Interpolation means increasing the sampling rate,
or filling in in-between samples. Equivalent to sampling a
bandlimited analog signal LL
times faster. For the ideal interpolator,
X
1
ω={
X
0
Lω if |ω|<πL0 if πL≤|ω|≤π
X
1
ω
X
0
L
ω
ω
L
0
L
ω
(1)
We wish to accomplish this digitally. Consider
Equation 2 and
Figure 1.
ym={
X
0
mL if m=0±L±2L…0 otherwise
y
m
X
0
m
L
m
0
±
L
±
2
L
…
0
(2)
The DTFT of
ym
y
m
is
Yω=∑m=−ω∞yme−(jωm)=∑n=−∞∞
x
0
ne−(jωLn)=∑n=−∞∞xne−(jωLn)=
X
0
ωL
Y
ω
m
ω
y
m
ω
m
n
x
0
n
ω
L
n
n
x
n
ω
L
n
X
0
ω
L
(3)
Since
X
0
ω
′
X
0
ω
′
is periodic with a period of
2π
2
,
X
0
Lω=Yω
X
0
L
ω
Y
ω
is periodic with a period of
2πL
2
L
(see
Figure 2).
By inserting zero samples between the samples of
x
0
n
x
0
n
, we obtain a signal with a scaled frequency response
that simply replicates
X
0
ω
′
X
0
ω
′
LL times over a
2π
2
interval!
Obviously, the desired
x
1
m
x
1
m
can be obtained simply by lowpass filtering
ym
y
m
to remove the replicas.
x
1
m=ym*
h
L
m
x
1
m
y
m
h
L
m
(4)
Given
H
L
m={1 if |ω|<πL0 if πL≤|ω|≤π
H
L
m
1
ω
L
0
L
ω
In practice, a finite-length lowpass filter is designed using
any of the methods studied so far (
Figure 3).
