Partial Recovery of Loss of Array Gain due to Quantization in Digital Beamformers by Post Filtering

Abstract

By assuming a sinusoidal wave embedded in Gaussian noise as array input signal, the output autocorrelation function of a digital beamformer can be expressed as the sum of infinite series Rsxs, Rnxn, and Rsxn, involving signal and its harmonics, ascending powers of noise cross-correlation functions, and “quantizer functions” of various orders [H. S. C. Wang, J. Acoust. Soc. Am. 53, 929 (1973)]. It is demonstrated that the effect of bandpass filtering of post-beamforming signal can be rigorously analyzed, without transforming to the frequency domain, by applying certain factors attributed first to Davenport [W. B. Davenport, Jr., J. Appl. Phys. 24, 702 (1953)] to the successive terms of the series Rnxk and Rsxn-In the special case of DIMUS processor, the series of quantizer functions reduce exactly to Davenport's confluent hyper-geometric series for his “bandpass rooter” with n → ∞. For a 12-channel beamformer, the array gain improvement or partial recovering of quantizing (clipping) loss by post filtering in correlated and uncorrelated noise fields at different values of input SNR is numerically evaluated and displayed.