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

12.2 Receiver Channel Equalization 671
Figure 12.3
Optimal matched
filter receiver.
y(t)
Matched filter z(t) z[n]
q(-t)
t=nT
where
h(t) = q(t) * q(-t)
(12.11)
If we denote the samples of h(t)
h[n] = h(nT)
then Eq. (12.10) can be simplified
z[n] = L Skh[n - k] = h[n] * s[n]
k
(12.12)
In short, the channel (input-output) signals are related by a single-input-single-output (SISO)
linear discrete channel with transfer function
H(z) = L h[n] z - n (12.13)
n
The SISO discrete representation of the linear QAM signal leads to the standard T-spaced
equalizer (TSE). The term T-spaced equalization refer to processing of the received signal
sampled at the rate of 1/T. Therefore, the time separation between successive samples equals
the baud (symbol) period T.
The optimal matched filter receiver faces a major practical obstacle that the total pulse
shape response q(t) depends on the multipath channel environment. In reality, it is practically
difficult to adjust the receiver filter according to the time-varying q(t) because channel environment
may undergo significant and possibly rapid changes. Moreover, the receivers generally
do not have a priori information on the channel that affects q(t). As a result, it does not make
sense to implement the optimum receiver filter q(-t) in a dynamic channel environment. It
makes better sense to design and implement a time-invariant receiver filter. Therefore, the
important task is to select a receiver filter without losing any signal information in y(t) .
To find a solution, recall the QAM channel input signal
x(t) = L Skp(t - kT)
k
We have learned from Section 7 .2 [ see Eq. (7. 9)] that the power spectral density of an amplitudemodulated
pulse train is
(12. 14a)
(12. 14b)
by simply substituting the pulse amplitude ak with the QAM symbol Sk , The signal spectrum
of Eq. (12.14a) shows that the signal component in y(t) is limited by the bandwidth of p(t)
or P(f).