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A Group-Delay Interpretation of Polyphase Filters

Module by: Phil Schniter. E-mail the author

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Summary: Discusses how polyphase filters can be used as intersample delay filters.

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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).

Figure 1
Figure 1 (m10438fig1.png)

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

H p z=1L∑l=0L−1Vⅇ-ⅈ2πLlz1L 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=1L∑l=0L−1ⅇ-ⅈ2πLlpzpLHⅇ-ⅈ2πLlz1L 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=1L∑l=0L−1ⅇⅈω−2πlLpHⅇⅈω−2πlL=∀ω,|ω|≤π:1LⅇⅈωLpHⅇⅈωL=∀ω,|ω|≤π:ⅇ-ⅈd−pLω 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 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 2
Figure 2 (m10438fig2.png)

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 ).

Figure 3
Figure 3 (m10438fig3.png)

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