In article <418159f7$0$28014$5402220f@news.sunrise.ch>,
Sam  <totalsam-n.o-s.p.a.m@hotmail.com> wrote:
>I'm playing a little with matlab's fdatool and I noticed this : when I 
>design a filter, as a 30th order one, with second order sections, the 
>sections doesn't have the same coefficients. It is not as if I just 
>designed a 2nd order filter and applied it 15 times.

On the page http://unicorn.us.com/alex/allpolefilters.html I
describe how to break down a 4 pole Bessel filter into two 2-pole
stages.  It's a derivation for laymen, and it becomes obvious when
you see it why the coefficients for each stage will be different.

-A

On Thu, 28 Oct 2004 22:45:36 +0200, Sam
<totalsam-n.o-s.p.a.m@hotmail.com> wrote:

>Hi all !
>
>I'm playing a little with matlab's fdatool and I noticed this : when I 
>design a filter, as a 30th order one, with second order sections, the 
>sections doesn't have the same coefficients. It is not as if I just 
>designed a 2nd order filter and applied it 15 times.
>
>What is the difference ? Couldn't I just apply the second method 
>described here ??
>
>Thanks in advance !
>
>Sam
Sam,
I won't repeat what Jon and Jerry have already said.  They are
correct.  I want to add something that is implied very strongly by
their comments.  Let's say the 30th order filter is butterworth.  To
implement the filter you use 15 Second Order Sections.  As was stated
the transfer function of the 15 sections in series has a butterworth
response.  The response of any of the second order sections will most
likely not have the reponse of a 2nd order butterworth filter of
similar type (HP,LP,BP etc).  The butterworth function generates the
poles at regular intervals (determined by the order of the filter)
around the unit circle. What the SOS function of matlab does is split
up the 30 poles and zeros into pairs.  One pair per SOS in a very
orderly well defined way.   One pair may be at the 2nd order pole
position depending on the original filter order.  So each SOS will
look very different in frequency response when looked at 1 SOS at a
time.  Perhaps you have already discovered this while investigating
you problem. Hope this helps.

Mark Allie

Sam wrote:

> Hi all !
> 
> I'm playing a little with matlab's fdatool and I noticed this : when I
> design a filter, as a 30th order one, with second order sections, the
> sections doesn't have the same coefficients. It is not as if I just
> designed a 2nd order filter and applied it 15 times.
> 
> What is the difference ? Couldn't I just apply the second method
> described here ??
> 
> Thanks in advance !
> 
> Sam

An nth-order Butterworth low-pass has a magnitude response of
1/|1 + jx^n|, where x = f/f_cutoff). A cascade of two order-n filters is
1/|(1+ jx^n)^2|. An order 2n filter is 1/|1 + jx^2n|. They are not the same.

All Butterworth filters are 3 dB down at x = 1. When two are cascaded,
the combination is 6 dB down. You could expect an order 30 filter
designed as you imagined to be 90 dB down. Hardly the edge of the passband.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

That is to be expected.  If you limit yourself to 15 identical second-order
sections, you can't generate certain types of filters.  For example, to get a
filter with a sharp but smooth cut-off, say Butterworth, each SoS will have a
different Q value.  Taken by themselves, many sections will show overshoot.  But
in the composite filter, the overshoot from some will be balanced by the
roll-off of others generating a smooth, sharp response.  In another example, in
a high order digital Bessel filter, each section has a different Q _and_ a
different cut-off frequency.  Stated mathematically, any higher order filter can
be broken up into second-order sections*, but they cannot necessarily be broken
up into _identical_ second-order sections.

So in summary, you certainly can create a 30th order filter with 15 identical
sections, but it may not be the kind of filter you are after.

Hope that helps.
-Jon

* and perhaps a single first-order section.

"Sam" <totalsam-n.o-s.p.a.m@hotmail.com> wrote in message
news:418159f7$0$28014$5402220f@news.sunrise.ch...
> Hi all !
>
> I'm playing a little with matlab's fdatool and I noticed this : when I
> design a filter, as a 30th order one, with second order sections, the
> sections doesn't have the same coefficients. It is not as if I just
> designed a 2nd order filter and applied it 15 times.
>
> What is the difference ? Couldn't I just apply the second method
> described here ??
>
> Thanks in advance !
>
> Sam

Hi all !

I'm playing a little with matlab's fdatool and I noticed this : when I 
design a filter, as a 30th order one, with second order sections, the 
sections doesn't have the same coefficients. It is not as if I just 
designed a 2nd order filter and applied it 15 times.

What is the difference ? Couldn't I just apply the second method 
described here ??

Thanks in advance !

Sam