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=0…L-1 p 0 … L 1 .

via k≔⌊nL⌋ ≔ k n L , p≔nmodL ≔ p n L via h p k≔hkL+p ≔ h p k h k L p 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.