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> For a 2-pole filter, yes, but not for higher order filters, where
> you can have inflection points in the frequency response curve. For
> example, I recently constructed a 6-pole highpass filter that had
> a more "desirable" (to me) temporal response to a step function.
> The frequency response was such that it leveled off at 14 dB about
> an octave away from the 3 dB point, stayed there for another half
> octave or so, and then fell off further from there. Actually the
> plateau was more of a slight hump. The filter had, essentially, two
> roll-off frequencies, at different attenuation levels.
>
> I don't think a measure like Q has any meaning in that case, with
> respect to desired filter performance -- the Q of my filter was
> basically the same as a 6-pole filter of some traditional type, but
> the frequency and time responses were quite different.
Yep, you're right Alex. Q is just shorthand for all-pole filters. If you
take a basic 2-pole filter and place the poles at different frequencies,
it will round the shoulder of the curve and lower the Q. You can
calculate the Q if you know where the poles are and vice-versa. But your
curve has zeros as well as poles so it gets more complicated.
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BTW, I'm sure you have already figured this out but I wrestled with the
whole pole and zero concept for a while before it finally clicked what
they mean in terms a trader can understand. Maybe this will help those
less mathematically inclined than yourself. The DSP dictionary says:
"Pole: Term used in the Laplace transform and z-transform. When the
s-domain or z-domain transfer function is written as one polynomial
divided by another polynomial, the roots of the denominator are the
poles of the system, while the roots of the numerator are the zeros."
Huh, okay, fine. :-) But all they really are for what we are doing are
the inflection points on a frequency response curve. A downward turn is
a pole and an upward turn is a zero. You can build just about any shape
frequency response curve with a combination of lowpass poles, lowpass
zeros, highpass poles and highpass zeros. Each pole and zero has the
basic frequency response shape of a 6dB/octave exponential moving
average flipped one way or another on the chart. Simple concept really,
of course the math can get sticky. :-)
--
Dennis
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