Deriving the Butterworth filter

[stratum]

Hi

Just out of curiosity I'm trying to understand how Mr. Butterworth himself derived his maximally flat amplitude response filter. From https://en.wikipedia.org/wiki/Butterworth_filter the original paper appears to be
http://www.gonascent.com/papers/butter.pdf

In page 536 (2nd in sequence) he describes the problem being:

An ideal electrical filter should not only completely reject the unwanted frequencies but should also have uniform sensitivity for the wanted frequencies

and in page 537 (3rd in sequence) he postulates that the desired filter has the frequency response:

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F = (1 + x^m)^(-1/2)  

(if I have read that correctly) and continues to develop his theory based on that.

What I wonder is, how did he know that this formula described the maximally flat passband filter with the fasted possible roll-off and anything better wasn't possible?

Thanks

[sault]

Okay, so I'm going to take a stab at this.

What he appears to be saying is that the ideal ratio is F = (1+x^m)^-1 (not -1/2). I started looking around, and came across http://www.ece.uah.edu/courses/ee426/Butterworth.pdf . We see again that same diagram, the F equation at various orders, where m is the order. So Butterworth used m for order, this document uses n. The graph is magnitude on the y and ratio on the x.

Anyways, I think the key here is that the first derivatives of the function are zero at Fc 0 and negative for all positive frequencies. If the derivative is always zero or negative it is monotonic, and thus has no ripples, which is the problem that he was trying to solve. The frequency of interest is 1 because the ratio is between the frequency of interest and the cut-off, and the ratio at the cut-off would be Fc/Fc, or 1, and I think he wanted the magnitude to be 1/2 there for design purposes.

I speculate that he chose that particular function specifically because of the zero derivative/negative derivative property and how it behaved at the x=1 point. In other words, kind of a "here's what I want, how do I express that in numbers, and what's a suitable equation that ties that together" thing.

Hope that helps...


Saul

[stratum]

Hi Saul,

Thanks for the explanation. It makes sense. I guess there is one further piece to the 'puzzle' and it is that how he knew there wasn't any other function like this? Perhaps a better one? Perhaps the circuits elements, when wired according to the topology he discusses (https://en.wikipedia.org/wiki/Butterwor ... ICauer.svg ) , always happen to give the same set of formulas as their frequency response, among which there was only one with the property he wanted?