[sdiy] 90-degree phase displacement network calculations
Brian Willoughby
brianw at audiobanshee.com
Sat Jan 9 22:42:17 CET 2021
Hi Dave,
Thank you for your patient summary. Calculating ideal filters is still a mystery that I leave to others.
Brian
On Jan 8, 2021, at 22:35, David G Dixon <dixon at mail.ubc.ca> wrote:
>
> Hi Brian,
>
> "Circular" trigonometric functions are just the regular trigonometric
> functions. So, "circular" sine is just sine. They trace the dimensions of
> triangles around circles.
>
> The Jacobi elliptic functions are generalizations of the trigonometric
> functions which trace triangles around ellipses rather than circles. They
> have two arguments instead of just one (the angle and the "modulus"), and
> they are generally not tabulated but have to be calculated by solving
> elliptic integrals parametrically (the answer is the unknown upper limit of
> the integral). However, there is this cool concept known as "Landen
> transformation" which plays a trick with the integral, and results in a very
> simple way to calculate the elliptic functions starting from the circular
> functions.
>
> There's a pretty good Wikipedia page about all this stuff.
>
> Some very very smart people figured out how to use all this crap to solve
> for the poles of various filter circuits back in the 30s, 40s and 50s, when
> radio science was in its heyday. Reading that stuff is very humbling.
>
> Cheers,
> Dave
>
>
> -----Original Message-----
> Sent: Friday, January 08, 2021 8:41 PM
> To: David G Dixon
> Cc: *SYNTH DIY
>
> On Jan 8, 2021, at 12:48, David G Dixon <dixon at mail.ubc.ca> wrote:
>> The Orchard paper has a numerical example (for a 4-stage network) and this
>> is what allowed me to figure out how to do the math. The math ultimately is
>> very simple. If the total number of stages is 2N (I'm restricting it to
>> even-numbered stages), then the pole values are given as:
>>
>> (1/k')^(0.5) * sn(2n*pi/4N,k) / cn(2n*pi/4N,k)
>>
>> where n goes from 1 to N. k' = f1/f2 where f1 and f2 are the endpoint
>> frequencies of the desired bandwidth. sn and cn are Jacobi's elliptic sine
>> and cosine. k (the modulus of the Jacobi functions) = (1 - k'^2)^(0.5). To
>> calculate these Jacobi functions, one uses the Landen transformation, which
>> allows the reduction of the k and k' values using a recursive formula until
>> k = 0 and k' = 1. This formula is simply that the next value of k in the
>> table = (1 - k')/(1 + k'). One then calculates the new value of k' = (1 -
>> k^2)^(0.5) and continues until k = 0 and k' = 1 (which I have confirmed
>> takes no more than 8 steps regardless of the initial value of k'). Once k'
>> = 1, the Jacobi sn function is equivalent to the circular sine function.
>
> I have never heard the terms, elliptic sine, or circular sine. Is there a
> simple definition of these?
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