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

670 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
Fi g ure 12.2
Baseband
representation of
QAM transmission
over a
linear timeinvariant
channel
with ISi.
Input
-----
L Sk o(t - kT + to)
LTI channel
q(t)
Output
+l----►
y(t)
nc (t) Noise
12.2 RECEIVER CHANNEL EQUALIZATION
It is convenient for us to describe the problem of channel equalization in the stationary channel
case. Once the fundamentals of linear time-invariant (LTI) channel equalization is understood,
adaptive technology can handle time-varying channels.
When the channel is LTI, we use the simple system diagram of Fig. 12.2 to describe the
problem of channel equalization. In general, channel equalization is studied for the (spectrally
efficient) digital QAM systems. The baseband model for a typical QAM (quadrature amplitude
modulated) data communication system consists of an unknown LTI channel q(t), which
represents the physical interconnection between the transmitter and the receiver in baseband.
The baseband transmitter generates a sequence of complex-valued random input data {sd,
each element of which belongs to the constellation A of QAM symbols. The data sequence
{sk } is sent through the baseband channel that is LTT with impulse response q(t). Because
QAM symbols {sd are complex-valued, the baseband channel impulse response q(t) is also
complex-valued in general.
Under the causal and complex-valued LTI communication channel with impulse response
q(t), the input-output relationship of the QAM system can be written as
00
y(t) = L skq(t - kT + to) + n e (t) Sk E A
k=-oo
(12.9)
Typically the baseband channel noise ne(t) is assumed to be stationary, Gaussian, and independent
of the channel input sk , Given the received baseband signal y(t) at the receiver, the
job of the channel equalizer is to estimate the original data {sk } from the received signal y(t).
In what follows, we present the common framework within which channel equalization
is typically accomplished. Without loss of generality, we let t 0 = 0.
12.2. 1 Antialiasing Filter vs. Matched Filter
We showed in Secs. 10. 1 and I 0.6 that the optimum receiver filter should be matched to the total
response q(t). This filter serves to maximize the SNR of the sampled signal at the filter output.
Even if the response q(t) has ISi, Forney 1 has established the optimality* of the matched filter
receiver, as shown in Fig. 12.3. With a matched filter q(-t) and symbol (baud) rate sampling
at t = nT, the receiver obtains an output sequence relationship between the transmitter data
{sk } and the receiver samples as
z[n] = L skh(nT - kT)
k
(12. 10)
* Forney proved 1 that sufficient statistics for input symbol estimation is retained by baud rate sampling at t = nT of
matched f i lter output signal. This result forms the basis of the well-known single-input-single-output (SISO) system
model obtained by matched filter sampling. However, when q(t) is unknown, the optimality no longer applies.