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I wrote:
>> A unique property of the Butterworth filter is that a single
>> even-order (2n-pole) Butterworth is identical to a cascade of n
>> 2-pole Butterworths. So you can construct a high-order filter using
>> the basic 2-pole filter. This isn't true for other filter types,
>> but it works here.
Dennis replied:
>Hmmmm, that may be true with your code but it's not true of Butterworth
>filters in general. In the frequency domain, a Butterworth of any order
>is by definition -3dB at Fc with Q=0.707. When you stack them, each
>filter contributes -3dB at Fc and the Q factors multiply. So, 2 second
>order Butterworths gives you a filter with Q=0.5 that is -6dB at at Fc,
>and so on as you add more stages.
According to the derivation of the Butterworth coefficients
at http://kwon3d.com/theory/filtering/fil.html any even-order
Butterworth coefficients can be derived from cascading the
calculations for a 2nd-order filter. "For example, a 6th-order
low-pass filter has 3 (6 divided by 2) elementary 2nd-order filters.
Passing the data through these 3 2nd-order filters consecutively is
the same to passing them through the 6th-order filter once."
It is true that the 3 dB point shifts if you stack the filters.
That's why my web page and source code include a correction term for
the cutoff frequency when cascading multiple filters (c in the web
page and cc in the code).
http://unicorn.us.com/trading/buttercrit.html
http://unicorn.us.com/trading/src/_buttercrit.txt
--
,|___ Alex Matulich -- alex@xxxxxxxxxxxxxx
// +__> Director of Research and Development
// \
// __) Unicorn Research Corporation -- http://unicorn.us.com
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