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Rate-changing appears expensive computationally, since for both decimation and interpolation the lowpass filter is implementedat the higher rate. However, this is not necessary.
For the interpolator, most of the samples in the upsampled signal are zero, and thus require no computation. ( [link] )
For $m=L\lfloor \frac{m}{L}\rfloor +m\mod L$ and $p=m\mod L$ ,
We only want every $M$ th output, so we compute only the outputs of interest. ( [link] ) $${x}_{1}(m)=\sum_{k={N}_{1}}^{{N}_{2}} {x}_{0}(Lm-k)h(k)$$
The decimation structures are flow-graph reversals of the interpolation structure. Although direct implementation ofthe full filter for every $M$ th sample is obvious and straightforward, these polyphasestructures give some idea as to how one might evenly partition the computation over $M$ cycles.Interpolate by $L$ and decimate by $M$ ( [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.
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