Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 38

2 cos ω t
c

modulated real
QAM signal
x ( t )

n

s ( n ) p ( t − nT )
c

lowpass
filter

x

baseband PAM
signals

ω

σ

lowpass
filter

x

n

−

s ( n ) p ( t − nT )
s

2 sin ω t
c

Figure 2.26 . Demodulation of a real QAM signal x ( t ) to obtain the baseband PAM

components n s c ( n ) p ( t − nT )and − n s s ( n ) p ( t − nT ).

Thus the complex QAM symbol can be extracted from the real QAM signal
x ( t ) by using the structure of Fig. 2.26 at the receiver. In practice, only a noisy
and distorted version of x ( t ) is received at the receiver, because of the effects of
the channel. So, the outputs of Fig. 2.26 are also distorted. The symbols s c ( n )
and s s ( n ) can be extracted from these versions with certain predictable error
probabilities, as explained in Sec. 2.3.2.

Complex baseband equivalent. As mentioned at the beginning of Sec. 2.4.3, the
acutal QAM signal x ( t ) is real. When a real bandpass signal is transmitted over
a real channel, it is often convenient to represent this with an equivalent lowpass
signal and a lowpass channel. Such equivalent signal and equivalent channel are,
in general, complex. Appendix 2.A at the end of this chapter describes this in
detail. This complex baseband equivalent is commonly used in all discussions and
is in fact very useful when we discuss digital implementations of the transmitter
and receiver.

2.5 Matched filtering

We now review the concept of matched filtering, which lies at the heart of receiver
designs in digital communication systems. Matched filters were first introduced
by North [1963], and have since been used widely in communication and radar
systems. The basic idea will be described here, and some deeper details will be
postponed to Chap. 5 (Sec. 5.2).