Correlation Statistics of Quantized Noiselike Signals

Abstract

I calculate the statistics of correlation of two digitized noiselike signals, which are drawn from complex Gaussian distributions, sampled, quantized, correlated, and averaged. Averaged over many such samples, the correlation r approaches a Gaussian distribution. The mean and variance of r fully characterize the distribution of r. The mean corresponds to the reproducible part of the measurement, and the variance corresponds to the random part, or noise. I investigate the case of nonnegligible covariance rho between the signals. Noise in the correlation can increase or decrease, depending on quantizer parameters, when rho increases. This contrasts with the correlation of continuously valued or unquantized signals, for which the noise in phase with rho increases with increasing rho, and noise out of phase decreases. Indeed, for some quantizer parameters, I find that the correlation of quantized signals provides a more accurate estimate of rho than would correlation without quantization. I present analytic results in exact form and as polynomial expansions, and compare these mathematical results with results of computer simulations.