Optimal detection of a target return contaminated by signal-dependent interference, as well as additive channel noise, requires the design of a transmit pulse f(t) and a receiver impulse response h(t) jointly maximizing the output signal to interference plus noise ratio (SINR). Despite the highly nonlinear nature of this problem, it has been possible to show that f(t) may always be chosen minimum-phase. A full analysis concludes with the construction of an effective numerical procedure for the determination of optimal pairs (f,h) that appears to converge satisfactorily for most values of input SINR. Extensive simulation reveals that the shape of f(t) can be a critical factor. In particular, the performance of a chirp-like pulse is often unacceptable, especially when clutter and channel noise are low-pass dominant and comparable.

In many signal-processing and system-theoretic applications, such as image restoration [9, 10, 15–18] and adaptive control [1–3, 6–8], it is necessary to consider the convergence properties of iterations of relaxed projection operators drawn from either a finite or an infinite pool. Moreover, in an adaptive environment it is invariably true that the choice of iteration sequence is unknown or totally unrestricted. Unfortunately, relatively little is known about this subject, especially if the operators are nonlinear and the setting is infinite-dimensional. In addition to summarizing some of the more pertinent recent literature, we also present some new results which can serve as a starting point for further study. A fairly extensive discussion of ideas and theorems is followed by detailed proofs.

An n-dimensional digital filter is said to be normal if it is linear shift-invariant, first-quadrant causal, and bounded-input-bounded output (BIBO) stable. The transfer function of such a filter is holomorphic in the open polydisc U/sup n/ and continuous in its closure, making it a member of W. Rudin's polydisc algebra A(U/sup n/) (see W. Rudin) (Function Theory in Polydiscs, New York: W.A. Benjamin, 1969). It has been shown by Rudin that, nevertheless, every all-pass function in A(U/sup n/) is rational, free of nonessential singularities of the second kind, and of finite norm. This result appears to have been completely overlooked in the engineering literature. The material covered here is mainly tutorial, comprising a leisurely expanded account of the key ideas behind the Fejer and Rudin constructions.

Many problems in electrical engineering, such as the synthesis of linear n ports and the detection and filtration of multivariable systems corrupted by stationary additive noise, depend for their successful solution upon the factorization of a matrix-valued function of a complex variablep. This paper presents several algorithms for affecting such decompositions for the class of rational matricesG(p), i.e., matrices whose entries are ratios of polynomials inp. The methods employed are elementary in nature and center around the Smith canonic form of a polynomial matrix. Several nontrivial examples are worked out in detail to illustrate the theory.

In many of the applications of probability theory to problems of estimation and detection of random functions an eigenvalue integral equation of the type \begin{equation} \phi(x) = \lambda \int_0^T K(x - y)\phi(y) dy, \qquad 0 \leq x \leq T, \end{equation} is encountered whereK(x)represents the covariance function of a continuous stationary second-order process possessing an absolutely continuous spectral density. In this paper an explicit operational solution is given for the eigenvalnes and eigenfunctions in the special but practical case when the Fourier transform ofK(x)is a rational function of\omega^2, i.e., \begin{equation} K(x) \doteqdot G(s^2) = \frac{N(s^2)}{D(s^2)}, \qquad s=i\omega, \end{euation} in whichN(s^2)andD(s^2)are polynomials ins^2. It is easy to show by elementary methods that the solutions are of the form \begin{equation} \phi(x)= \sum C_r e^{-\alpha_r x} \cos (\beta_r x + \gamma_r), \end{equation} the constantsC_r, \alpha_r, \beta_r, and\gamma_rbeing linked together by the integral equation. It is precisely the labor involved in their determination that in practice often causes the problem to assume awesome proportions. By means of the results given herein, this labor is diminished to the irreducible minimum-the solving of a transcendental equation.

A signal is received in the time interval (t - T ≤ τ ≤ t). It is known that this signal is composed of noise plus intelligence a(t) which is statistical in nature and which has been modulated in some known way. Assuming that both intelligence and noise are Gaussian (although not necessarily stationary) time series, the analog of the classical maximum-likelihood estimate for α(t) is derived. The advantage of this approach is that it can handle arbitrary types of modulation. For unmodulated stationary intelligence and stationary noise, the solution reduces to that of Zadeh and Ragazzini. In the general case, the optimum estimate is given as the solution of a pair of integral equations. The amplitude-modulated case is treated in some detail. The application of the maximum-likelihood technique to problems involving arbitrary modulation was first suggested, as far as the author is aware, by F. W. Lehan and R. J. Parks of the Jet Propulsion Laboratory.