Page 181 - Fundamentals of Communications Systems

Amplitude Modulation 6.3

This implies E B = 50%. Because of this characteristic the modulation is often
known as double sideband-amplitude modulation (DSB-AM). An efficiency of
50% is wasteful of the precious spectral resources but obviously the simplicity
of the modulator is a positive attribute.
The power of a DSB-AM modulation is often of interest (e.g., to specify the
characteristics of amplifiers or to calculate the received signal-to-noise ratio).
To this end the power is given as
1 T m /2 2
= lim x (t)dt
c
P x c = P r c
T m →∞ T m −T m /2
1 T m /2
2
2
2
2
= lim A m (t)2 cos (2π f c t)dt = P m A = P x z (6.3)
c
c
T m →∞ T m −T m /2
For a DSB-AM modulated waveform the output power is usually given as the
2
product of the power associated with the carrier amplitude (A ) and the power
c
in the message signal (P m ).
EXAMPLE 6.1
Linear modulation with
β 2 β 2
m(t) = β sin(2π f m t) G m (f ) = δ( f − f m ) + δ( f + f m )
4 4
produces
√
x c (t) = A c β sin(2π f m t) 2 cos(2π f c t)
and
2 2
A β
c
G x c (f ) = (δ( f − f m − f c ) + δ( f + f m − f c ) + δ( f − f m + f c ) + δ( f + f m + f c ))
8
The output power is
2 2
A β
c
=
P x c
2
The mesage signal and the output modulated time domain signal are plotted in
√
Figure 6.3(a) for f c = 20 f m and A c = 1/ 2. Note that both the message signal and
the modulated signal are periodic with a period of T = 1/f m . The plot of the energy
spectrum of the message and the modulated signals are plotted in Figure 6.3(b) for
√
= 0.25.
f c = 20 f m , β = 1, and A c = 1/ 2. It should be noted that P x z = P r z
EXAMPLE 6.2
The computer generated voice signal given in Chapter 2 (W = 2.5 kHz) is used to DSB-
AM modulate a 7-kHz carrier. A short time record of the message signal and the resulting
modulated output signal is shown in Figure 6.4(a). The energy spectrum of the signal is
shown in Figure 6.4(b). Note the bandwidth of the carrier modulated signal is 5 kHz.