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<P><FONT size=2 face="Courier New">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.<BR><BR>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:<BR><BR><IMG border=0 hspace=0 alt="" align=baseline
src="cid:212144623@11012021-181E"></FONT></P>
<P><FONT size=2 face="Courier New">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.</FONT></P>
<P><FONT size=2 face="Courier New">How's that for an analytical and
thought-provoking response?</FONT></P>
<P><BR><BR><BR><BR><FONT size=2 face="Courier New">-----Original
Message-----<BR>From: Michael E Caloroso [</FONT><A
href="mailto:mec.forumreader@gmail.com"><FONT size=2
face="Courier New">mailto:mec.forumreader@gmail.com</FONT></A><FONT size=2
face="Courier New">]<BR>Sent: Monday, January 11, 2021 3:29 PM<BR>To: David G
Dixon<BR>Cc: Ian Fritz; Bernard Arthur Hutchins, Jr; synth-diy@synth-diy.org;
Brian Willoughby<BR>Subject: Re: [sdiy] 90-degree phase displacement network
calculations<BR><BR>[CAUTION: Non-UBC Email]<BR><BR>Well that was a highly
analytical and thought provoking conclusion<BR><BR>MC<BR><BR>On 1/11/21, David G
Dixon <dixon@mail.ubc.ca> wrote:<BR>> Hello Ian,<BR>><BR>> Well,
I'm getting a bit tired about arguing about this, so my official<BR>> answer
is... whatever.<BR>><BR>> Cheers<BR>> Dave<BR>><BR>>
-----Original Message-----<BR>> From: Ian Fritz [</FONT><A
href="mailto:ijfritz@comcast.net"><FONT size=2
face="Courier New">mailto:ijfritz@comcast.net</FONT></A><FONT size=2
face="Courier New">]<BR>> Sent: Monday, January 11, 2021 6:44 AM<BR>> To:
David G Dixon; 'Bernard Arthur Hutchins, Jr';<BR>>
synth-diy@synth-diy.org<BR>> Cc: 'Brian Willoughby'<BR>> Subject: Re:
[sdiy] 90-degree phase displacement network calculations<BR>><BR>>
[CAUTION: Non-UBC Email]<BR>><BR>> That looks not to be true. The
difference between two successive k'(i)<BR>> values clearly can not be zero.
The process is a (rapidly) converging<BR>> iterative one.<BR>><BR>> In
case you can't see this, the proof is trivial:<BR>> Suppose k'(i) =
k'(i-1)<BR>> Then from the second equation, k(i) = k(i-1) Now the first
equation<BR>> yields 0 = k(i)-k(i-1) = [1-k'(i-1)]/[1+k'(i-1)]
-<BR>> [1-k'(i-2)]/[1+k'(i-2)]<BR>> This can be generally true
only if k'(i-2) = k'(i-1) So by induction,<BR>> all the k'(i) values are the
same.<BR>><BR>> A sequence either iterates or it doesn't -- it can't just
drop dead in<BR>> the middle of the street.<BR>><BR>> Ian<BR>> (math
minor, including some pretty tough analysis courses)<BR>><BR>><BR>> On
1/11/2021 2:23 AM, David G Dixon wrote:<BR>><BR>>> ........ There
are no "approximate" answers, and this problem is not<BR>>> one where one
gets closer and closer to the true solution with each<BR>>> step.
That would be an iterative solution, and as I've said ad<BR>>> nauseum,
this is not that problem.<BR>><BR>><BR>><BR>>
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