# Re: [music-dsp] "Factorization" of filter kernels

```Hello,

a technique that allows something similar to what you are suggesting
is to use polyphase filters. The difference is that you will not
process contiguous vectors, but (for a 2-phase decomposition example)
process the even samples with one stage of the filter and the odd
samples with another stage. It is generally used for multirate filter
design, but it makes sense to use this kind of decomposition if you
can process the stages in parallel... or at least it is what I think
makes sense.```
```
You can search for references to this technique here [1] and here [2].
A full section on how to perform the decomposition is presented on
"Digital Signal Processing: a Computer-based approach" by Sanjit K.
Mitra.

[1]
http://www.ws.binghamton.edu/fowler/fowler%20personal%20page/EE521_files/IV-05%20Polyphase%20FIlters%20Revised.pdf
[2] https://ccrma.stanford.edu/~jos/sasp/Multirate_Filter_Banks.html

--
João Felipe Santos

On Tue, Jan 18, 2011 at 5:46 AM, Uli Brueggemann
<uli.brueggem...@gmail.com> wrote:
> Hi,
>
> a convolution of two vectors with length size n and m gives a result
> of length n+m-1.
> So e.g. two vectors of length 512 with result in a vector of length 1023.
>
> Now let's assume we have a vector (or signal or filter kernel) of size
> 1024, the last taps is 0.
> How to decompose it to two vectors of half length? The smaller vectors
> can be of any arbitrary contents but their convolution must result
> must be equal to the original vector.
>
> It would be even interesting to "factorize"  given kernel into n
> smaller kernels. Again the smaller kernels may have any arbitrary but
> senseful contents, they can be identical but this is not a must.
>
> Is there a good method to carry out the kernel decomposition? (e.g.
> like calculating n identical factors x of a number y by x =
> Exp(Log(y)/n) with x^n = x*x*...*x = y)
>
> Uli
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