Power spectrum tests of random numbers

Abstract

Three nonrandomness tests having very low correlation between them and utilizing some statistical properties of the spectral estimates of the random number sequences having any shape of probability distribution function, are given. The spectral density is estimated as 1/ N times the periodogram (magnitude square of the discrete Fourier transform) of an N-term sequence of the random numbers. From M such spectral estimates, M peak values, M spacings between the ordered (sorted) frequencies of the peak values, and the peak value of the ensemble average of the estimates, are found and tested to decide if they closely fit their cumulative probability distributions as derived for random numbers. The sensibility thresholds of the tests are measured on random binary sequences artificially contaminated by substitution of periodic binary signals or substitution of random binary signals having periodic mean or shortening long runs.