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

10. 9 Nonwhite (Colored) Channel Noise 577
Substituting Eq. (10.128) into Eq. (10.129) and observing that the set <pi is orthonormal, we
have
(10.130)
Hence, for the same performance ( error probability), the mean energy of the simplex signal set
is l - l / M times that of the orthogonal signal set. For M » 1, the difference is not significant.
For this reason and because of the simplicity in generating, orthogonal signals, rather than
simplex signals are used in practice whenever M exceeds 4 or 5.
Tn Sec. 13.6, we shall show that in the limit as M -+ oo, the orthogonal (as well as the
simplex) signals attain the upper bound of performance predicted by Shannon's theorem.
10. 9 NONWHITE (COLORED) CHANNEL NOISE
Thus far we have restricted our analysis exclusively to white Gaussian channel noise. Our
analysis can be extended to nonwhite, or colored, Gaussian channel noise. To proceed,
the Karhunen-Loeve expansion of Eq. (10.57) must be solved for the colored noise with
autocorrelation function R x (t, ti). This general solution, however, can be quite complex to
implement. 4
Fortunately, for a large class of colored Gaussian noises, the power spectral density Sn (f) is
nonzero within the message signal bandwidth B. This property provides an effective alternative.
We use a noise-whitening filter H (f) at the input of the receiver, where
H (f) = __ l_e-J2:n:ftd
Jsn (f)
The delay tc1 is introduced to ensure that the whitening filter is causal (realizable).
Consider a signal set {si (t)} and a channel noise n(t) that is not white [Sn (f) is not constant].
At the input of the receiver, we use a noise-whitening filter H (f) that transforms the colored
noise into white noise (Fig. 10.34). But it also alters the signal set {si (t)} to {s;(t)}, where
s; (t) = s;(t) * h(t)
We now have a new signal set {s; (t)} mixed with white Gaussian noise, for which the optimum
receiver and the corresponding error probability can be determined by the method discussed
earlier.
Figure 10.34
Optimum Mary
receiver for
nonwhite
channel noise.
n(I)
/s1 (t) ) !
Noise-whitening Optimum receiver
filter
for
S; (t) + H(f)
{s';
n(t)
s '; (t) + n,Jt)
(t) }in white noise
Decision