Page 154 - Fundamentals of Communications Systems

4.30 Chapter Four

2
The amplitude imbalance is denoted γ = (B/A) in practice.
/2 and find the values of A and B that achieve
(a) Assume that P x I = P x Q = P x z
.
a specified γ and P x z = P ˜x z
(b) For the transformation detailed in (a) find the output complex envelope,
˜ x z (t), when x z (t) = exp[ j 2π f m t]. Plot the signal spectrum before and after
the amplitude imbalance.

Problem 4.28. (PD) Let x c (t) be a bandpass signal with

|| f |− f a |≤ W
X 0
X c (f ) = (4.55)
0 0 elsewhere
.
(a) Find E x c
(b) Plot X z (f ) for f c = f a .Is x z (t) a real valued signal?
(c) Plot X z (f ) for f c = f a + W.Is x z (t) a real valued signal?

4.9 Example Solutions

Problem 4.2.
(a) Using sin(a − b) = sin(a) cos(b) − cos(a) sin(b) gives

x c (t) = sin(2π f c t) cos(2π f m t) − cos(2π f c t) sin(2π f m t) (4.56)

By inspection we have
−1 −1
x I (t) = √ sin(2π f m t) x Q (t) = √ cos(2π f m t) (4.57)
2 2
√
(b) Recall x c (t) = x A(t) 2 cos(2π f c t + x P (t)) so by inspection we have
1 1
x z (t) = √ exp( j 2π f m t) x I (t) = √ cos(2π f m t)
2 2
1
x Q (t) = √ sin(2π f m t) (4.58)
2
√
(c) Recall x c (t) = x A(t) 2 cos(2π f c t + x P (t)) so by inspection we have
1 1 1
x z (t) = √ exp( j φ p ) x I (t) = √ cos(φ p ) x Q (t) = √ sin(φ p ) (4.59)
2 2 2
Problem 4.6. We can write
f Af f

X I (f ) = Arect and X Q (f ) = j rect
2 f 1 f 1 2 f 1