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 H⁢zHz is an ideal lowpass filter with gain L L, cutoff π L L, and constant group delay of dd: H⁢ej⁢ω={L⁢e−(j⁢d⁢ω)  if  ω∈ −πL πL 0  if  ω∈ −π −πL & πL π H ω L d ω ω L L 0 ω L L

Recall that the polyphase filters are defined as h p ⁢k=h⁢k⁢L+p  ,   p∈0…L−1    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 h⁢n h n (see Figure 1).

Figure 1

The DTFT of the pth pth polyphase filter impulse response is then

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 x⁢n= x c ⁢n⁢T 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−pL)⁢T y p n x c n d p L T (see Figure 2).

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 12⁢T⁢Hz 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 h⁢n h n ).

Figure 3

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Last edited by Charlet Reedstrom on Sep 16, 2005 2:30 pm -0500.