B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

10.5 Vector Decomposition of White Noise Random Processes 531
For a set of deterministic signals, the basis signals can be derived via the Gram-Schmidt
orthogonalization procedure. However, Gram-Schmidt is invalid for random processes.
Indeed, a random process x(t) is an ensemble of signals. Thus, the basis signals {<Pk (t)} must
also depend on the characteristics of the random process.
The full and rigorous description of the decomposition of a random process can be found
in some classic references. 4 Here, it suffices to state that the orthonormal basis functions must
be solutions of the following integral equation
(10.57)
The solution Eq. (10.57) is known as the Karhunen-Loeve expansion. The auto-correlation
function R x (t, t1) is known as its kernel function. Indeed, Eq. (10.57) is reminiscent of the
linear algebra equation with respect to eigenvalue A and eigenvector cp:
in which </> is a column vector and R x is a positive semidefinite matrix; A; are known as the
eigenvalues, whereas the basis functions <Pi (t) are the corresponding eigenfunctions.
The Karhunen-Loeve expansion clearly establishes that the basis functions of a random
process x(t) depend on its autocorrelation function R x (t, t1 ). We cannot arbitrarily select a
CON function set. In fact, solving the Karhunen-Loeve expansion can be a nontrivial task.
10.5.2 Geometrical Representation of
White Noise Processes
For a stationary white noise process x(t), the autocorrelation function is luckily
For this special kernel, the integral equation Eq. (10.57) is reduced to a simple form of
fn To N
N
A; · q>;(t) = -o(t - t1 ) · q>;(t1) dt1 = - q>;(t) t E (0, T 0 ) (10.58)
o 2 2
This result implies that any CON set of basis functions can be used to represent stationary
white noise processes. Additionally, the eigenvalues are identically A; = N /2.
This particular result is of utmost importance to us. In most digital communication applications,
we focus on the optimum receiver design and performance analysis under white noise
channels. In the case of M-ary transmissions, we have an orthonormal set of basis functions
{q>k (t)} to represent the M waveforms {s; (t) }, such that
s;(t) = L si,k<Pk (t)
k
i = I, ... , M
(10.59a)
Based on Eq. (10.58), these basis functions are also suitable for the representation of the white
channel noise nw(t) such that
nw(t) s . L nkcpk (t)
k
(10.59b)