William L. Root

An introduction to the theory of the detection of signals in noise

An introductory discussion is presented in the general field of detection theory, with particular emphasis on the problems of detecting signals in Gaussian noise and of estimating parameters of signals in Gaussian noise.

Ill-posedness and precision in object-field reconstruction problems

The inherent stability or instability in reconstructing an object field, in the presence of observation noise, for a class of ill-posed problems is investigated for situations in which constraints are imposed on the object fields. The class of ill-posed problems includes inversion of truncated Fourier transforms. Two kinds of constraint are considered. It is shown that if the object field is restricted to a subset of L2 space over Rn that is bounded, closed, convex, and has nonempty interior, then a (nonlinear) least-squares estimate always exists but is unstable. It is also shown that if one is primarily concerned with the situation in which the object field belongs to a compact parallelepiped in L2, aligned in a natural way, there is a satisfactory, stable linear estimate that is optimal according to a min–max criterion. This also leads to a nonlinear modification for the case in which the object field is actually restricted to the parallelepiped. A summary of some relevant mathematical background is included.

The Operator Pseudoinverse in Control and Systems Identification

Publisher Summary This chapter discusses the pseudoinverse of a linear transformation between Hilbert spaces. It presents the basic theory and shows its use for applications to systems identification and the quadratic regulator problem. It also presents an exposition of the basic theory of the pseudoinverse for densely defined linear closed operators with arbitrary range. The theory includes the case of operators that do not have closed range. The chapter discusses the Gauss-Markov theorem on statistical estimation that is shown to be proved under the hypothesis that both the quantity to be estimated and the observations are elements of Hilbert spaces. This theorem applies only to nonsingular covariance operators and reduces to the classical Gauss-Markov theorem when the spaces are finite dimensional. In one classical form of the quadratic regulator problem, it is required to find the minimum energy input that will move a system from some initial state to the origin at a designated time. The chapter presents a reformulation that generalizes this problem to admit a greater variety of linear constraints, possibly including some of which are incompatible and/or unattainable by the system. The solution always exists as a pseudoinverse and reduces to the classical result if the system is capable of meeting the constraint.

Some remarks on statistical detection

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.

On the measurement and use of time-varying communication channels

In this paper the beginnings of a theory are established concerning time-varying a nd random linear channels with the intent of characterizing classes of channels which can be determined exact]y or approximately by measurement, showing how the measurements can be made, analyzing the errors, and applying the results to the theory of signal detection. The notion of a determinable class of channels is defined and general examples are given. These include classes of channels that are time-invariant, periodic, and which vary with a known trend. The measurement of slowly-varying channels by approximation by time-invariant ones belonging to a known determinable class is discussed. Relation between almost-time-inva rianee of a channel a nd the correlation properties of a kind ef stationary random channel are developed and tied-in with the ehanne! measurement theory. An application is made to the problem of detecting sure signals in noise when the channel is slowly-varying.

Estimation of constrained parameters in a linear model with multiplicative and additive noise

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. >

Remarks, mostly historical, on signal detection and signal parameter estimation

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.

A weak stochastic integral in Banach space with application to a linear stochastic differential equation

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved.A class of linear, time-invariant, stochastic differential equations in real, separable, reflexive Banach spaces is formulated in such fashion that a solution of the equation is a cylindrical process. An existence and uniqueness theorem is proved. A stochastic version of the problem of heat conduction in a ring provides an example.

Considerations regarding input and output spaces for time-varying dynamical systems

A small topic in the abstract theory of system modelling is investigated in this paper. As understood here, a system is a map F from an input space U to an output space Y. The systems with which we are concerned are causal dynamical systems for which inputs and outputs are functions of time. The topic under consideration is the choice and topologization of input and output spaces and their relation to system behavior. The concepts described below, which underlie the work in [1] and [2], provide intuitive background for the development here. Let y = F(u) be the input-output map of a causal (in general of a nonlinear, time-varying) system, where u and y are functions of time belonging to suitable function spaces U and Y, respectively, and F is causal, continuous and bounded. Let Pt, t ~ R , denote projection on the past; i.e. for any function of time z, (Ptz)(s) = z(s) or 0 according as s t. Fix T > 0 and define fit for all t ~ R by

Identifiability of slowly varying systems

A system is conceived of as being slowly varying if it changes slowly enough to permit identification to within a specified error. A generic model is developed to study the identifiability and identification of slowly varying systems. The model is suitable for a large variety of nonlinear, time-varying, causal, bounded memory systems; it has finitely many parameters and is linear in its parameters. Results are obtained with the use of this general model that give guaranteed accuracy of identification as a function of the prior knowledge of the unknown system, the maximum rate of time variation of the system, and the characteristics of output observation noise. To derive these results, a recursive estimation procedure is developed for time-discrete linear dynamical system structures in which the observation noise is statistical but the dynamic equation noise is nonstatistical and is known only to be bounded.