Monotone gain, first-order autocorrelation and zero-crossing rate

autocorrelation of a weakly stationary time series is discussed. When the gain is monotone increasing, the first-order autocorrelation cannot increase. Otherwise, when the gain is monotone decreasing, the correlation cannot decrease. Further, when the gain is strictly monotone, the first-order autocorrelation is unchanged if and only if the process is a pure sinusoid with probability 1. Under the Gaussian assumption, the zero-crossing rate moves oppositely from the first-order autocorrelation. 1. Introduction. In this paper it is shown that when a weakly stationary process is filtered with a time-invariant linear filter possessing a monotone increasing (decreasing) gain, the first-order autocorrelation cannot increase (decrease). If, in addition, the gain is strictly monotone, the first-order autocorrelation is unchanged if and only if the process is a pure sinusoid with probability 1. When the relationship between the autocorrelation and the zero-crossing rate is known, this can be translated into statements concerning the zero-crossing rate. The Gaussian case is the best example [Kedem (1984)]. Let {Zt; t = O + 1, ? 2, ... 1 be a real-valued zero-mean stationary process with autocorrelation Pk and spectral distribution function FGo), 0 < cl < wr. Let .( ) denote a time-invariant linear filter with transfer function H(G) satisfying the conditions that HG) = H(-w), and that the squared gain IHGo)12 is integrable with respect to the distribution function F. Let pk(H) be the kth-order autocorrelation of the filtered process {Y(Z)t; t = 0 ? 1,