1.11 A group-delay interpretation of polyphase filters

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

Assume that $H(z)$ is an ideal lowpass filter with gain $L$ , cutoff $\frac{\pi }{L}$ , and constant group delay of $d$ : $$H(e^{i\omega })=\begin{cases}Le^{-(id\omega )} & \text{if $\omega \in \left[-\left(\frac{\pi }{L}\right) , \frac{\pi }{L}\right)$}\\ 0 & \text{if $\omega \in (\left[-\pi , -\left(\frac{\pi }{L}\right)\right)\land \left[\frac{\pi }{L} , \pi \right))$}\end{cases}$$

Recall that the polyphase filters are defined as $$\forall p, p\in \{0, \dots , L-1\}\colon h_p(k)=h(kL+p)$$

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 $p^{\mathrm{th}}$ polyphase filter impulse response is then

Thus, the ideal $p^{\mathrm{th}}$ polyphase filter has a constant magnitude response of one and a constant group delay of $\frac{d-p}{L}$ samples. The implication is that if the input to the $p^{\mathrm{th}}$ polyphase filter is the unaliased $T$ -sampled representation $x(n)=x_c(nT)$ , then the output of the filter would be the unaliased $T$ -sampled representation $y_p(n)=x_c((n-\frac{d-p}{L})T)$ (see ).

shows the role of polyphase interpolation filters assume zero-delay( $d=0$ ) processing. Essentially, the $p^{\mathrm{th}}$ filter interpolates the waveform $\frac{p}{L}$ -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 $\frac{1}{2T}\mathrm{Hz}$ , perfect polyphase filtering yields a perfectly interpolated output. In practice, we use the casual FIRapproximations of the polyphase filters $h_p(k)$ (which which correspond to some casual FIR approximation of the master filter $h(n)$ ).