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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.

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 H⁢z H z into its upsampled polyphase components H p ⁢zL H p z L , p=0…L−1 p 0 … L 1 .

H⁢z=∑nnh⁢n⁢z−n=∑kk∑p=0L−1h⁢k⁢L+p⁢z−(k⁢L+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≔〈n〉L ≔ p n L
H⁢z=∑p=0L−1∑kk h p ⁢k⁢z−(k⁢L)⁢z−p H z p 0 L1 k k h p k z k L z p
(2)
via h p ⁢k≔h⁢k⁢L+p ≔ h p k h k L p
H⁢z=∑p=0L−1 H p ⁢zL⁢z−p H z p 0 L1 H p z L z p
(3)
Above, ⌊·⌋· denotes the floor operator and 〈·〉M · M the modulo-MM operator. Note that the pthpth polyphase filter h p ⁢k h p k is constructed by downsampling the "master filter" h⁢n 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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