Digital Signal Processing Reference
In-Depth Information
x
(
t
)
c
x
(
n
)
d
y
(
t
)
c
x
(
t
)
c
y
(
n
)
y
(
n
)
j
ω
j
ω
T
P
(
e
)
P
(
e
)
C/D
T
C/D
T
Figure G.1
. The noble identity for C/D converters.
x
(
t
)
c
y
(
t
)
c
x
(
n
)
y
(
t
)
c
y
(
n
)
x
(
n
)
j
ω
T
j
ω
P
(
e
)
P
(
e
)
D/C
T
D/C
T
Figure G.2
. The noble identity for D/C converters.
Note that the first noble identity can also be written in the form of an equation,
namely
P
(
e
jω
)
X
c
(
jω
)
=
X
c
(
jω
)
P
(
e
jωT
)
.
(
G.
6)
↓T
↓T
G.3 The generalized alias-free(T) band
We know that if a signal
x
(
t
) is bandlimited to the interval
σ<ω<σ
,thenit
can be sampled at the rate
ω
s
=2
σ
without aliasing. This is because the terms
in
−
∞
X
s
(
jω
)=
1
T
X
(
j
(
ω
+
kω
s
))
k
=
−∞
do not overlap for any two values of
k.
Equivalently, for any fixed
ω
,wedonot
have more than one nonzero term in the summation. We say that the interval
−
σ<ω<σ
represents an alias-free(
T
) band for the sampling rate
ω
s
.Here
T
represents the sample spacing
T
=
2
π
ω
s
as usual. The lowpass band
σ<ω<σ
shown in Fig. G.3(a) is not the only
possible band of this kind. For example, the bandpass region
−
−
σ
+
c<ω<σ
+
c
is a valid alias-free(
T
) band for any constant
c.
See Fig. G.3(b). A third example
is shown in part (c) of the figure, where the band of length 2
σ
has been split
into two halves. If the individual halves are appropriately positioned, then the
copies of the left half shifted in multiples of 2
σ
do not overlap with similar shifted
copies of the right half. This happens if and only if
σ
1
(equivalently
σ
2
)isa
multiple of
σ
. In this case the combination of the two bands is an alias-free(
T
)
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