Rate-changing appears expensive computationally, since for both
decimation and interpolation the lowpass filter is implementedat the higher rate. However, this is not necessary.
Interpolation
For the interpolator, most of the samples in the upsampled
signal are zero, and thus require no computation. (
[link] )
For
and
,
Pictorially, this can be represented as in
[link] .
These are called
polyphase structures , and the
are called
polyphase filters .
Computational cost
If
is a length-
filter:
No simplification:
Polyphase structure:
where
is the number of filters,
is the taps/filter, and
is the rate.
Thus we save a factor of
by
not being dumb.
For a given precision,
is
proportional to
, (why?), so
the computational cost does increase with the interpolationrate.
Can similar computational savings be obtained with IIR
structures?
Efficient decimation structures
We only want every
th output,
so we compute only the outputs of interest. (
[link] )
Polyphase decimation structure
The decimation structures are flow-graph reversals of the
interpolation structure. Although direct implementation ofthe full filter for every
th
sample is obvious and straightforward, these polyphasestructures give some idea as to how one might evenly partition
the computation over
cycles.
Efficient l/m rate changers
Interpolate by
and decimate by
(
[link] ).
Combine the lowpass filters (
[link] ).
We can couple the lowpass filter either to the interpolator or
the decimator to implement it efficiently (
[link] ).
Of course we only compute the polyphase filter output selected
by the decimator.
Computational cost
Every
, compute one polyphase filter of length
, or
However, note that
is
proportional to
.
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