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Efficient Multirate Filter Structures

Module by: Douglas L. Jones. E-mail the author

Rate-changing appears expensive computationally, since for both decimation and interpolation the lowpass filter is implemented at 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. (Figure 1)

Figure 1
Figure 1 (imag001.png)
For m=LmL+mmodL m L m L m L and p=mmodL p m L ,
x 1 m=m= N 1 N 2 h L p mym=k= N 1 L N 2 L g p k x 0 mLk x 1 m m N 1 N 2 h L p m y m k N 1 L N 2 L g p k x 0 m L k
(1)
g p n=hLn+p g p n h L n p Pictorially, this can be represented as in Figure 2.
Figure 2
Figure 2 (imag002.png)
These are called polyphase structures, and the g p n g p n are called polyphase filters.

Computational cost

If hm h m is a length-NN filter:

  • No simplification: N T 1 =LN T 0 computationssec N T 1 L N T 0 computations sec
  • Polyphase structure: (LLN1 T 0 o )computationssec=N T 0 L L N 1 T 0 o computations sec N T 0 where LL is the number of filters, NL N L is the taps/filter, and 1 T 0 1 T 0 is the rate.
Thus we save a factor of LL by not being dumb.

Note:

For a given precision, NN is proportional to LL, (why?), so the computational cost does increase with the interpolation rate.

Question:

Can similar computational savings be obtained with IIR structures?

Efficient Decimation Structures

We only want every MMth output, so we compute only the outputs of interest. (Figure 3) x 1 m=k= N 1 N 2 x 0 Lmkhk x 1 m k N 1 N 2 x 0 L m k h k

Figure 3
Polyphase Decimation Structure
Polyphase Decimation Structure (imag003.png)
The decimation structures are flow-graph reversals of the interpolation structure. Although direct implementation of the full filter for every MMth sample is obvious and straightforward, these polyphase structures give some idea as to how one might evenly partition the computation over MM cycles.

Efficient L/M rate changers

Interpolate by LL and decimate by MM (Figure 4).

Figure 4
Figure 4 (imag004.png)
Combine the lowpass filters (Figure 5).
Figure 5
Figure 5 (imag005.png)
We can couple the lowpass filter either to the interpolator or the decimator to implement it efficiently (Figure 6).
Figure 6
Figure 6 (imag006.png)
Of course we only compute the polyphase filter output selected by the decimator.

Computational Cost

Every T 1 =ML T 0 seconds T 1 M L T 0 seconds , compute one polyphase filter of length NL N L , or NL T 1 =NLML T 0 =NM T 0 multipliessecond N L T 1 N L M L T 0 N M T 0 multiplies second However, note that NN is proportional to maxLM L M .

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