Recipes for non-ripple 2-pole filters available

For the newbies here (like me):

I have compiled recipes (step by step instructions) for coding three
kinds of 2-pole digital filters, both low-pass and high-pass, complete
with correction factors to ensure that the 3 dB cutoff frequency stays
where you put it when you cascade filters together.

       http://unicorn.us.com/alex/2polefilters.html

The recipes cover Butterworth, Critically-Damped, and Bessel
filters.  I also include test results; i.e. plots of actual
frequency response and step-function temporal response for each
filter.

I was interested only in filters that exhibited no ripple in
the frequency response.  Therefore I didn't include Chebyschev,
Elliptical, and other ripply types.

What brought this on?  Well, I spent many frustrating hours looking
in vain for a general purpose "recipe" for a particular filter.
I wanted formulas for the end result (the filter coefficients),
that didn't require me to understand much about DSP.  All I could
ever find were numeric tools that let me construct one-of-a-kind
filters with specific, numeric performance specifications, which
wasn't useful to me.  I needed the formulas!  The Bessel filter in
particular has a lamentable dearth of information available.

With the patient help of the good people in this newsgroup, I
gradually grew to understand more about DSP, to the point where I
was finally able to derive a filter independently, all by myself
(Bessel lowpass).  Rather than do it over again each time I need it,
I tabulated the formulas in an easy-to-use step-by-step table.

I find it ironic that I now have filter recipes that one can use
without understanding how to derive filters, but I had to learn
something about how to derive filters to complete my set of recipes.

You'd think something like this would be covered in the comp.dsp
FAQ.  That was actually one of the first places I went when I began
this quest.

-Alex

axlq wrote:

> I have compiled recipes (step by step instructions) for coding three
> kinds of 2-pole digital filters, both low-pass and high-pass, complete
> with correction factors to ensure that the 3 dB cutoff frequency stays
> where you put it when you cascade filters together.
> 
>        http://unicorn.us.com/alex/2polefilters.html

   ...

You're a good man, Alex! You ought to be strutting around with your
thumbs under your armpits around; you can be justly proud.

You're right: that page ought to make it into the FAQ. As Tom Lehrer
attributed to Werner Von Braun, "That's not my department." Let's hope
someone knows how to arrange it.

Jerry
-- 
... the worst possible design that just meets the specification - almost
a definition of practical engineering.                     .. Chris Bore
������������������������������������������������������������������������

Nice work, Alex! I second the motion to get it added to the FAQ. 
-- 
%  Randy Yates                  % "Ticket to the moon, flight leaves here today 
%% Fuquay-Varina, NC            %  from Satellite 2"
%%% 919-577-9882                % 'Ticket To The Moon' 
%%%% <yates@ieee.org>           % *Time*, Electric Light Orchestra
http://home.earthlink.net/~yatescr

In article <411f87db$0$5916$61fed72c@news.rcn.com>,
Jerry Avins  <jya@ieee.org> wrote:
>axlq wrote:
>> I have compiled recipes (step by step instructions) for coding three
>> kinds of 2-pole digital filters, both low-pass and high-pass, complete
>> with correction factors to ensure that the 3 dB cutoff frequency stays
>> where you put it when you cascade filters together.
>> 
>>        http://unicorn.us.com/alex/2polefilters.html
>
>You're a good man, Alex! You ought to be strutting around with your
>thumbs under your armpits around; you can be justly proud.

Thanks Jerry, I am.  And I'm grateful for all the advice and teaching,
to which you contributed in a major way.

I had wanted to include a Gaussian filter in there also, but that
one stymied me.  Bilinear transformation doesn't seem applicable in
that case.

I'd also like to extend the page to cover 4th order filters.  The
4th-order Butterworth and Critically-damped ones are easy to
implement in 2-pole sections.  However, the Bessel is giving
me trouble; I can't figure out how to decompose the s-domain
polynomial s^4 + 10s^3 + 45s^2 + 105s + 105 into two 2nd-order
polynomials.  What's worse, I recall solving such problems in high
school.  It might be easier if I knew how the poles are supposed
to be distributed in polar coordinates in the z plane, but that
information seems impossible to come by.

>You're right: that page ought to make it into the FAQ. As Tom Lehrer
>attributed to Werner Von Braun, "That's not my department." Let's hope
>someone knows how to arrange it.

I found a FAQ submission address.  I'll see what they say.

-Alex

"axlq" <axlq@spamcop.net> wrote in message
news:cfo700$ckm$1@blue.rahul.net...
> I'd also like to extend the page to cover 4th order filters.  The
> 4th-order Butterworth and Critically-damped ones are easy to
> implement in 2-pole sections.  However, the Bessel is giving
> me trouble; I can't figure out how to decompose the s-domain
> polynomial s^4 + 10s^3 + 45s^2 + 105s + 105 into two 2nd-order
> polynomials.  What's worse, I recall solving such problems in high
> school.  It might be easier if I knew how the poles are supposed
> to be distributed in polar coordinates in the z plane, but that
> information seems impossible to come by.
> -Alex

The roots for 1 10 45 105 105 are:

-2.896212-0.867234i
-2.896212+0.867234i
-2.103789+2.657418i
-2.103789-2.657418i

This should allow you to implement a Bessel filter as two two pole filters.
You need to get a math package.

Peter Nachwey

In article <HtCdncY_3otRfILcRVn-vg@comcast.com>,
Peter Nachtwey <peter@deltacompsys.com> wrote:
>"axlq" <axlq@spamcop.net> wrote in message
>news:cfo700$ckm$1@blue.rahul.net...
>> I'd also like to extend the page to cover 4th order filters.  The
>> 4th-order Butterworth and Critically-damped ones are easy to
>> implement in 2-pole sections.  However, the Bessel is giving
>> me trouble; I can't figure out how to decompose the s-domain
>> polynomial s^4 + 10s^3 + 45s^2 + 105s + 105 into two 2nd-order
>> polynomials.
>
>The roots for 1 10 45 105 105 are:
>
>-2.896212-0.867234i
>-2.896212+0.867234i
>-2.103789+2.657418i
>-2.103789-2.657418i

Groan.  I was kinda hoping they'd be nice whole numbers, or at least
have functional form like cos(pi*f(n)) or something.

>This should allow you to implement a Bessel filter as two two pole filters.
>You need to get a math package.

I know, but I can't afford one that does symbolic math.  SciLab
seems to be more for numerical applications.

-Alex