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# Polyphase Interpolation

Module by: Phil Schniter. E-mail the author

Summary: Implementation of a polyphase interpolation filter.

## Polyphase Interpolation Filter

Recall the standard interpolation procedure illustrated in Figure 1.

Note that this procedure is computationally inefficient because the lowpass filter operates on a sequence that is mostly composed of zeros. Through the use of the Noble identities, it is possible to rearrange the preceding block diagram so that operations on zero-valued samples are avoided.

In order to apply the Noble identity for interpolation, we must transform Hz H z into its upsampled polyphase components H p zL H p z L , p=0L1 p 0 L 1 .

Hz=nnhnzn=kkp=0L1hkL+pz(kL+p) H z n n h n z n k k p 0 L 1 h k L p z k L p
(1)
via knL k n L , pnmodL p n L
Hz=p=0L1kk h p kz(kL)zp H z p 0 L1 k k h p k z k L z p
(2)
via h p khkL+p h p k h k L p
Hz=p=0L1 H p zLzp H z p 0 L1 H p z L z p
(3)
Above, ·· denotes the floor operator and ·modM · M the modulo-MM operator. Note that the pthpth polyphase filter h p k h p k is constructed by downsampling the "master filter" hn h n at offset pp. Using the unsampled polyphase components, the Figure 1 diagram can be redrawn as in Figure 2.

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

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