It is demonstrated that in problems involving the estimation of linear regression parameters in colored Gaussian noise, the simple least-squares estimator can be significantly suboptimal. When the noise covariance function can be described as a known function of a finite number of unknown nonrandom parameters, it is possible to take advantage of this information to improve upon the least-squares estimator by an appropriate bootstrapping technique. Two examples are given, and comments that may lead to other examples are presented.

Computational procedures for obtaining minimum mean square error estimates of parameters are developed for the case in which the observation functions, describing the dependence of observed data on the unknown parameters, can be accurately approximated by expansions through quadratic terms in the unknown parameters. Analytical closed-form solutions are obtained for the minimum mean square error estimates of real valued functions of the unknown parameters. It is pointed out that even in cases where the statistical assumptions are not satisfied the resulting computational procedures may be applicable. In such cases adequate, although nonoptimum, estimates may still be obtained and the computational procedure may be more convenient than iterative least-square methods.

The analysis of radar detection probability in the presence of unknown parameters, given by Kelly, Reed, and Root for maximum-likelihood detection procedures, is extended in this paper to cases where the detection procedures still belong to the class of maximum-likelihood procedures, but the actual waveform, or noise covariance, or both, may differ from those for which the detection procedure would be maximum likelihood. The analysis covers detection procedures ranging from fully coherent, through partially coherent, to pulse-to-pulse incoherent cases. These cases are distinguished by changing the list of parameters treated as unknown by the detection procedure. The resulting equations for the characteristic function of the detection test variable lead to probability distribution functions which are incomplete Toronto functions or generalizations of them; this is convenient for performance analysis, since these functions have been extensively studied. The analysis is further extended to the case of signals having fluctuating amplitudes and phases, for cases ranging from pulse-to-pulse fluctuations, through partially correlated fluctuations, to the case of complete correlation for the duration of the received waveform on a single scan. In the case of partially correlated fluctuations of amplitude and phase, the situation is considered where such fluctuations impose the limit on the amount of coherent integration which can be accomplished. The analysis is brought within the Kelly, Reed, Root framework by modifying the maximum-likelihood procedure to take account of a priori knowledge of the correlation properties of the signal amplitude and phase fluctuations, so that the resulting detection procedure becomes a maximum a posteriori procedure. A classification of special cases is given, with attention focused on the question of when it is still true (as in the case considered by Kelly, Reed, and Root) that the detection probability is given by an incomplete Toronto function, and when generalizations of these must be used. Also, a classification is given of the appropriate definition of signal-to-noise ratio for various special cases.

Formulas are presented for the maximum accuracy with which waveform parameters can be estimated in the presence of Gaussian noise. These formulas are presented in a form which is not widely known and which is very simple to use, especially in the white noise case. The unknown parameters may include the conventional ones, such as range, range rate, and acceleration, as well as others depending on target motion and configuration, properties of the transmission medium, or even sensor bias errors. A discussion is given of the conditions of validity of the formulas. The relationship between the approach followed here and other approaches to the parameter estimation problem is discussed.

In this paper the techniques of sequential analysis are applied to a multiple-resolution-element search radar for different types of detection. By using sequential methods such a radar shows considerable saving compared to fixed-scan-rate detection radars. Many of the problems occurring in sequential radar detection are examined; the difficulties of obtaining analytical results for multiple-resolution-element sequential radars are discussed. Most of the results of this paper were obtained by simulation. Included are interesting results which suggest that truncation is not necessary for the tests which were considered. Curves are given describing the improvement obtained by sequential detection procedures. The curves are discussed in detail to make this paper useful to those whose primary interest is in the results which are obtained.

A class of procedures for detection of a pulsed signal having unknown frequency shift is described. From this class, procedures can be selected whose detection performance approximates that of coherent detection arbitrarily closely. Such procedures do not require the construction of a filter bank; on the other hand, they do not yield information as to the magnitude of the frequency shift of a detected signal. Envelope detection followed by video integration, and Reed's so-called semicoherent procedure are elements of this class of detection procedures. An iterative method for obtaining successively better approximations to coherent detection is described, the implementation of which would require a large number of mixing and delay operations. The number of iterations required to insure any given degree of approximation to coherent detection, for any number of integrated pulses, is estimated. Finally, the possible application of these ideas to reception of signals by array antennas is described.

A random surface is a sample function of a random process\{f(x, y)\}depending on two real parameters. Examples of random surfaces would be: photographs or television pictures; topographic maps; atmospheric pressure charts; and the like. A number of statistical properties of the contours of such surfaces are derived. An application of the results to the problem of obtaining bounds on the information content of quantized random surfaces is outlined.

This report considers the probability of detection off a target by a pulsed search radar, when the target has a fluctuating cross section. Formulas for detection probability are derived, and curves off detection probability vs, range are given, for four different target fluctuation models. The investigation shows that, for these fluctuation models, the probability of detection for a fluctuating target is less than that for a non-fluctuating target if the range is sufficiently short, and is greater if the range is sufficiently long. The amount by which the fluctuating and non-fluctuating cases differ depends on the rapidity of fluctuation and on the statistical distribution of the fluctuations. Figure 18, p. 307, shows a comparison between the non-fluctuating case and the four fluctuating cases considered.

This paper treats the detection of pulsed signals in the presence of receiver noise for the case of randomly fluctuating signal strength. The system considered consists of a predetection stage, a square law envelope detector, and a linear postdetection integrator. The main problem is the calculation of the probability density function of the output of the postdetection integrator. The analysis is carried out for a large family of probability density functions of the signal fluctuations and for very general types of correlation properties of the signal fluctuations. The effects of nonuniform beam shape and of nonuniform weighting of pulses by the postdetection integrator are also taken into account. The function which is actually evaluated is the Laplace transform of the probability density function of the integrator output. In many of the cases treated, the resulting Laplace transform has an inverse of known form. In such cases the evaluation of the probability density function would require the computation of a finite number of constants; in practice this would usually require the use of computing machinery, but would be perfectly feasible with presently available computing machinery.