Previously, polyphase interpolation and decimation were derived from the Noble identities and motivated for reasons of computational efficiency. Here we present a different interpretation of the (ideal) polyphase filter.

Assume that HzHz is an ideal lowpass filter with gain L L, cutoff π L L, and constant group delay of dd: Hⅇⅈω=Lⅇ-ⅈdωifω∈-πLπL0ifω∈-π-πL∧πLπ H ω L d ω ω L L 0 ω L L

Recall that the polyphase filters are defined as ∀p,p∈0…L-1: h p k=hkL+p p p 0 … L 1 h p k h k L p

In other words, h p k h p k is an advanced (by pp samples) and downsampled (by factor LL) version of hn h n (see Figure 1).

The DTFT of the pth pth polyphase filter impulse response is then where Vz=Hzzp Vz H z z p

Thus, the ideal pthpth polyphase filter has a constant magnitude response of one and a constant group delay of d-pL d p L samples. The implication is that if the input to the pthpth polyphase filter is the unaliased TT-sampled representation xn= x c nT x n x c n T , then the output of the filter would be the unaliased T T-sampled representation y p n= x c n-d-pLT y p n x c n d p L T (see Figure 2).

Figure 3 shows the role of polyphase interpolation filters assume zero-delay ( d=0 d 0 ) processing. Essentially, the pthpth filter interpolates the waveform pL p L -way between consecutive input samples. The LL polyphase outputs are then interleaved to create the output stream. Assuming that x c t x c t is bandlimited to 12THz 1 2 T Hz , perfect polyphase filtering yields a perfectly interpolated output. In practice, we use the casual FIR approximations of the polyphase filters h p k h p k (which which correspond to some casual FIR approximation of the master filter hn h n ).