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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:

Filter bank showing decimation by M points, application of filters $P_{k}(z)$, application of M-point DFT, output as signals y_{k}[m], and reconstruction in reverse

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

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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
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4 Answers 4

up vote 3 down vote accepted

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.

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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.

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

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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
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$W_M$ 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
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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.

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are you familiar with matlab? –  skipfer0712 Feb 7 '12 at 21:51
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You are clearly trying to implement DFT filter bank using polyphase structure. The filters in the diagram in analysis branch are Pk(z). U can choose whichever alphabet to represent filters. Hk(z) ie hk(n) are polyphase components of a low pass filter h(n) ie H(z). Polyphase components mean, for M=4 h0(n) will have samples [0 4 8 ...] h1(n) will have samples [1 5 9 ...] h2(n) will have samples [2 6 10...] & h3(n) will have samples [3 7 11...]. If u check the frequency response of each hk(n) u will get low pass response but expanded in frequency by 4 times.(Decimation in time equals expansion in frequency). All 4 responses will be similar as u are taking adjacent samples of LPF. For polyphase decompostion, the formula is hk(n)=h(nM+k) for k = 0 to M-1. Thus ur above formula represents not polyphase decomposition of low pass filter but direct implementation of DFT filterbank where each individual filters of the analysis bank are shifted versions(and not decimated versions) of low pass filter. But the diagram in ur question is of polyphase structure so that formula in ur question is not used there

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