Generating a Gaussian sample

The general theoretical difficulties in analyzing the effect of a random input signal on a known system are pointed out. Basically, if certain output statistics are computed directly, each statistic represents a complete, separate problem. An alternative analytical computational procedure is suggested, using a Monte Carlo type technique in which the output is obtained by numerical integration from sequences of values which represent members of the statistical ensemble of the input process. For such applications, or for other possible uses such as in testing, it is necessary to generate statistical sequences, analogous to tables of random numbers. Techniques are discussed for analytically generating such sequences, to correspond to gaussian probability distributions which are further characterized by arbitrarily specified power spectra or autocorrelation functions. The procedure makes use of the standard tables of random numbers, these numbers being distributed uniformly and without correlation. The exact statistical generation of values of a sequence is shown to require, in general, the diagonalization (or solution for the eigenvalues and eigenvectors) of an th order matrix; two simpler approximate procedures are also described.