Quantizer functions and their use in the analyses of digital beamformer performance

The formulation of output autocorrelation function of a digital beamformer into an infinite series leads naturally to the introduction of “quantizer functions” of various orders as a convenient vehicle in the evaluation of beamformer performance corresponding to different input quantizer designs. For arrays operating in a Gaussian ambient noise field, the quantizer function of a general nonuniform quantizer is defined and evaluated in terms of “reduced Hermite polynomials,” which facilitate less costly machine computation via their unique recursion formulae. A rigorous proof of the convergence of the series of quantizer functions is given to ensure confidence in numerical results. Examples of the use of quantizer functions given are (i) determination of optimum step sizes of 2‐, 3‐, and 4‐bit uniform quantizers which yield maximum array gain for a typical array, (ii) calculation of directivity patterns of digital beamformer with input signals of arbitrary bandwidth, and (iii) evaluation of the effect of p...