Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 37

cos ω t
c

x s ( n )

c

D/C

P ( j ω )

from PAM
constellation

T

+

x ( t )

QAM signal

− 1

s ( n )
s

x

D/C

P ( j ω )

from PAM
constellation

T

sin ω t
c

Figure 2.25 . Generating the real QAM signal x ( t ) from s c ( n )and s s ( n ) , which are
the real and imaginary parts of the complex QAM symbol s c ( n )+ js s ( n ).

2.4.4 Extracting the complex QAM signal from the real version

We now explain how the components s c ( n )and s s ( n ) can be extracted from the
real QAM signal shown in Eq. (2.40). For this, refer to Fig. 2.26 where the real
QAM signal x ( t ) is multiplied with 2 cos ω c t and 2 sin ω c t separately. Using Eq.
(2.40) we see that

2 x ( t )cos ω c t
=2
n

2
n

nT )cos 2 ω c t s c ( n ) p ( t − − s s ( n ) p ( t − nT )sin ω c t cos ω c t

=
n

s c ( n ) p ( t − nT )(1 + cos 2 ω c t ) − s s ( n ) p ( t − nT )sin2 ω c t.

n

The term p ( t nT ) cos 2 ω c t has its energy mostly concentrated around the high

frequency 2 ω c (since p ( t ) is a baseband signal with bandwidth 2 σ<< 2 ω c ).
Similarly, the term p ( t−nT )sin2 ω c t has its energy mostly concentrated around
2 ω c . The lowpass filter in the figure therefore retains only the term

−

s c ( n ) p ( t − nT ) , (2 . 41)

n

which contains the real part of the QAM symbol s c ( n )+ js s ( n ) . Similarly, the
lowpass filter in the lower branch extracts the signal

s s ( n ) p ( t − nT ) , (2 . 42)

n

which contains the imaginary part of the QAM symbol s c ( n )+ js s ( n ) .