Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

8.4 EIGENFUNCTIONS OF RXX(τ)

Consider the integral equation for the Karhunen-Loève expansion (KLE) in (7.258), but with the finite limits [0, T] extended to :

(8.54)

where are eigenfunctions and {λ n} are eigenvalues. Since X(t) is wide-sense stationary, the argument of the autocorrelation function depends on the difference of the two time instants. As a result, this integral has the form of a convolution, corresponding to the LTI system in Figure 8.1, given by

(8.55)

For an LTI system, complex exponentials are eigenfunctions:

(8.56)

where we have changed variables to v = t−τ, and

(8.57)

is the frequency response of h(t) evaluated at the particular frequency ω n. This is a fundamental result covered in basic courses on signals and systems: the output has the same frequency as the input, and only its magnitude and phase are possibly different, as determined by H(ω n). Since the autocorrelation ...