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Polyphase Decimation Filter

Module by: Phil Schniter

Summary: Implementation of polyphase decimation filters.

Polyphase Decimation

Recall the standard decimation method in Figure 1.
m10433fig1.png
Figure 1
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=0…M-1 p 0 … M 1 , defined previously in the context of polyphase interpolation.
Hz=∑nhnz-n=∑k∑p=0M-1hkM+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 k≔⌊nM⌋ ≔ k n M , p≔nmodM ≔ p n M
Hz=∑p=0M-1∑k h p kz-kMz-p H z p 0 M1 k k h p k z k M z p (2)
via h p k≔hkM+p ≔ h p k h k M p
Hz=∑p=0M-1 H p zMz-p 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.
m10433fig2.png
Figure 2
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.
m10433fig2.png
Figure 3

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