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Polyphase Decimation Filter

Module by: Phil Schniter. E-mail the author

Summary: Implementation of polyphase decimation filters.

Polyphase Decimation

Recall the standard decimation method in Figure 1.

Figure 1
Figure 1 (m10433fig1.png)
Note that this procedure is computationally inefficient because it discards the majority of the computed filter outputs. Through the use of the Noble identities, it is possible to rearrange Figure 1 so that filter outputs are not discarded.

In order to apply the Noble identity for decimation, we must transform HzHz into its upsampled polyphase components H p zM H p z M , p=0M1 p 0 M 1 , defined previously in the context of polyphase interpolation.

Hz=nnhnzn=kkp=0M1hkM+pz((kM))p H z n n h n z n k k p 0 M 1 h k M p z k M p
(1)
via knM k n M , pnmodM p n M
Hz=p=0M1k h p kz(kM)zp H z p 0 M1 k k h p k z k M z p
(2)
via h p khkM+p h p k h k M p
Hz=p=0M1 H p zMzp H z p 0 M1 H p z M z p
(3)
Using these unsampled polyphase components, the preceding block diagram can be redrawn as Figure 2.

Figure 2
Figure 2 (m10433fig2.png)

Applying the Noble identity for decimation to Figure 2 yields Figure 3. The ladder of delays and downsamplers on the left below accomplishes a form of serial-to-parallel conversion.

Figure 3
Figure 3 (m10433fig3.png)

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