[sdiy] Filter terminology question

[ASSI]

On Thursday 20 November 2003 20:02, Mididood at aol.com wrote:
[...]
> I could never get my mind around this term: 'pole'.
>
> I then, quite recently came across a fantastic explanation of how
> this term came to be used in the first place and it all cleared up
> for me...I hope this isn't too naive or basic on my part...
[...]

The point however is that the concept of poles combined with that 
explanation makes sense only in the complex plane (which is a four 
dimensional space really, so much for the tent analogy) and in the case 
of a continous time filter you get there by doing a laplace transform 
and the space is commonly called the s-plane (depending on the book 
you've cracked open it might be a different variable, in german books 
you often find p instead). There's literally a twist to that: each pole 
will also twist the "cloth", very unlike a real tentpole. Zeros are 
likewise points where the tent is nailed down (and twisted again). Of 
course trying to imagine that could seriously twist your mind, but I 
digress.

You get the frequency response by cutting the tent apart along the 
imaginary axis of the complex plane, the height of the skin of the 
"tent" gives the amplitude response and the twist gives the phase 
response. If the poles and zeros are either spaced "far" apart or 
stacked in the same place (multiple poles and zeros) you get away with 
doing the famous approximation of +-n*6dB/oct or +-n*20dB/dec (the "-" 
is for the zeros, and the n gives the multiplicity) for the magnitude 
response. Extracting phase is a little bit more complicated, but each 
pole or zero moves the phase by 90° at most - here's another 
explanation of why you need a second order system to get oscillation - 
you need n*180° phase shift at the oscillation frequency.

[ Still with me?
. Two functions meet in the complex plane. The first shouts: "Out of
. my way or I'll integrate you! The second says: "Go ahead, I'm e^x!"
]


Achim.

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