An approximate expression is derived for the probability density of the sample magnitude-squared coherence function obtained using overlapped Fourier transforms. For this approximation it is shown that the expression obtained with overlap is identical to the expression obtained without overlap but with a reduced number of degrees of freedom. In addition, it is shown that, regardless of the amount of overlap, the bias and variance of the sample statistic cannot be reduced to less than 2/3 of their no-overlap values. Numerical examples are presented to illustrate the accuracy of the approximate expression.

Determination of the amplitude distribution and density of shot processes is a problem of longstanding interest. In this paper we present approximations for these distributions and densities whose salient features are accuracy and ease of computation. The parameters that characterize the approximations are determined by the lowerorder moments or cumulants of the shot process. To illustrate the accuracy of the approximations, they are compared with the exact functions in three cases. A logarithmic error measure is introduced and a simple approximation for it is given.

Approximately optimum detectors for problems in which either the signal or noise is a linear process are developed. An approximate expression for the characteristic function of a class of linear processes is inverted to obtain a convergent series approximation for the joint probability density. From this series, a detection algorithm is designed that is optimum when applied to the series and approximately optimum when applied to the original process. This technique is applied and evaluated in several cases that model coherent or instantaneous amplitude signal detection in reverberation noise, multipath transmission, and similar problems.

This correspondence is concerned with the problem of choosing an approximation for the random variable obtained by operating on a stochastic processx(t)with a zero-memory non-linearity followed by a linear transformation. It is desired to approximate the nonlinearity with a simpler function and two different criteria are considered for selecting this approximation, viz., the mean-square error obtained before and after the linear transformation. A sufficient condition is presented-for an approximation to simultaneously minimize both error criteria; in addition, it is shown that if the two minima coincide for every linear transformation and every nonlinearity, this condition is also necessary. The condition appears as a restriction on the class of approximating functions and is related to the second-order distributions ofx(t). For several processes of interest, this restriction is satisfied when the approximating functions are polynomials, the most notable example being the Gaussian process.

Intersymbol interference and additive Gaussian noise are two important sources of distortion in digital systems, and a principal goal in the analysis of such systems is the determination of the resulting probability of error. Earlier related work has sought to estimate the error probability either by calculating an approximation based upon a truncated version of the random pulse train or by obtaining an upper bound which results from consideration of the worst case intersymbol interference. In this paper a new upper bound is derived for the probability of error which is computationally simpler than the truncated pulse-train approximation and which never exceeds the worst case bound. Moreover, the new bound is applicable in a number of cases where the worst case bound cannot be used. The bound is readily evaluated and depends upon three parameters: the usual signal-to-noise ratio; the ratio of intersymbol interference power to total distortion power; and the ratio of the maximum intersymbol interference amplitude to its rms value. To illustrate the utility of the bound, it is compared with the earlier methods in three cases which are representative of the most important situations occurring in practice.

Band-limited pulse trains derived from a discrete stochastic process are considered. An upper bound is obtained for the average probability that the magnitude of the departure of the instantaneous signal value from the mean signal value exceeds\lambdatimes the standard deviation, as a function of\lambda. This upper bound is useful in the analysis and prediction of intersymbol interference, quantizing errors, and digit errors in digital data communication systems.