563489578934

Sec. 4–4 Evaluation of Power 247
where, because m(t) is real, M*( f ) = M(-f ) and d ( f ) = d (-f ) (the delta function was defined to be
even) were used. Suppose that the magnitude spectrum of the modulation happens to be a triangular
function, as shown in Fig. 4–2a. This spectrum might arise from an analog audio source in which
the bass frequencies are emphasized. The resulting AM spectrum, using Eq. (4–20a), is shown in
Fig. 4–2b. Note that because G(f - f c ) and G * (-f - f c ) do not overlap, the magnitude spectrum is
ƒ S(f) ƒ = e
1
2 A cd(f-f c ) + 1 2 A c |M(f-f c )|, f 7 0
1
2 A cd(f+f c ) + 1 2 A c |M(-f-f c )|, f 6 0
(4–20b)
The 1 in g(t) = A c [1 + m(t)] causes delta functions to occur in the spectrum at f = ; f c , where f c
is the assigned carrier frequency. Using Eq. (4–17), we obtain the total average signal power
P s = 1 2 A c 2 8|1 + m(t)| 2 9 = 1 2 A c 2 81 + 2m(t) + m 2 (t)9
= 1 2 A c 2 [1 + 28m(t)9 + 8m 2 (t)9]
If we assume that the DC value of the modulation is zero, as shown in Fig. 4–2a, the average signal
power becomes
P s = 1 2 A c 2 [1 + P m ]
(4–21)
1
2 A c 2
where P is the power in the modulation m(t), is the carrier power, and 2 A c 2 m = 8m 2 (t)9
P m is
the power in the sidebands of s(t).
1
|M(f)|
1.0
–B B
f
(a) Magnitude Spectrum of Modulation
1
2
|S(f)|
Discrete carrier term
1
with weight = – A
2 c
Weight = – A c
A c
Lower
2
sideband
Upper
sideband
–f c -B – f c –f c +B f c -B f c f c +B
(b) Magnitude Spectrum of AM Signal
Figure 4–2
Spectrum of AM signal.
f