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

684 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
It is quite clear from comparing Eqs. (12.45) and (12.43c) that under a short training sequence
(preamble), the optimum equalizer can be obtained by replacing the exact values of the correlation
function with their time average approximations. If matrix inverse is to be avoided for
complexity reasons, adaptive channel equalization is a viable technology. Adaptive channel
equalization was first developed by Lucky at Bell Labs 4 • 5 for telephone channels. It belongs
to the field of adaptive filtering. Interested readers can refer to the book by Ding and Li 6 and
the references therein.
12.4 LINEAR FRACTIONALLY SPACED
EQUALIZERS (FSE)
We have shown that when the channel response is unknown to the receiver, TSE is likely to
lose important signal information. In fact, this point is quite clear from the sampling theory.
As shown by Gitlin and Weinstein,7 when the transmitted signal (or pulse shape) does have
spectral content beyond a frequency of 1 / (2T) Hz, baud rate sampling at the frequency of 1 /T
is below the Nyquist rate and can lead to spectral aliasing. Consequently, receiver performance
may be poor because of information loss.
In most cases, when the transmission pulse satisfies Nyquist's first criterion of zero ISi,
the received signal component is certain to possess frequency content above l/(2T) Hz. For
example, when a raised-cosine (or a root-raised-cosine) pulse PrrcU) is adopted with roll-off
factor r [Eq. (12.23)], the signal component bandwidth is
l + r
-- Hz
2T
For this reason, sampling at 1 / T will certainly cause spectral aliasing and information loss
unless we use the perfectly matched filter q( - t) and the ideal sampling moments t = kT.
Hence, the use of faster samplers has great significance. When the actual sampling period
is an integer fraction of the baud period T, the sampled signal under linear modulation can
be equivalently represented by a single-input-multiple-output (SIMO) discrete system model.
The resulting equalizers are known as the fractionally spaced equalizers (or FSE).
12.4.1 The Single-Input-Multiple-Output (SIMO) Model
An FSE can be obtained from the system in Fig. 12.6 if the channel output is sampled at a rate
faster than the baud or symbol rate 1/T. Let m be an integer such that the sampling interval
becomes fl = T /m. In general, because of the (root) raised-cosine pulse has bandwidth B:
1 1 + r 1
-<B =--<-
2T - 2T - T
Figure 12.6
Fractionally
spaced sampling
receiver front
end for FSE.
y (t)
Receiver filter
p(-t)
t = nL'.l
z(nL'.l)