Maximum Likelihood Phase Demodulation for Spatially Turbulent Channels

The likelihood equation for the maximum likelihood estimate (MLE) of a temporal phase modulation (a sample function of a wide-sense stationary zero-mean Gaussian random process) received in a spatial aperture after transmission through a spatially turbulent channel and addition to a white Gaussian noise field is derived for unknown, Gaussian, and limiting cases of log-normal received signal fields. The received signal field estimation performed by the demodulator structure is explicitly exhibited and other clarifications made. The likelihood equation is linearized and solved and the resulting mean-square estimation error calculated. The Cramer-Rao lower bound is given. In an example it is shown that the linear estimate is efficient and has variance inversely proportional to
M
, the number of coherence areas required to disjunctly cover the receiver spatial aperture. A heterodyne receiver with an optimized aperture weighting is shown to use essentially only one coherence area; its asymptotic mean-square error is also calculated.