Invisible wrote:

> A Chebyshev low-pass filter arranges poles in a semicircle. The 
> resonance comes from deforming this circle to make it elliptical. As you 
> do so, the poles get nearer to the line, and the "ripples" you see are 
> actually the edges of the peak surrounding each pole. I'll draw you a 
> graph if you like... ;-)

...and what *is* the most advanced graphing tool known?

Yeah, that's right! ;-)

The first image shows a plot of the transfer function for a 3-pole 
Butterworth filter. (I.e., a lowpass filter with 0% ripple.) You can't 
see it terribly well from this angle, but those three mountain peaks 
(poles) are arraned in a semicircle.

The second image is the same thing, but I've cut a slice through the 
graph along the critical line to reveal the filter's frequency response 
(red). As you can see, it's flat in the middle, and tapers down at 
either side. (The X-axis is frequency, with 0 Hz in the middle and 
negative/positive frequencies on each side.) Normally you'd look at the 
s-domain the other way round, but in 3D you can't see the cut very well, 
so I've rotated it.

The third image is the same as the second, except that I've made the 
semicircle elliptical. (You can't see this very much with this camera 
angle. Feel free to change it...) Notice how the frequency response now 
has "ripples" - corresponding to the edges of the peaks.

The radius of the circle is the cutoff frequency, and the 
ellipticallness is the ripple (and cutoff slope). The parts to fiddle 
with are the lines that #define the R1-R3 and I1-I3 variables. Set the 
radius of both equal for a Butterworth filter, and then decrease R# only 
for Chebyshev.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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