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

7.11 KARHUNEN–LOÈVE EXPANSION

In the final section of this chapter, we return to continuous-time random processes and describe a decomposition that is a weighted sum of orthonormal functions known as the Karhunen–Loève expansion (KLE).

Theorem 7.21 (Karhunen-Loève). Let X(t) be a zero-mean random process with autocorrelation function RXX(t1, t2) defined on the finite interval . It can be decomposed as follows:

(7.256)

where {Xn} are random variables generated as follows:

(7.257)

and are orthonormal eigenfunctions that satisfy

(7.258)

with eigenvalues .

The sum in (7.256) and the integral in (7.257) are defined in the MS sense, whereas the last integral is over the product of deterministic functions. The autocorrelation function in (7.258) is also called the kernel of the integral equation.

Proof. Let the sum in (7.256) be finite with N elements so that it is a sequence. For MS convergence, ...