Covariance matrix properties and multiple symbol/soft decision detection in slow flat Rician fading

For a small number of symbols N and slow flat fading channels, it is shown that covariance matrices encountered in practice have two nonnegligible eigenvalues, the first much larger than the second, with a symmetric eigenvector associated with the first eigenvalue, and a skew symmetric eigenvector associated with the second eigenvalue. The first eigenvector is well approximated by a conditional mean, and the second eigenvector represents a small drift about the mean. The eigenvalues and eigenvectors of the slow flat fading channel covariance matrix are shown to be strongly related to those of a certain conditional covariance matrix. The maximum likelihood (ML) rules for block hard decision and symbol-by-symbol hard decision, and a rule for soft decision detection of M-DPSK, all using multiple symbol information, are obtained for the Rician channel as a function of N. The eigenvalue-eigenvector results lead to practical implementations of all rules. For small to moderate N, it is shown that a simple open-loop algorithm, of complexity N log N, attains the performance of the ML decision rules for an E_s/N ₀ range of interest for several land mobile satellite systems. The ML decision rules are seen to give rapidly diminishing returns as N increases, showing that simple noncoherent techniques can have very effective performance for the Rician fading channel. Lastly, several conclusions are drawn about the asymptotic channel behavior, including the Rayleigh channel. The work is directly applicable to the Australian and North American land mobile satellite systems