[sdiy] Filter terminology question

[Michael Baxter]

Hi Tom,

There's relatively easy answers to your question, but I will explain some 
of the system theory that supports it.

On Wed, 19 Nov 2003, Tom Arnold wrote:

> Okay. Here's a question that I thought I knew the answer to and when I
> realised I didnt, I started getting bogged down in the answers I was
> finding.
> 
> When talking about a filter, what is the difference between Poles and Order?

Poles and Order are different first of all. Poles are related to Zeros, 
but not directly to Order. Every filter has a transfer function, and 
usually this is described as a frequency-dependant ratio between input (a 
forcing function) and output. A pole is neighborhood in the frequency 
response where the ratio becomes very large, e.g. a peak in the frequency 
response. A zero is similarly a neighborhood in the frequency response 
where the ratio becomes zero -- no output.

A pole would be represented for instance by a band-pass filter, and a zero 
by a notch filter.

Now it's easy to see how this might be slightly confusing. Order has to do
with the number of energy storage elements, or cascaded sections. A
"second order filter" is somewhat like a "two pole filter," but only if
the transfer function has poles... it might have zeros instead. The poles 
or zeros might be place at the same or different frequencies. If they're 
the same frequency, this generally indicates that the filter has more 
selectivity. A 2-pole bandpass filter generally has a narrower "skirt" or 
bandwidth (which can be defined as either the upper and lower frequencies 
where the transfer function is -3 dB or -6 dB, depending on context) than 
a single-pole bandpass filter. Similarly, a 2-pole low-pass filter also 
has a greater fall-off rate as frequency increases, than does a 1-pole 
low-pass filter.

So, sometimes "Order" can be in effect mean the same thing as the number 
of "Poles," it doesn't always have to.

A more detailed elaboration beyond this would involve forcing functions 
that are complex exponentials, v = exp(st), where s is a complex operator, 
and exp(st) represents complex frequency. Also confusing, complex 
exponentials are just a way to say sine and cosine waves at the same time. 
This notation, and way of working, involves what is called the S-plane, 
and is based on the computational artifice called Laplace transforms, 
which greatly simplify manipulation of the differential equations involved 
in R-L-C networks. Basically, Laplace transforms turn differential 
equations involving complex quantities into simpler algebra. And, there 
are some graphical methods with the S-plane that tell you a lot about how 
a filter (or a system) is going to behave.

In the Laplace notation, poles and zeros also become much more obvious,
because each typical type of filter has a characteristic equation in
simple algebra form. A zero is where all or part of numerator goes to
zero, rendering and products in the numerator as zero, thus the transfer 
function becomes zero -- the filter has no output. A pole then is one or 
more products in the denominator of a charactertistic equation that also 
go to zero at some specific frequency -- 1/n where n is vanishingly small 
becomes essentially infinite, yielding a "pole" in the transfer function.

I hope I made some sense here, and this is useful information!

Kindly yours,
Michael