionless response beamforming in the circular harmonics domain. It is found that the well-known phase-mode beamformer for circular arrays can be viewed as an ideal minimum variance distortionless response beamformer against the planarly isotropic noise in the circular harmonics domain. This interesting result is useful for analyzing and evaluating the performance of the phase-mode circular arrays. In this circular-harmonics-domain beamformer, additional constraints can be imposed to improve its robustness and control the sidelobes, which is important in practical applications. The developed approach can include both the delay-and-sum and phase-mode beamformers as special cases, which leads to very flexible designs. Simulation and experimental results show excellent performance of the proposed beamforming approach.

Most of the existing spherical array modal beamformers are implemented in the frequency domain, where a block of snapshots is required to perform the discrete Fourier transform. In this paper, an approach to real-valued time-domain implementation of the modal beamformer for broadband spherical microphone arrays is presented. The microphone array data are converted to the spherical harmonics domain by using the discrete spherical Fourier transform, and then steered to the look direction followed by the pattern generation unit implemented using the filter-and-sum structure. We derive the expression for the array response, the beamformer output power against both isotropic noise and spatially white noise, and the mainlobe spatial response variation in terms of the finite impulse response (FIR) filters' tap weights. A multiple-constraint problem is then formulated to find the filters' tap weights with the aim of providing a suitable tradeoff among multiple conflicting performance measures such as directivity index, robustness, sidelobe level, and mainlobe response variation. Simulation and experimental results show good performance of the proposed time-domain broadband modal beamforming approach.

An approach to optimal array pattern synthesis based on spherical harmonics is presented. The array processing problem in the spherical harmonics domain is expressed with a matrix formulation. The beamformer weight vector design problem is written as a multiply constrained problem, so that the resulting beamformer can provide a suitable trade-off among multiple conflicting performance measures such as directivity index, robustness, array gain, sidelobe level, mainlobe width, and so on. The multiply constrained problem is formulated as a convex form of second-order cone programming which is computationally tractable. We show that the pure phase-mode spherical microphone array can be viewed as a minimum variance distortionless response (MVDR) beamformer in the spherical harmonics domain for the case of spherically isotropic noise. It is shown that our approach includes the delay-and-sum beamformer and a pure phase-mode beamformer as special cases, which leads to very flexible designs. Results of simulations and experimental data processing show good performance of the proposed array pattern synthesis approach. To simplify the analysis, the assumption of equidistant spatial sampling of the wavefield by microphones on a spherical surface is used and the aliasing effects due to noncontinuous spatial sampling are neglected.

The directivity pattern of an array is known to degrade in the presence of errors in the array manifolds, with respect to the desired nominal array pattern. This paper describes a new robust pattern synthesis approach to arrays with manifold vectors perturbation. This synthesis technique optimizes the worst case performance by minimizing the worst case sidelobe level while maintaining a distortionless respect to the worst case signal steering vector. The possible values of the manifold are covered by an ellipsoid that describes the uncertainty in terms of errors in element gains and phase angles. The pattern synthesis parameters can be optimally chosen based on known levels of uncertainty in the manifold vectors. Two optimization criteria, l 2 regularization and l 1 regularization, of a robust beamformer are proposed. Both criteria of the robust beamformer problem can be reformulated in a convex form of second-order cone programming, which is computationally tractable. A simple lower bound on the difference between the worse case sidelobe level of the robust beamformer and the sidelobe level of the nominal optimal beamformer with no array manifold uncertainty is derived. This robust approach is applicable to arrays with arbitrary geometry. Its effectiveness is illustrated through its application to a circular hydrophone array. An experiment is performed to measure the manifold vectors uncertainty set of hydrophone arrays. Results of applying the algorithms to both simulated and experimental data are presented and they show good performance of the proposed robust pattern synthesis approach.

Broadband beamformers with constant mainlobe response over the frequency of interest are desirable in many applications including underwater acoustics, ultrasonics, acoustic imaging and communications, and so on. Solutions to this problem have been presented for specific array geometry often requiring a larger number of sensors. And the array pattern synthesis error minimization is employed for the whole field of view, which leads to suboptimal designs. In this paper, a broadband array pattern synthesis approach to designing time-domain constant mainlobe response beamformer is proposed. By imposing constraints both on the mainlobe spatial response variation over frequency and on the sidelobes of the beamformer, several optimization criteria are presented and the corresponding convex second-order cone programming implementations are given. In this approach, no preliminary desired beampattern is required and the beam responses variation minimization is employed only in the mainlobe region and not in the sidelobe regions, which improves the beamformer mainlobe synthesis accuracy. Equally, one can obtain lower sidelobes at the same mainlobe synthesis accuracy. This approach is applicable to arrays with arbitrary geometry. Simulation and experimental results are presented to illustrate the effectiveness of this approach. Performance comparisons of the proposed beamformers and the existing beamformer are also provided.