Page 135 - Fundamentals of Communications Systems

Complex Baseband Representation of Bandpass Signals 4.11

This can be rearranged to give

X I (f − f c ) + jX Q (f − f c ) X I (f + f c ) − jX Q (f + f c )
X c (f ) = √ + √ (4.13)
2 2
Using Eq. (4.11) in Eq. (4.13) gives

1 1
∗
X c (f ) = √ X z (f − f c ) + √ X (− f − f c ) (4.14)
z
2 2
This is a very fundamental result. Equation (4.14) states that the Fourier
transform of a bandpass signal is simply derived from the spectrum of the
complex envelope. For positive values of f , X c (f ) is obtained by translating
√
X z (f )to f c and scaling the amplitude by 1/ 2. For negative values of f , X c (f )
is obtained by flipping X z (f ) around the origin, taking the complex conjugate,
√
translating the result to − f c , and scaling the amplitude by 1/ 2 . This also
demonstrates that if X c (f ) only takes values when the absolute value of f is in
[ f c − B T /2, f c + B T /2], then X z (f ) only takes values in [−B T /2, B T /2]. The
energy spectrum of x c (t) can also be expressed in terms of the energy spectrum
of x z (t)as
1 1
(f ) = (f − f c ) + (− f − f c ) (4.15)
G x c G x z G x z
2 2
∞ ∞

= (f ) df = (f ) df, Eq. (4.15) guarantees that the en-
Since E x c G x c G x z
−∞ −∞
ergy of the complex envelope is identical to the energy of the bandpass signal.
Additionally, Eq. (4.15) guarantees that the energy spectrum of the bandpass
signal is an even function of frequency as it should be for a real signal. Consid-
ering these results, the spectrum of the complex envelope of the signal shown in
Figure 4.1 will have a form shown in Figure 4.9 when f c = f C . Other values of
f c would produce a different but equivalent complex envelope representation.
This discussion of the spectral characteristics of x c (t) and x z (t) should reinforce
the idea that the complex envelope contains all the information in a bandpass
waveform.

G ( f )
x
z

B
T
Figure 4.9 The complex envelope
energy spectrum of the bandpass
f signal in Figure 4.1 with f c = f C .