Linear mean square estimation of a Rayleigh variable corrupted with noise

A linear mean-squared (LMS) estimation technique is proposed for the estimation of the true amplitude of a carrier in a Rayleigh fading channel in terms of the envelope of the noise-corrupted signal available at the receiver. The proposed method takes a finite number of equal-time spaced samples from the envelope of the noise-corrupted signal and then applies the Yule-Walker equations. A prerequisite in applying these equations is the a priori knowledge of two specific correlation functions: the autocorrelation function of the envelope of the noise-corrupted signal, and the cross-correlation function of the envelope of the noise-corrupted signal as well as the true amplitude of the signal. First the autocorrelation function of a stochastic signal with a Rayleigh probability density function is determined in terms of the autocorrelation function of its two orthogonal components. This expression is then used to determine the autocorrelation function of the envelope of the noise-corrupted signal. An expression is then derived for the required cross-correlation function. The results are implemented using the Yule-Walker equations and a sample quantity of ten