ABSTRACT
Wide sense stationary randomprocesses are often characterized by their first-and second-ordermoments, that is by the mean value
m = Ex(t) (60.1) and the covariance function
r(τ) Δ= E[x(t + τ)−m][x(t)−m], (60.2)
where t, τ take integer values 0,±1± 2, . . .. For a wide sense stationary processes the expected values in Equations 60.1 and 60.2 are independent of t. As an alternative to the covariance function one can use its discrete Fourier transform, that is, the spectrum,
φ(z) = ∞∑
n=−∞ r(n)z−n. (60.3)
Evaluated on the unit circle it is called the spectral density,
φ(eiω) = ∞∑
n=−∞ r(n)e−inω. (60.4)
As
r(τ) = 1 2π
φ(eiω)eiτωdω = 1 2πi
∮ φ(z)zτ
dz
z (60.5)
(where the last integration is counterclockwise around the unit circle) the spectral density describes how the energy of the signal is distributed over different frequency bands (set τ = 0 in Equation 60.5).
