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Baseband Pulse and Digital Signaling Chap. 3
the filters easier to build. The Ke -jvT d
factor(s) would not affect the zero ISI result, but, of
course, would modify the level and delay of the output waveform.
Nyquist’s First Method (Zero ISI)
Nyquist’s first method for eliminating ISI is to use an equivalent transfer function, H e (f), such
that the impulse response satisfies the condition
C, k = 0
h e (kT s + t) = u
0, k Z 0
(3–66)
where k is an integer, T s is the symbol (sample) clocking period, t is the offset in the
receiver sampling clock times compared with the clock times of the input symbols, and
C is a nonzero constant. That is, for a single flat-top pulse of level a present at the input to
the transmitting filter at t = 0, the received pulse would be ah e (t). It would have a value of
aC at t = t but would not cause interference at other sampling times because h e (kT s + t) = 0
for k Z 0.
Now suppose that we choose a (sin x)x function for h e (t). In particular, let t = 0, and
choose
h e (t) = sin pf st
pf s t
(3–67)
where f s = 1T s . This impulse response satisfies Nyquist’s first criterion for zero ISI,
Eq. (3–66). Consequently, if the transmit and receive filters are designed so that the overall
transfer function is
H e (f) = 1 f s
ßa f f s
b
(3–68)
there will be no ISI. Furthermore, the absolute bandwidth of this transfer function is B = f s 2.
From our study of the sampling theorem and the dimensionality theorem in Chapter 2 and Sec.
3–4, we realize that this is the optimum filtering to produce a minimum-bandwidth system. It
will allow signaling at a baud rate of D = 1T s = 2B pulses/s, where B is the absolute bandwidth
of the system. However, the (sin x)x type of overall pulse shape has two practical difficulties:
• The overall amplitude transfer characteristic H e ( f) has to be flat over -B 6 f 6 B and
zero elsewhere. This is physically unrealizable (i.e., the impulse response would be
noncausal and of infinite duration). H e (f) is difficult to approximate because of the steep
skirts in the filter transfer function at f = ; B.
• The synchronization of the clock in the decoding sampling circuit has to be almost perfect,
since the (sin x)x pulse decays only as 1x and is zero in adjacent time slots only
when t is at the exactly correct sampling time. Thus, inaccurate sync will cause ISI.
Because of these difficulties, we are forced to consider other pulse shapes that have a
slightly wider bandwidth. The idea is to find pulse shapes that go through zero at adjacent