# Fractional decimation

Started by March 1, 2009
```Hello,

I've been studying polyphase decomposition, decimation, interpolation and
fractional decimation. I think I have a handle on what I'm doing, but
there's one thing I don't understand...

Do you have to have greater than an L*Mth order filter in order to use
polyphase decomposition to put the compressor and expander in their most
efficient places (first and last, respectively)?

For example: Let's say I want to reduce my sample rate to Fs'= 0.8333Fs. L
= 5 and M = 6, so they're relatively prime, which is good.

But then I do Type II decomposition and get R0(z^5), R1(z^5)...R4(z^5).
What if all I needed was a 3rd order FIR filter in the first place? Then
wouldn't R4 =0?

Whats more, when I go to do the TypeI decompsition, I'm going to get R00,
01, etc, but won't all but the first row of the R filter coefficient matrix
equal zero?

So, either
(A)it's okay to have zero-terms in the polyphase elements, or
(B)the minimum order of the original filter, H(z) has to be L*M (per the
example, N=30).

Which is correct?

Thank you for your kind consideration of my question.

```
```On Mar 1, 5:41&#2013266080;pm, "tharris00" <ted.harris.h...@gmail.com> wrote:
> Hello,
>
> I've been studying polyphase decomposition, decimation, interpolation and
> fractional decimation. I think I have a handle on what I'm doing, but
> there's one thing I don't understand...
>
> Do you have to have greater than an L*Mth order filter in order to use
> polyphase decomposition to put the compressor and expander in their most
> efficient places (first and last, respectively)?
>
> For example: Let's say I want to reduce my sample rate to Fs'= 0.8333Fs. L
> = 5 and M = 6, so they're relatively prime, which is good.
>
> But then I do Type II decomposition and get R0(z^5), R1(z^5)...R4(z^5).
> What if all I needed was a 3rd order FIR filter in the first place? Then
> wouldn't R4 =0?
>
> Whats more, when I go to do the TypeI decompsition, I'm going to get R00,
> 01, etc, but won't all but the first row of the R filter coefficient matrix
> equal zero?
>
> So, either
> (A)it's okay to have zero-terms in the polyphase elements, or
> (B)the minimum order of the original filter, H(z) has to be L*M (per the
> example, N=30).
>
> Which is correct?
>
> Thank you for your kind consideration of my question.

I find the polyphase terminology in books to be a huge distraction
from the very simple and obvious concept of avoiding unnecessary
decimator and elimatate wasted work, you'll end up at the right place.
The only trick is the book-keeping.

John
```
```Not 100% sure (would need to think for that :-)). But here is what I
(don't) think.
Since polyphase structure is exactly (theortically and practically)
equivalent to the normal interpolation/filtering/decimation structure
without polyphase, having some terms equal to 0 should be OK if the
filter is very small order.
But, maybe I would really think over it more seriously sometime.

Regards
Piyush

On Mar 2, 3:41&#2013266080;am, "tharris00" <ted.harris.h...@gmail.com> wrote:
> Hello,
>
> I've been studying polyphase decomposition, decimation, interpolation and
> fractional decimation. I think I have a handle on what I'm doing, but
> there's one thing I don't understand...
>
> Do you have to have greater than an L*Mth order filter in order to use
> polyphase decomposition to put the compressor and expander in their most
> efficient places (first and last, respectively)?
>
> For example: Let's say I want to reduce my sample rate to Fs'= 0.8333Fs. L
> = 5 and M = 6, so they're relatively prime, which is good.
>
> But then I do Type II decomposition and get R0(z^5), R1(z^5)...R4(z^5).
> What if all I needed was a 3rd order FIR filter in the first place? Then
> wouldn't R4 =0?
>
> Whats more, when I go to do the TypeI decompsition, I'm going to get R00,
> 01, etc, but won't all but the first row of the R filter coefficient matrix
> equal zero?
>
> So, either
> (A)it's okay to have zero-terms in the polyphase elements, or
> (B)the minimum order of the original filter, H(z) has to be L*M (per the
> example, N=30).
>
> Which is correct?
>
> Thank you for your kind consideration of my question.

```