Polyphase filter notation

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

-
Could you rephrase what you are asking? There is no $H(z)$ or $H_k(z)$ or indication as to what $k$ is on your diagram. –  Dilip Sarwate Feb 5 '12 at 2:46
Fixed it... $H(z) = P(z)$ which corresponds to the Low pass filter –  skipfer0712 Feb 5 '12 at 2:48

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.

-

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.

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

-
What is W? the dft of a window? –  skipfer0712 Feb 6 '12 at 20:42
W is often used to represent the filter itself. I'm not certain in your example since usually these are defined near the equation. My confusion in the terminology is because the example is talking about polyphase and the equations should really be accompanied by a matching diagram or definitions of how the variables are used. –  sleeves Feb 14 '12 at 17:49
$W_M$ is the Fourier matrix, i.e. $\left[W_M\right]_{pq}=e^{-\frac{2\pi j}{M} p q}$. So the product $W_M x$ is another way to compute the DFT of the vector $x$. By the way, I have seen $W_M^\ast$ (the conjugate of $W_M$) instead of $W_M$ in some books. –  Arrigo May 16 '13 at 21:47