# 6.2 Efficient multirate filter structures

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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.

## Interpolation

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 $p=m\mod L$ ,
${x}_{1}(m)=\sum_{m={N}_{1}}^{{N}_{2}} {h}_{Lp}(m)y(m)=\sum_{k=\frac{{N}_{1}}{L}}^{\frac{{N}_{2}}{L}} {g}_{p}(k){x}_{0}(\lfloor \frac{m}{L}\rfloor -k)$
${g}_{p}(n)=h(Ln+p)$ Pictorially, this can be represented as in [link] . These are called polyphase structures , and the ${g}_{p}(n)$ are called polyphase filters .

## Computational cost

If $h(m)$ is a length- $N$ filter:
• No simplification: $\frac{N}{{T}_{1}}=\frac{LN}{{T}_{0}}\frac{\mathrm{computations}}{\mathrm{sec}}$
• Polyphase structure: $L\frac{L}{N}\frac{1}{{T}_{0}^{o}}\frac{\mathrm{computations}}{\mathrm{sec}}=\frac{N}{{T}_{0}}$ where $L$ is the number of filters, $\frac{N}{L}$ is the taps/filter, and $\frac{1}{{T}_{0}}$ is the rate.
Thus we save a factor of $L$ by not being dumb.
For a given precision, $N$ is proportional to $L$ , (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 $M$ 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.

## Efficient l/m rate changers

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.

## Computational cost

Every ${T}_{1}=\frac{M}{L}{T}_{0}\mathrm{seconds}$ , compute one polyphase filter of length $\frac{N}{L}$ , or $\frac{\frac{N}{L}}{{T}_{1}}=\frac{\frac{N}{L}}{\frac{M}{L}{T}_{0}}=\frac{N}{M{T}_{0}}\frac{\mathrm{multiplies}}{\mathrm{second}}$ However, note that $N$ is proportional to $\max\{L , M\}$ .

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