[sdiy] Filter topology: n-order == n-poles?

[Eric Brombaugh]

Amos wrote:
> Filter users typically refer to a
> filter's number of poles, where each pole seems to add -6dB/Octave to
> the cutoff slope.  Filter designers and mathematical types tend to
> refer to the "order" of the filter, e.g. "2 cascaded, buffered
> 1st-order sections" in the Korg MS filters.  Is a 2-pole filter
> necessarily a 2nd-order filter, or is the relationship more
> complicated? 

In general, yes.

Linear filters can be mathematically described by equations of the form

F(s) = B(s) / A(s)

where B and A are polynomial equations, like this

A(s) = a0 + a1*s + a2*s^2 + ... an*s^n

These equations can be factored out like this

A(s) = (s + r0)*(s + r1)*(s + r2)...(s + rn)

The roots of these polynomials are the places where A(s) and B(s) have a 
value of 0. Roots of A are called 'poles' because A is in the 
denominator of the F(s) equation and cause it to go to infinity (think 
of tent poles 'holding up' the surface of the equation). Roots of B(s) 
are called zeros because B is the numerator of F(s) and cause it to go 
to zero.

Typically a polynomial of order N will have N roots, so usually the 
number of poles and the order are the same.

Eric