Fourier Series and Karhunen-Loeve Expansions
A process x(t) is MS periodic with period T if for all t. A Wide sense stationary (WSS) process is MS periodic if its autocorrelation R(τ) is periodic with period T = 2π/ω0. Expanding R(r) into a Fourier series, we obtain
where (1)
Given a WSS periodic process x(t) with period T. we form the sum
where (2)
This sum equals x(t) in the MS sense:
(3)
Furthermore, the random variables cn are uncorrelated with zero mean for n ≠ 0, and their variance equals γn:
and (4)
The Karhunen-Loeve Expansion
The Fourier series is a special case of the expansion of a process x(t) into a series of the form
(5)
For this purpose, we introduce a random variable x with uniform distribution in the interval (0, 1) and we form the random variable y = g(x). As we know,
(6)
where φn(t) is a set of orthonormal functions in the interval (0, T);
(7)
and the coefficients cn are random variables given by
(8)
In this development, we consider the problem of determining a set of ortbononnal functions φn(t) such that: (a) the sum in Eq.9 equals x(t); (b) the coefficients cn are orthogonal. To solve this problem, we form the integral equation
(9)
where R(t1,t2) is the autocorrelation of the process x(t). It is well known from the theory of integral equations that the eigenfunctions cn(t) of Eq.12 are orthonormal as in Eq.10 and they satisfy the identity
(10)
The sum in Eq.9 is called the Karhunen-Loeve (K-L) expansion of the process x(t). In this expansion, x(t) need not be stationary. If it is stationary then the origin can be chosen arbitrarily. For further reading on theoretical concepts the reader is referred to Athanasios Papoulis, S. Unnikrishna Pillai - Probability, Random Variables and Stochastic Processes-McGraw-Hill Europe (2002).