Digital Signal Processing Reference
In-Depth Information
channel input
ω
(a)
−ω
c
0
ω
c
H
(
j
ω)
portion seen by receiver
channel
(b)
ω
−ω
c
0
ω
c
channel output
(c)
ω
−ω
c
0
ω
c
Figure 2.44
. Examples of Fourier transforms of (a) the channel input, (b) the channel,
and (c) channel output which does not have Hermitian symmetry in the neighborhood
ω
c
or
−ω
c
.
2σ
X
(
j
ω
)
(a)
bp
ω
−ω
c
0
ω
c
X
(
j
ω)
lp
(b)
ω
−σ
σ
0
Figure 2.45
. Fourier transforms of (a) a
bandpass
signal, and (b) the corresponding
(lowpass)
baseband
signal.
Given any real bandpass signal
x
bp
(
t
)with
ω
c
>σ
as in Fig. 2.45(a), consider
the lowpass signal
x
lp
(
t
) constructed as shown in Fig. 2.45(b). Thus, the band
around
ω
c
is moved to zero frequency, and the band around
ω
c
is discarded.
We say that
x
lp
(
t
) is the (lowpass)
baseband equivalent
of
x
bp
(
t
) (with respect
to carrier frequency
ω
c
). Similarly,
x
bp
(
t
)isthe
bandpass equivalent
of
x
lp
(
t
)
,
and can be expressed analytically as
x
bp
(
t
)=Re
x
lp
(
t
)
e
jω
c
t
.
−
(2
.
93)
Sometimes a factor of 2 is included on the right-hand side, but we shall keep
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