[sdiy] 90-degree phase displacement network calculations

David G Dixon dixon at mail.ubc.ca
Tue Jan 12 20:57:55 CET 2021

You are talking about solving equiripple Chebyshev filter problems.  That is
not the problem I am solving.  I am finding the lowest equiripple phase
error in a pair of allpass networks.  This is a completely different
problem.  Those things you mention below are UNNECESSARY for solving this
The issue here is that you are consistently failing to understand what it is
that I have done.  If you understood, then you would realize that the
question of "3 rather than 8 iterations" has nothing to do with solving the
allpass filter problem.  It is simply a matter of obtaining an accurate
value for the elliptic sine and its period.  HENCE, one would NEVER USE
FEWER ITERATIONS THAN WERE NECESSARY to drive the modulus of sn to zero and
thereby achieve the goal of the Landen transformation.
IN OTHER WORDS, if 8 iterations are necessary to achieve the Landen
transformation, then using 3 iterations is NOT "approximating the solution
with less computational effort."  It IS simply "doing the problem wrong."
It's like saying "I know there are 8 steps to get from the first floor to
the second, but I'm going to see what happens when I only climb 3 steps and
then magically am on the second floor."  It makes just as little sense as

From: Bernard Arthur Hutchins, Jr [mailto:bah13 at cornell.edu] 
Sent: Tuesday, January 12, 2021 11:08 AM
To: synth-diy at synth-diy.org; David G Dixon; Ian Fritz
Subject: [sdiy] 90-degree phase displacement network calculations 

[CAUTION: Non-UBC Email] 


The proper terms to contrast/compare here are perhaps CLOSED-FORM and

Equiripple FILTER designs, of necessity, employ well-studied ALGORITHMS
(Remez Exchange, Parks-McClellan) which converge TOWARD the UNIQUE-BEST
solution (Alternation Theorem). 

Beyond this, we really do need a current (complete, corrected/updated,
commented) version of David's code, with phase-error graphs, showing what
happens, for example, when we use perhaps just 3 rather than 8 iterations. 


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