Polyphase filter notation

Question

Hi I am a little confused on what the notation of the following statement means.

$$ H_{k}(z)= H(W_{4}^{k} z), k = 0,...,3$$

It comes from a question in which I have designed a FIR low-pass filter $H(z)$ and my goal is to implement a DFT filter bank scheme like this:

Exchange $P(z)$ for $H(z)$ and k would correspond to the subscript of P and M in this case is equal to 4

I guess I am confused on how to find $H_{k}(z)$ or what exactly a polyphase filter is

Answer 1

Hk are modulations of the low pass filter (band pass instead of low pass).

$$ (W_{4}^{k}) = e^{-2j\pi k /4}. $$ For $z = e^{j\omega}$: $$H(W_{4}^{k} z) = H(e^{j(\omega-2\pi k/4)})$$ This means that the filters $H_k$ are shifted in frequency- these are the band pass filters you want to get using your filter bank. For $k=0$ $H_0$ will pass the frequencies $[\frac{-\pi}{4} \frac{\pi}{4}]$, for $k=1,[\frac{\pi}{4} \frac{3\pi}{4}]$ etc..

In the DFT filter bank scheme, $y_0[m]$ are the outputs from $H_0$ , $y_1[n]$ are the outputs from $H_1$ and so on.

Answer 2

You are going to end up with 4 filters. $$ H_0(z), H_1(z), H_2(z), H_3(z) $$

These are constructed by taking your original filter W(z) and dividing it into 1/4ths.

I believe this terminology is telling you to take W(z) and skip every 4th item, starting at the kth item.

$$ H(W_{4}^{k} z), k = 0,...,3 $$

Although I am not certain about the terminology, I do know this is how you would split up a polyphase filter.

I.E.

$$ H(W_{stride}^{start} z) $$

Answer 3

One usage of the term polyphase filter is for a set of related FIR filters designed by sampling an impulse response (finite or windowed). You can sample a waveform using slightly different starting points or offsets less than the sample rate spacing, or "phases". Use of a differently phased set of samples of the impulse response can be used in a FIR filter to create sub-sample delay effects, or as an interpolation method.