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−
2
π
W
2
π
W
σ
0
ω
F I GU R E 12 . 10
Fourier transform of the sampled function.
−
2
π
W
2
π
W
ω
−
2
π
W
2
π
W
ω
F I GU R E 12 . 11
Effect of sampling at a rate less than 2W samples per second.
−
2
π
W
2
π
W
ω
F I GU R E 12 . 12
Aliased reconstruction.
Thus, the Fourier transform of
f
S
(
t
)
is
n
=−∞
δ(w
−
F
S
(ω)
=
F
(ω)
⊗
n
σ
0
)
(53)
n
=−∞
=
(ω)
⊗
δ(w
−
σ
0
)
(54)
F
n
n
=−∞
=
F
(ω
−
n
σ
0
)
(55)
where the last equality is due to the sifting property of the delta function.
Pictorially, for
F
(ω)
as shown in Figure
12.8
,
F
S
(ω)
is shown in Figure
12.10
. Note that
1
if
T
is less than
2
W
,σ
0
is greater than 4
π
W
, and as long as
σ
0
is greater than 4
π
W
, we can
recover
F
(ω)
by passing
F
S
(ω)
through an ideal low-pass filter with bandwidth
W
Hz (2
π
W
radians).
