Estimation of a Hilbert-space valued parameter in a linear model with compact linear transformation is considered with both multiplicative and additive noise present. The unknown parameter is assumed a priori to lie in a compact rectangular parallelepiped oriented in a certain way in the Hilbert space. Linear estimators are devised that minimize reasonable upper bounds on mean-squared error depending on conditions on the noise. Under prescribed conditions the estimators are minimax in the class of linear estimators. With the prior constraint on the unknown parameter removed, the estimation problem is ill-posed. Restricting the unknown provides a regularization of the basically ill-posed estimation. It turns out the estimators developed here belong to a well-known class of regularized estimators. With the interpretation that the constraint is soft, the procedure is applicable to many signal-processing problems.>

The stability with respect to model uncertainty of linear estimators of the coefficients of a linear combination of deterministic signals in noise is investigated. A class of estimators having nominal performances constrained to be close to that of the nominal linear, unbiased, minimum-variance (LUMV) estimator is specified. Two estimator stability indexes are defined, one based on a worst-case estimate mean-square error and the other on a type of signal-to-noise ratio. The estimator minimizing each index, subject to the optimality constraints, is found by reference to related LUMV estimation results. In most cases, the minimizing (or most stable) estimator is the same under the two indexes.>

Historical remarks outlining some of the evolution of the statistical theory of detecting signals in noise and of estimating signal parameters make up the greater part of the paper. The time period covered is roughly from the years of World War II to the middle 1960s. The remarks are rather severely limited in scope; they do not touch on nonparametric statistical methods, or filtering, or prediction theory, for example. The last part of the paper is a discussion of one aspect of a signal-processing problem area that is of current interest in optics and radar, and can be regarded as one of many descendants of the early parameter estimation theory. The area is object-field reconstruction or restoration, and the aspect of interest is the intrinsic relation between the inherent instability of solutions of many such problems and the precision or resolution of detail that can be attained in the construction.

The stability of linear detectors for signals nominally known except for amplitude in correlated nominally Gaussian noise is considered. Stability is defined in terms of bounds on the distribution of the test statistic as the true probability measure on the observation space varies in a Prokhorov neighborhood of the nominal; the derived stability index is independent of the neighborhood size. An optimization problem is formulated and solved to give the most stable linear detector meeting a Constraint on the nominal level of detectability.

A class of finite-alphabet discrete-time nonprobabilistic communication channels with finite nonzero memory is defined. The unknown input/output mapping is required to stay fixed for a specified number of channel uses and then can change arbitrarily within a given set of mappings. The frequency with which the input/output mapping can change gives a rate of time-variation for the channel. The class of channels is parametrized by this rate of time-variation, by alphabet size, by memory duration, and by the set of permissible mappings. Zero-error capacity in the sense of Shannon is computed for a collection of special cases within this class.

The following simple abstract model for a class of communications problems is adopted: the set of possible transmitted signalsxis taken to be the unit ball in theL_{2}space of functions defined on[-T, T](bounded energy); the transmitted signal is assumed to be operated on by a convolution operatorH; and the final observed received signalzisz = Hx + n, wherenis an unknown error, caused either by additive noise, lack of complete knowledge ofH, or other causes, of norm less than some specified\epsilon(not necessarily small). The problem is to determine how many "distinguishable" signals can be sent, i.e., how manyx_{i}there are such that they_{i} = Hx_{i}are separated in norm by at least\epsilon. The chief results are asymptotic upper and lower bounds on the rate of error-free transmission possible, i.e., the ratio of the logarithm of the number of distinguishable signals to the time interval2TasT \rightarrow \infty. These estimates are in terms of the Fourier transform of the kernel of the convolution operatorH. The suitability of the model and the nature of the results are discussed.

A particular type of communications detection problem is considered: the problem of specifying a detector to decide which one of two sure signals is being transmitted when the signals are perturbed randomly both by Gaussian noise and multipath transmission. If the delays in the various channels are known, but the strengths are random, a maximum likelihood detector may be specified by methods which are a simple extension of known methods. If the delays are random, the problem is more difficult. One possible solution is first to estimate certain channel parameters from the received signal and then to use these estimates in a likelihood test. It is shown how to make consistent unbiased estimates for appropriate channel parameters under certain assumptions on the nature of the signal.