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

360 PRINCIPLES OF DIGITAL DATA TRANSMISSION
Figure 7.22
Zero-forcing
equalizer
analysis.
(a)
p,(1)
-37i,
a_3
(b)
a,
1-
P 0 (t )
(N- l ) 7i, NTb (N+ l ) 7i,
(c)
filter equalizers are easily adjustable to compensate against different channels or even slowly
time-varying channels. The design goal is to force the equalizer output pulse to have zero ISi
values at the sampling (decision-making) instants. In other words, the equalizer output pulses
satisfy the Nyquist or the controlled ISi criterion. The time delay T between successive taps
is chosen to be T b , the interval between pulses.
To begin, set the tap gains co = 1 and q = 0 for all other values of k in the transversal filter
in Fig. 7.22a. Thus the output of the filter will be the same as the input delayed by NT b . For a
single pulse Pr (t) (Fig. 7 .22b) at the input of the transversal filter with the tap setting just given,
the filter output p 0 (t) will be exactly Pr Ct - NT b ), that is, p ,. (t) delayed by NT b . This delay has
no practical effect on our communication system and is not relevant to our discussion. Hence,
for convenience, we shall ignore this delay. This means that Pr (t) in Fig. 7 .22b also represents
the filter output p 0 (t) for this tap setting (co = 1 and q = 0, k I- 0). We require that the
output pulse p 0 (t) satisfy the Nyquist's criterion or the controlled ISi criterion, as the case may
be. For the Nyquist criterion, the output pulse p 0 (t) must have zero values at all the multiples
of T b . From Fig. 7 .22b, we see that the pulse amplitudes a1 , a_ 1 , and a2 at T b , -T b , and 2Tb,
respectively, are not negligible. By adjusting the tap gains (ck), we generate additional shifted
pulses of proper amplitudes that will force the resulting output pulse to have desired values at
t = 0, ±T b , ±2T b , ....
The output p 0 (t) (Fig 7.22c) is the sum of pulses of the form CkPr(t - kT b ) (ignoring the
delay of NT b ). Thus
N
Po (t) = L Cnp,(t - nT b )
n=-N
(7.50)