Introduction
The reconstruction process begins by taking a sampled signal,
which will be in discrete time, and performing a few
operations in order to convert them into continuous-time and,
with any luck, into an exact copy of the original signal. A
basic method used to reconstruct a
-ππ
bandlimited signal from its samples on the integer
is to do the following steps:
-
turn the sample sequence
fsn
fs
n
into an impulse train
fimpt
fimp
t
-
lowpass filter
fimpt
fimp
t
to get the reconstruction
f
~
t
f
~
t
(cutoff freq. = π)
The lowpass filter's impulse response is
gt
g
t
. The following equations allow us to reconstruct
our signal (
Figure 2),
f
~
t
f
~
t
.
f
~
t=gtfimpt=gt∑n=-∞∞fsnδt-n=
f
~
t=∑n=-∞∞fsngtδt-n=∑n=-∞∞fsngt-n
f
~
t
g
t
fimp
t
g
t
n
fs
n
δ
t
n
f
~
t
n
fs
n
g
t
δ
t
n
n
fs
n
g
t
n
(1)
Examples of Filters g
Example 1: Zero Order Hold
This type "filter" is one of the most basic types of
reconstruction filters. It simply holds the value that is
in
fsn
fs
n
for
ττ seconds.
This creates a block or step like function where each
value of the pulse in
fsn
fs
n
is simply dragged over to the next pulse. The
equations and
illustrations
below depict how this reconstruction filter works
with the following
gg:
gt=
1if0<t<τ0otherwise
g
t
1
0
t
τ
0
fsn=∑n=-∞∞fsngt-n
fs
n
n
fs
n
g
t
n
(2)
question:
How does
f
~
t
f
~
t
reconstructed with a zero order hold compare
to the original
ft
f
t
in the frequency domain?
Example 2: Nth Order Hold
Here we will look at a few quick examples of variances to
the Zero Order Hold filter discussed in the previous
example.
Ultimate Reconstruction Filter
question:
What is the ultimate reconstruction filter?
If
Gⅈω
G
ω
has the following shape (
Figure 6):
then
f
~
t=ft
f
~
t
f
t
Therefore, an ideal lowpass filter will give us perfect
reconstruction!
In the time domain, impulse response
gt=sinπtπt
g
t
t
t
(3)
f
~
t=∑n=-∞∞fsngt-n=∑n=-∞∞fsnsinπt-nπt-n=ft
f
~
t
n
fs
n
g
t
n
n
fs
n
t
n
t
n
f
t
(4)
Amazing Conclusions
If
ft
f
t
is bandlimited to
-ππ
, it can be reconstructed perfectly from its samples on
the integers
fsn=ft|t=n
fs
n
t
n
f
t
ft=∑n=-∞∞fsnsinπt-nπt-n
f
t
n
fs
n
t
n
t
n
(5)
The above equation for perfect reconstruction deserves a
closer look, which you
should continue to read in the following section to get a
better understanding of reconstruction. Here are a few things
to think about for now:
-
What does
sinπt-nπt-n
t
n
t
n
equal at integers other than n?
-
What is the support of
sinπt-nπt-n
t
n
t
n
?
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