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

530 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
of luck. On the other hand, Gram-Schmidt orthogonalization is a systematic approach to
extract the basis signals from the known signal set. The details of this approach are given in
Appendix C.
10.5 VECTOR DECOMPOSITION OF WHITE NOISE
RANDOM PROCESSES
In digital communications, the message signal is always one of the M possible wavefonns. It
is therefore not difficult to represent all M waveforms via a set of CON basis functions. The
real challenge, in fact, lies in the vector decomposition of the random noise n(t) at the receiver.
A deterministic signal can be represented by one vector, a point in a signal space. Is it possible
to represent a random process as a vector of random variables? If the answer is positive, then
the detection problem can be significantly simplified.
Consider a complete orthonormal (CON) set of basis functions { <f!k (t)} for a signal space
defined over [0, T 0 ]. Then any deterministic signal s(t) in this signal space will satisfy the
following condition:
(10.56a)
This implies that for t E [0, T 0 ], we have the equality*
s(t) = I>k<f!k (t)
k
However, for random processes defined over [0, T 0 ], this statement is generally not true.
Certain modifications are necessary.
10.5. 1 Determining Basis Functions for a Random Process
First of all, a general random process x(t) cannot strictly satisfy Eq. (10.56a). Instead, a proper
convergence requirement is in the mean square sense, that is,
E {
f lx(t) -
x,c, (t{ dt } = O
(10.56b)
This equality can be denoted as
x(t) s . I >k<f!k (t)
k
(10.56c)
If x(t) and y(t) are equal in the mean square sense, then physically the difference between
these two random processes have zero energy. As far as we are concerned in communications,
signals ( or signal differences) with zero energy have no physical effect and can be viewed as 0.
* Strictly speaking. this equality is true not for the entire interval [O, T0]. The set of points for which equality does
not hold is a measure zero set.