The problem of estimating a member of a discrete-time vector process from past and present quantum mechanical measurements is considered; specifically, the minimum-variance linear estimator based on optimal present measurements is studied. Necessary and sufficient conditions that characterize the optimal processing matrix coefficients and the optimal measurements are discussed and interpreted. The optimal linear filter is compared to the optimal quantum estimator without postprocessing of past data. When the signal sequence is pairwise Gaussian and the optimal quantum measurement without postproccssing has the properties that it is linear in a specific sense and that its outcome and the corresponding element of the signal sequence are jointly Gaussian, then the optimal linear filter separates. That is, the optimal measurement can be taken to be the same as thc optimal measurement without regard to past data, and the past and present data are processed classically. Thc results are illustrated by considering the filtering of the in-phase and quadrature amplitudes of a laser field received in a single-mode cavity along with thermal noise. In this case, when the random signal sequence satisfies a linear recursion, the estimate can be computed recursively in a very efficient manner.

The problem of estimating a member of a scalar random signal sequence with quantum-mechanical measurements is considered. The minimum variance linear estimator based on an optimal present quantum measurement and optimal linear processing of past measurements is found. When the average optimal measurement without postprocessing, for a fixed signal, is linear in the random signal and the signal sequence is pairwise Gaussian, the optimal processing separates: the optimal measurement is the same as the optimal measurement without regard to past data, and the past and present data are processed classically. The results are illustrated by considering the estimator of the real amplitude of a laser signal received in a single-mode cavity along with thermal noise; when the random signal sequence satisfies a linear recursion, the estimate can be computed recursively. For a one-step memory signal sequence it is shown that the optimal observable generally differs from the optimal observable disregarding the past; the optimal measurement can be computed recursively.

A model that is sufficiently general to describe the predominant statistical characteristics of the output of many real optical detectors is formulated. This model is used to study the optimum receiver processing for direct-detection optical communication systems. In particular, the structures of detectors and estimators for randomly filtered doubly stochastic Poisson processes observed in additive white Gaussian noise are considered. Representations for the posterior statistics of a vector-valued Markov process that modulates the intensity of the doubly stochastic Poisson process are obtained. Quasi-optimum estimators and detectors are specified in general terms and specialized for several important applications. These include a demodulator for subcarrier angle modulation, a detector structure for binary signaling with known intensities, and a detector structure for binary signaling in the turbulent atmosphere.

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 toM, 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.

An upper bound on the minimum probability of error for an equal-strength diversity channel is simply derived that improves a previously known bound by the factor[4(1 - p)]^(-1),0 \leq p \leq \frac{1}{2}.

It is shown that two covariance-type functions can completely characterize lognormal random processes and their likelihood functional can be a simple modification of the likelihood functional of their associated normal random processes.

The problem of optimally processing data with unknown focus is investigated by the method of maximum likelihood. It is assumed that the signal prior to defocusing is a zero-mean Gaussian process with a known spectral density and a very short correlation interval relative to the observation interval, and that it is received (recorded) with additive, white Gaussian noise. One of the most interesting data systems that satisfy the assumptions and approximations made is side-looking, synthetic-aperture radar. An electro-optical system that computes the maximum-likelihood estimate is shown, and the modifications to estimate also the signal sample function are discussed.