Series Solution of Digital Beamformer Problems

Abstract

Due to the nonlinear characteristics of the input quantizers, the statistical average-power output of a digital beamformer corresponding to Gaussian random input signal and/or noise can best be expressed as an infinite series of a “quantizer function” Qm(K,q̃), where m is the term order and K, q̃ are quantizer parameters. To ensure valid calculations of important performance criteria as array gain, directivity patterns, etc., the convergence of such series is rigorously proved and numerically demonstrated. Quantizer functions of orders up to 150 corresponding to 1–4 bits of quantization per input channel have been calculated by means of “reduced Hermite polynomials.” With such a formulation, values of on-target array gain are computed as a function of the quantizer step size, which clearly exhibits a maximum representing optimum design condition for each specific number of quantization bits. This optimum array gain increases with the number of bits per channel, but approaches the ideal linear array gain for the same number of receiving elements when 4 bits is reached. For arbitrary signal incident angle θ, the output power of an array of N elements spaced d/λ wavelengths apart assumes the form (q̃2/2π)Σm−0∞[Qm2(K,q̃)Θm(N,d/λ,θ)]. Directivity patterns of line and circular arrays corresponding to narrowband and broadband signals of various bandwidths are calculated with relative ease and presented. The special case of 1 bit/channel DIMUS beamformer is discussed in considerable detail and compared with known closed-form solutions.