Page 133 - Fundamentals of Communications Systems

Complex Baseband Representation of Bandpass Signals 4.9

integer, e.g., 2 cos(2πn) = 2 and sin(2πn) = 0. Likewise t = n/f m + 1/(4 f m ) corresponds
to the point x z (t) = (0, 1).

4.4 Spectral Characteristics of the Complex Envelope
4.4.1 Basics
It is of interest to derive the spectral representation of the complex baseband
signal, x z (t), and compare it to the spectral representation of the bandpass
signal, x c (t). Assuming x z (t) is an energy signal, the Fourier transform of x z (t)
is given by

X z (f ) = X I (f ) + jX Q (f ) (4.9)

where X I (f ) and X Q (f ) are the Fourier transform of x I (t) and x Q (t), respec-
tively, and the energy spectrum is given by

2 ∗
Q
G x z (f ) =|X z (f )| = G x I (f ) + G x Q (f ) + 2 [X I (f )X (f )] (4.10)
(f ) are the energy spectrum of x I (t) and x Q (t), respectively.
where G x I (f ) and G x Q
The signals x I (t) and x Q (t) are lowpass signals with a one-sided bandwidth of
(f ) can only take nonzero values
less than B T /2 so consequently X z (f ) and G x z
for | f | < B T /2.

EXAMPLE 4.3
Consider the case when x I (t) is set to be the message signal from Example 2.3 (com-
puter generated voice saying “bingo”) and x Q (t) = cos(2000πt). X I (f ) will be a lowpass
spectrum with a bandwidth of 2500 Hz while X Q (f ) will have two impulses located
at ±1000 Hz. Figure 4.8 shows the measured complex envelope energy spectrum for
these lowpass signals. The complex envelope energy spectrum has a relation to the
voice spectrum and the sinusoidal spectrum exactly as predicted in Eq. (4.10). Note
here B T = 5000 Hz.

Equation (4.9) gives a simple way to transform between the lowpass signal
spectrums to the complex envelope spectrum. A similar simple formula exists
for the opposite transformation. Note that x I (t) and x Q (t) are both real signals
so that X I (f ) and X Q (f ) are Hermitian symmetric functions of frequency. It is
straightforward to show

∗
∗
X z (− f ) = X (f ) + jX (f )
Q
I
(4.11)
∗
X (− f ) = X I (f ) − jX Q (f )
z