Correlation autoregressive processes with application to helicopter noise

Abstract

This paper introduces a new class of random processes X( t) the autocorrelations R x ( t 1, t 2) of which satisfy a linear relation of the type R x(t 1, t 2)= ∑ j=1 N a jR x(t 1+τ j, t 2+τ j) for all t 1 and t 2 in some interval of the time axis. Such random processes are denoted as correlation autoregressive. This class is shown to include the familiar stationary and periodically correlated processes as well as many other, both harmonizable and non-harmonizable, non-stationary processes. When a process is correlation autoregressive for all times and harmonizable, its two-dimensional power spectral density S x ( ω 1, ω 2) is shown to take a particularly simple form, being non-zero only on lines such that ω 1 − ω 2 = r k , where the r k s are (not necessarily equally spaced) real roots of a characteristic function. The relationship of such processes to the class of stationary processes is examined. In addition, the application of such processes in the analysis of typical helicopter noise signals is described.