Fourier Analysis of Stochastic Processes

Abstract

1. Time series Given a discrete time process (X n) n2Z , with X n : R or X n : C 8n 2 Z, we de…ne time series a realization of the process, that is to say a series (x n) n2Z of real or complex numbers where x n = X n (!)8n 2 Z. In some cases we will use the notation x(n) instead of x n , with the same meaning. For convenience, we introduce the space l 2 of the time series such that P n2Z jx n j 2 < 1. The …nite value of the sum can be sometimes interpreted as a form of the energy, and the time series belonging to l 2 are called …nite energy time series. Another important set is the space l 1 of the time series such that P n2Z jx n j < 1. Notice that the assumption P n2Z jx n j < 1 implies P n2Z jx n j 2 < 1, because P n2Z jx n j 2 sup n2Z jx n j P n2Z jx n j and sup n2Z jx n j is bounded when P n2Z jx n j converges. Given two time series f (n) and g(n), we de…ne the convolution of the two time series as h(n) = (f g)(n) = X k2Z f (n k) g (k) 2. Discrete time Fourier transform Given the realization of a process (x n) n2Z 2 l 2 , we introduce the discrete time Fourier transform (DTFT), indicated either by the notation b x (!) or F [x] (!) and de…ned by b x (!) = F [x] (!) = 1 p 2 X n2Z e x n ; ! 2 [0; 2 : Note that the symbol b is used to indicate both DTFT and empirical estimates of process parame-ters; the variable ! is used to indicate both the independent variable of the DTFT and the outcomes (elementary events) in a set of events. Which of the two meanings is the correct one will be always evident in both cases. The sequence x n can be reconstructed from its DTFT by means of the inverse Fourier transform x n = 1 p 2 Z 2