Page 155 - Fundamentals of Communications Systems

Complex Baseband Representation of Bandpass Signals 4.31

Then

f
2
2
(f ) =|X I (f )| = A rect
G X I
2 f 1
2
A f 2 f
2
(f ) =|X Q (f )| = rect
G X Q 2
f 2 f 1
1
2
jA f f
∗
X I (f )X (f ) =− rect
Q
f 1 2 f 1
2
A f f
! "
∗
X I (f )X (f ) =− rect
Q
f 1 2 f 1
Hence
(f ) + 2 X I (f )X (f )
∗
Q
G X z (f ) = G X I (f ) + G X Q

f f f f f
2
2
= A rect + rect − 2 rect
2 f 1 f 1 2 f 1 f 1 2 f 1
f f
2
(f ) = f − rect
G X z
f 1 2 f 1
Problem 4.11. x I (t) is periodic with period T . Because of this x I (t) can be repre-
sented in a Fourier series expansion
∞
j 2πkt
x I (t) = x k exp (4.60)
T
k=−∞
If the bandwidth of the signal is less than WHz then the Fourier series will be
truncated to a finite summation. Define k m to be the largest integer such that
k m /T ≤ W then
k m
j 2πkt
x I (t) = x k exp (4.61)
T
k=−k m
The bandpass signal will have the form
√
x c (t) = x I (t) 2 cos(2π f c t)
1 1

= x I (t) √ exp( j 2π f c t) + √ exp(− j 2π f c t) (4.62)
2 2
k m
1 k k
= √ x k exp j 2π + f c t + exp j 2π − f c t
2 T T
k=−k m
Since the bandpass signal has a representation as a sum of weighted
sinusoids there is a possibility that the bandpass signal will be periodic.