Examines ideas of ideal reconstruction of signals for a sampled, noisy original.
Reconstruction of sampled signals
How do we go from
x
n to CT (
)?
Step 1
Place
x
n into CT on an impulse train
s
t (
).
Step 2
Pass
x
s
t through an idea lowpass filter
H
LP
(
).
If we had no aliasing then
x
r
t
x
c
t , where
x
n
x
c
n
T .
Ideal reconstruction system
In frequency domain:
X
s
X
T where
X
T is the DTFT of
x
n at digital frequency
T .
X
r
H
LP
X
s
In time domain:
x
s
t
n
∞
∞
x
n
t
n
T
x
r
t
n
∞
∞
x
n
t
n
T
h
LP
t
h
LP
t
sinc
T
t
x
r
t
n
∞
∞
x
n
sinc
T
t
n
T
h
LP
t "interpolates" the values of
x
n to generate
x
r
t (
).
Sinc interpolator
x
r
t
n
∞
∞
x
n
T
t
n
T
T
t
n
T