Some schemes for detecting moving point targets against structured backgrounds from observations on the output of an imaging system are investigated. When the velocity of the target and the background image are considered as known, it is shown that the uniformly most powerful detector, invariant with respect to image intensity variations, consists of specific spatial-temporal differencing schemes. This places such schemes on a rigorous foundation.

Various estimators of a dynamical state vector are optimally combined to obtain new estimators. The performance o! these new estimators is evaluated by comparing the traces of their covariance matrices. A nontrivial example is given to illustrate the techniques.

Let\cal ybe a mean zero complex stationary Gaussian signal process depending on a vector parameter\theta \prime = \{ \theta_{1}, \theta_{2}, \theta_{3} \}whose components represent parameters of the covariance function R(r) of\cal y. These parameters are chosen as\theta_{1} = R(0), \theta_{2} = |R( \tau )| /R(0), \theta_{3} =phase ofR( \tau), and they are simply related to the parameters of the spectral density of\cal y. This paper is concerned with the determination of most powerful (MP) tests that distinguish between random signals having different covariance functions. The tests are based uponNcorrelated pairs of independent observations on\cal y. Although the MP test that distinguishes between\theta = \theta_{o}and the alternative hypothesis\theta = \theta_{1}has been solved previously [11], the problem of identifying the random signals is often complicated by the fact that the signal power\theta_{1} = R(0)is not a distinguishing feature of either hypothesis. This paper determines the MP invariant test that delineates between the composite hypothesis\lambda \equiv R( \tau)/R(0) = \lambda_{0}and the composite alternative\lambda = \lambda_{1}. In addition, the uniformly MP invariant test that distinguishes between the composite hypotheses\theta_{2} <_{=} | \lambda_{o} |and\theta_{2} > | \lambda_{0} |has also been found. In all cases, exact probability distributions have been obtained.

Methods for estimating the normalized moments of the power spectrum of a stationary stochastic process via both spectral and covariance approaches are outlined. Various relationships between these two techniques are examined and typical extensions which submit to similar analyses are listed.

We are interested in estimating the moments of the spectral density of a comp[ex Gaussian signal process\{ q^{(1)} (t) \}when the signal process is immersed in independent additive complex Gaussian noise\{q^{(2)} (t) \}. Using vector samplesQ = \{ q(t_1),\cdots ,q(t_m)\}, whereq(t) = q^{(1)}(t) + q^{(2)}(t), estimators for determining the spectral moments or parameters of the signal-process power spectrum may be constructed. These estimators depend upon estimates of the covariance functionR_1 (h)of the signal process at only one value ofh \neq 0. In particular, ifm = 2, these estimators are maximum-likelihood solutions. (The explicit solution of the likelihood equations form > 2is still an unsolved problem.) using these solutions, asymptotic (with sample size) formulas for the means and variances of the spectral mean frequency and spectral width are derived. It is shown that the leading term in the variance computations is identical with the Cramér-Rao lower bound calculated using the Fisher information matrix. Also considered is the case Where the data set consists ofNsamples Of continuous data, each of finite duration. In this case asymptotic (withN) formulas are also derived for the means and variances of the spectral mean frequency and spectral width.

Let\{q^(1) (t)\}, the signal, be a complex Gaussian process corrupted by additive Gaussian noise\{q^(2) (t) \}. Observations onp(t)q(t)andp(t) q^(2) (t)are assumed to be available wherep(t)is a smooth weighting function andq = q^(1) + q^(2). Using the Fourier transform of the samples ofp(t)q(t)andp(t) q^(2) (t), estimators are derived for estimating the mean frequency and spectral width of the unknown power spectrum of the unweighted signal process. The means and variances of these statistics are computed in general, and explicitly for nontrivial practical examples. Asymptotic formulas for the moment estimators as a function of the number of realizations, frequency resolution, signal-to-noise ratio and spectral width, and consistency of the estimators are some of the results that are discussed in detail.

LetRy_{t} = u_{t}be a stochastic difference equation. Various relations between the input and output covariances and spectral densities are deduced under the hypotheses thatRis time dependent and thatu_{t}is a member of a nonstationary random process.

A technique is developed for the straightforward simulation of the transfer function of a certain class of linear systems. This method is particularly well adapted to the analysis of systems with fixed transfer function and variable initial conditions and forcing functions. In particular, a single simulation, minimal in its use of integrators, will suffice to handle forcing functions and initial conditions on both input and output.

An important characteristic of coherent integrators is that their effective bandwidth decreases as the integration time increases. If it is only known that a weak signal occurs somewhere in a given frequency range, then the number of integration channels required to cover the specified range increases as the amount of coherent integration is increased. However, each integration channel can independently cause a false alarm, although only the particular channel in which the signal appears can cause a true alarm. The question arises therefore whether it is profitable to lengthen the coherent integration period to increase the signal-to-noise ratio when doing so requires an increase in the number of integration channels. This problem is investigated analytically. Numerical results appropriate for system design are presented as a series of graphs of missed-signal probability vs number of integration channels, with initial signal-to-noise ratio and over-all false alarm probability as parameters. Also included is a detailed analysis of statistical properties of ideal and approximate ideal coherent integrators.

In many of the theories of prediction and detection developed during the past decade, one encounters linear integral equations which can be subsumed under the general form\int_a^b R(t, \tau) x(\tau) d\tau = f(t), a \leqq t \leqq b. This equation includes as special cases the Wiener-Hopf equation and the modified Wiener-Hopf equation\int_0^T R(\mid t - \tau \mid ) x(\tau) d\tau = f(t), 0 \leqq t \leqq T. The type of kernel considered in this note occurs when the noise can be regarded as the result of operating on white noise with a succession of not necessarily time-invariant linear differential and inverse-differential operators. For this type of noise, which is essentially a generalization of the stationary noise with a rational spectral density function, it is shown that the solution of the integral equation can be expressed in terms of solution of a certain linear differential equation with variable coefficients.