Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 60

Appendix to Chapter 2

2.A The baseband signal and the passband signal

The basic message x ( t ) transmitted by a communication system is usually a
lowpass signal called the baseband signal (Fig. 2.43(a)). When this is modulated
with a carrier the result becomes a bandpass signal. For example, the PAM/DSB
signal has the form x ( t )cos ω c t , and its Fourier transform magnitude is as in Fig.
2.43(b).

baseband signal

(a)

ω

σ

−σ

0

(b)

bandpass signal
ω

−ω c

0

ω c

Figure 2.43 . Fourier transform magnitudes of (a) a baseband signal, and (b) the
corresponding bandpass signal obtained by modulation.

We always assume σ<ω c so that each copy of the baseband signal shown in
Fig. 2.43(b) is confined entirely to one side of the frequency axis. Note that if
x ( t ) is real, its Fourier transform has Hermitian symmetry around ω = 0 (i.e.,
X ( jω )= X ∗ (

jω )) so the Fourier transform of x ( t )cos ω c t has such symmetry in
the neighborhood of ω c (and

−

ω c ). Imagine such a signal is communicated over
a channel with some frequency response H ( jω ). Assuming the channel impulse
repsonse is real, H ( jω ) has Hermitian symmetry around ω =0 , but of course,
not around ω c in general (see Fig. 2.44(b)). So, the channel output, which is still
bandpass, does not usually have Hermitian symmetry in the neighborhood of ω c
(Fig. 2.44(c)), even if the channel input does. So, in communication theory, a
bandpass signal typically signifies a real signal with passbands centered around
±

−

ω c and with σ<ω c , but the passbands do not necessarily exhibit any symmetry
around ω c or ω c . 13

−

13 Another example is a signal of the form x ( t ) sin ω c t (which is one of the two terms in
the QAM signal x ( t ) cos ω c t − y ( t ) sin ω c t ). This has Fourier transform − 0 . 5 jX ( j ( ω − ω c )) +
0 . 5 jX ( j ( ω + ω c )) . So the copy of the Fourier transform around ω c is phase-shifted by −π/ 2,
which means the phase response has no antisymmetry around ω c (and similarly around −ω c ).
So the Hermitian symmetry of X ( jω ) (around ω = 0) is not present in the modulated version
(around ω = ω c or ω = −ω c ).