ABSTRACT
Let X(t) be a stationary stochastic process that is periodic with period T . Because X(t) is periodic, we find that:
RXX(τ + T ) = E(X(t)X(t+ τ + T )) = E(X(t)X(t+ τ)) = RXX(τ).
That is, the autocorrelation “inherits” periodicity from the stochastic process. As X(t) is periodic, we can expand it into its Fourier series (as long as X(t)
behaves in a “reasonable” fashion). That is:
X(t) = ∞∑
n=−∞ ane
2pijnft, f = 1 T , an =
1 T
X(t)e−2pijnft dt.
