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2.
Sampling and reconstruction.
Continuous time signal
discretetime signal
The sample values of x(t) are equal to the values x(nT),
n=0,
±
1,
±
2, ….
.
Represent x(t) by x(nT) – under what condition the representation is unique?
For analysis, we construct a continuoustime signal x
s
(t) from x(nT):
()
( ) (
)
(
)
() ()
s
nn
x
t
x nT
t
nT
x t
t
nT
x t p t
δδ
∞∞
=−∞
=−
=
−
=
∑∑
where p(t) is the impulse train given by
∑
∞
−∞
=
−
=
n
nT
t
t
p
)
(
)
(
δ
To determine the Fourier transform of x(t)p(t), first express p(t) using FS.
∑
∞
−∞
=
=
k
t
jk
k
s
e
c
t
p
ω
)
(
where
T
s
π
2
=
is the sampling frequency.
k
c
=
Fourier transform of p(t) is
P
=
( )
s
xt pt
X
↔=
Let X(t) be a bandlimited signal, X(
ω
) = 0 ,
for 
ω
 > B
If
ω
s
> 2B
there is no overlap between X(
ω
k
ω
s
)
Sampling
Reconstruction
0
ω
X(
ω
)
B
B
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View Full Document x(t
H
(
ω
)
x
r
(t)
∑
−
=
n
nT
t
t
p
)
(
)
(
δ
x
s
(t)=
x(t)p(t)
X
If
ω
s
< 2B
there is overlap
When there is no overlap, x(
ω
) can be recovered from X
s
(
ω
) by a lowpass
filter.
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This note was uploaded on 10/15/2010 for the course BIM BIM108 taught by Professor Qiu during the Winter '09 term at UC Davis.
 Winter '09
 Qiu

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