Interpolation means increasing the sampling rate,
or filling in in-between samples. Equivalent to sampling abandlimited analog signal
times faster. For the ideal interpolator,
We wish to accomplish this digitally. Consider
[link] and
[link] .
The DTFT of
is
Since
is periodic with a period of
,
is periodic with a period of
(see
[link] ).
By inserting zero samples between the samples of
, we obtain a signal with a scaled frequency response
that simply replicates
times over a
interval!
Obviously, the desired
can be obtained simply by lowpass filtering
to remove the replicas.
Given
In practice, a finite-length lowpass filter is designed using
any of the methods studied so far (
[link] ).
Interpolator block diagram
Decimation: sampling rate reduction (by an integer factor m)
Let
(
[link] )
That is, keep only every
th
sample (
[link] )
In frequency (DTFT):
Now
for
as shown in homework #1, where
is the DFT of one period of the periodic sequence.
In this case,
for
and
.
so
i.e. , we get
digital
aliasing .(
[link] )
Usually, we prefer not to have aliasing, so the downsampler
is preceded by a lowpass filter to remove all frequencycomponents above
(
[link] ).
Rate-changing by a rational fraction l/m
This is easily accomplished by interpolating by a factor of
, then decimating by a factor
of
(
[link] ).
The two lowpass filters can be combined into one LP filterwith the lower cutoff,
Obviously, the computational complexity and simplicity of
implementation will depend on
:
will be easier to implement than
!
