The family of distributions introduced by G.H. Hansmann in 1934 is considered. Apparently, it has gone unnoticed for over fifty years that there is a major error in Hansmann's work and that his solutions are incomplete in some cases. It is shown how to complete them and the necessity of doing so is demonstrated by means of two examples.>

A method is developed for calculating the probability density functionp(y)of the output of anRCfilter when the input is a particular kind of random telegraph wave. The method makes use of a computer to determine the numerical values of the coefficients in two series solutions, one of which contains a logarithm, of a fourth-order linear differential equation. The constants of integration are determined from the boundary conditions by a procedure that involves a summation of the series. Representative values and curves ofp(y)are presented, and the generality of the computer method is discussed briefly,

The problem of calculating the probability density function of the output of anRCfilter driven by a binary random process with intervals generated by an equilibrium renewal process is studied. New integral equations, closely related to McFadden's original integral equations, are derived and solved by a matrix approximation method and by iteration. Transformations of the integral equations into differential equations are investigated and a new closed-form solution is obtained in one special case. Some numerical results that compare the matrix and iteration solutions with both exact solutions and approximate solutions based upon the Fokker-Planck equation are presented.

The probability distribution of the phase angle between two vectors perturbed by correlated Gaussian noises is studied in detail. Definite integral expressions are derived for the distribution function, and its asymptotic behavior for large signal-to-noise is found for "small," "near\pi/2," and "large" angles. The results are applied to obtain new formulas for the symbol error rate in MDPSK, to calculate the distribution of instantaneous frequency, to study the error rate in digital FM with partial-bit integration in the postdetection filter, and to obtain a simplified expresion for the error rate in DPSK with a phase error in the reference signal. In the degenerate case in which one of the vectors is noise free, the results lead to the symbol error rate in MPSK.

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.

Some new results are presented on the harmonics and noise in the output of a linear envelope detector whose input is an amplitude-modulated carrier accompanied by noise. The transition region between the cases of weak and strong modulation is also studied. The signal-to-noise ratio is assumed throughout to be large and the index of modulation to be near unity.

The problem of computing the distributions of the envelope and instantaneous value of the sum of several sine waves plus Gaussian noise is reviewed. It is pointed out that the associated integrals can be efficiently evaluated by the trapezoidal rule except when the noise is small. Approximations which hold when the noise is small are given.

The problem of determining the first-order probability density of a filtered version of hard-limitedRCnoise is considered. An integral equation for the density is derived by using Siegert's results on zero-crossing distributions forRCnoise. We also show how the integral equation can be obtained from a two-dimensional Fokker-Planck equation. The exact solution of the integral equation has not been found, but some general information is available. An iterative scheme for determining the moments has been found, and some information has been obtained regarding the behavior of the density as the variable approaches the ends of its range. One result of this portion of the investigation is a disproof of au expression for the density conjectured by Pawula and Tsai. Finally, curves for the density are obtained by solving the integral equation numerically.