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# A Group-Delay Interpretation of Polyphase Filters

Module by: Phil Schniter. E-mail the author

Summary: 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 HzHz is an ideal lowpass filter with gain L L, cutoff π L L, and constant group delay of dd: Heiω={Le(idω)  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 p,p0L1: 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

H p z=1Ll=0L1Ve(i)2πLlz1L H p z 1 L l 0 L 1 V 2 L l z 1 L
(1)
where Vz=Hzzp Vz H z z p
H p z=1Ll=0L1e(i)2πLlpzpLHe(i)2πLlz1L H p z 1 L l 0 L 1 2 L l p z p L H 2 L l z 1 L
(2)
H p eiω=1Ll=0L1eiω2πlLpHeiω2πlL=ω,|ω|π:1L(eiωLpHeiωL)=ω,|ω|π:e(i)dpLω 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
(3)

Thus, the ideal pthpth polyphase filter has a constant magnitude response of one and a constant group delay of dpL 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 (ndpL)T 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 ).

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