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Polyphase Interpolation

Module by: Phil Schniter

Summary: Implementation of a polyphase interpolation filter.

Polyphase Interpolation Filter

Recall the standard interpolation procedure illustrated in Figure 1.

Figure 1
Figure 1 (m10431fig1.png)

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=0…L-1 p 0 … L 1 .

Hz=∑nhnz-n=∑k∑p=0L-1hkL+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 k≔⌊nL⌋ ≔ k n L , p≔nmodL ≔ p n L
Hz=∑p=0L-1∑k h p kz-kLz-p H z p 0 L1 k k h p k z k L z p (2)
via h p k≔hkL+p ≔ h p k h k L p
Hz=∑p=0L-1 H p zLz-p 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.

Figure 2:
Figure 2 (m10431fig2.png)

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.

Figure 3:
Figure 3 (m10431fig3.png)

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