Page 221 - Fundamentals of Communications Systems

Amplitude Modulation 6.43

The frequencies contained in the bandpass signal are f = 194, 198, 202,
206 so the bandpass signal will be periodic with period T = 1/2. Since

∞

x c (t) = x k exp[ j 2π2kt] (6.52)
k=−∞
the Fourier series coefficients are

8A c 8A c 8A c 8A c
x 97 = √ x −97 = √ x 99 = √ x −99 = √ (6.53)
9π 2 2 9π 2 2 π 2 2 π 2 2
8A c 8A c 8A c 8A c
x 101 = √ x −101 = √ x 103 = √ x −103 = √ (6.54)
π 2 2 π 2 2 9π 2 2 9π 2 2
2
2
= A P m = A 0.3325. If a transmitted power
c c
(c) The power of DSB-AM is P x z
2
of 50 W is to be achieved then A = 50/.3325 or A c = 12.26.
c
(d) See Figure 6.6.
Problem 6.23. For SSB-AM we have
x z (t) = x I (t) + jx Q (t) = A c [m(t) + jh Q (t) ∗ m(t)] (6.55)
where h Q (t) is the Hilbert transformer where H Q (f ) =− j sgn(f ).
(a) Since
8 8
m(t) = cos(2π(2)t) + cos(2π(6)t) (6.56)
π 2 9π 2
we have
8 8

x I (t) = A c 2 cos(2π(2)t) + 2 cos(2π(6)t) (6.57)
π 9π
The complex envelope of the transmitted bandpass signal is given as

x z (t) = x I (t) + jx Q (t)
8 8 j 8

= A c 2 cos(2π(2)t) + 2 cos(2π(6)t) + 2 sin(2π(2)t)
π 9π π
j 8
+ sin(2π(6)t)
9π 2
8 8

= A c 2 exp( j 2π(2)t) + 2 exp( j 2π(6)t) (6.58)
π 9π
The frequency domain representation of the complex envelope is
A c 8 A c 8
X z (f ) = 2 δ( f − 2) + 2 δ( f − 6) (6.59)
π 9π
The complex envelope is periodic with period T = 1/2.