A Group-Delay Interpretation of Polyphase Filters 2.12 2002/01/08 2005/09/16 14:30:27.394 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Adan Galvan jago@rice.edu Erin Maloney emaloney@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu decimation group-delay interpolation polyphase Discusses how polyphase filters can be used as intersample delay filters. 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 Hz is an ideal lowpass filter with gain L, cutoff L, and constant group delay of d: 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 h k L p In other words, h p k is an advanced (by p samples) and downsampled (by factor L) version of h n (see ).

The DTFT of the pth polyphase filter impulse response is then H p z 1 L l 0 L 1 V 2 L l z 1 L where Vz H z z p H p z 1 L l 0 L 1 2 L l p z p L H 2 L l z 1 L H p ω 1 L l 0 L 1 ω 2 l L p H ω 2 l L ω ω 1 L ω L p H ω L ω ω d p L ω Thus, the ideal pth polyphase filter has a constant magnitude response of one and a constant group delay of d p L samples. The implication is that if the input to the pth polyphase filter is the unaliased T-sampled representation x n x c n T , then the output of the filter would be the unaliased T -sampled representation y p n x c n d p L T (see ).

shows the role of polyphase interpolation filters assume zero-delay ( d 0 ) processing. Essentially, the pth filter interpolates the waveform p L -way between consecutive input samples. The L polyphase outputs are then interleaved to create the output stream. Assuming that x c t is bandlimited to 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 (which which correspond to some casual FIR approximation of the master filter h n ).