# [sdiy] 90-degree phase displacement network calculations

David G Dixon dixon at mail.ubc.ca
Tue Jan 12 00:56:07 CET 2021

```Well, Michael, we're basically arguing about the meaning of the words
"iteration" and "recursion" at this point, and I find this argument to be
utterly fruitless.

To me, iteration is something that is required when approximate solutions
are sought and the criterion is convergence.  The solution to the phase
displacement problem is exact.  The poles are given by the following
closed-form equation:

The only recursive part of this whole problem is that 2K is a function of k'
which must be determined by a recursive process (as far as I know).  This is
a feature of elliptic sines and has nothing to do with the filter pole
calculation, which is completely closed form.  Elliptic functions differ
from circular functions in that the latter have fixed periods (2*pi), but
the former have periods which are functions of their modulus k.  However,
the period is a unique function of the modulus.  Hence, when you specify the
modulus you are also specifying the period.  However, to actually calculate
the value of the period from the modulus, you need to engage in a recursive
calculation.  THIS IS NOT ITERATION.  We are not approximating or
approaching some idealized solution by iterating to the correct period.  The
period is determined by the modulus -- there is nothing approximate about
it.  The solution to this problem is exact.

How's that for an analytical and thought-provoking response?

-----Original Message-----
From: Michael E Caloroso [ <mailto:mec.forumreader at gmail.com>
Sent: Monday, January 11, 2021 3:29 PM
To: David G Dixon
Cc: Ian Fritz; Bernard Arthur Hutchins, Jr; synth-diy at synth-diy.org; Brian
Willoughby
Subject: Re: [sdiy] 90-degree phase displacement network calculations

[CAUTION: Non-UBC Email]

Well that was a highly analytical and thought provoking conclusion

MC

On 1/11/21, David G Dixon <dixon at mail.ubc.ca> wrote:
> Hello Ian,
>
>
> Cheers
> Dave
>
> -----Original Message-----
> From: Ian Fritz [ <mailto:ijfritz at comcast.net> mailto:ijfritz at comcast.net]
> Sent: Monday, January 11, 2021 6:44 AM
> To: David G Dixon; 'Bernard Arthur Hutchins, Jr';
> synth-diy at synth-diy.org
> Cc: 'Brian Willoughby'
> Subject: Re: [sdiy] 90-degree phase displacement network calculations
>
> [CAUTION: Non-UBC Email]
>
> That looks not to be true. The difference between two successive k'(i)
> values clearly can not be zero. The process is a (rapidly) converging
> iterative one.
>
> In case you can't see this, the proof is trivial:
> Suppose k'(i) = k'(i-1)
> Then from the second equation, k(i) = k(i-1) Now the first equation
> yields 0 = k(i)-k(i-1) = [1-k'(i-1)]/[1+k'(i-1)] -
>   [1-k'(i-2)]/[1+k'(i-2)]
> This can be generally true only if k'(i-2) = k'(i-1) So by induction,
> all the k'(i) values are the same.
>
> A sequence either iterates or it doesn't -- it can't just drop dead in
> the middle of the street.
>
> Ian
> (math minor, including some pretty tough analysis courses)
>
>
> On 1/11/2021 2:23 AM, David G Dixon wrote:
>
>> ........  There are no "approximate" answers, and this problem is not
>> one where one gets closer and closer to the true solution with each
>> step.  That would be an iterative solution, and as I've said ad
>> nauseum, this is not that problem.
>
>
>
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