<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META content="text/html; charset=us-ascii" http-equiv=Content-Type>
<STYLE style="DISPLAY: none" type=text/css>P {
MARGIN-TOP: 0px; MARGIN-BOTTOM: 0px
}
</STYLE>
<META name=GENERATOR content="MSHTML 8.00.6001.23562"></HEAD>
<BODY dir=ltr>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial>Bernie,</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial>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. </FONT></SPAN><SPAN
class=109253319-12012021><FONT color=#0000ff size=2 face=Arial>Those things you
mention below are UNNECESSARY for solving this problem.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial>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.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial>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 that.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN><SPAN class=109253319-12012021><FONT
color=#0000ff size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial>Cheers,</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial>Dave</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=109253319-12012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left>
<HR tabIndex=-1>
</DIV>
<DIV dir=ltr align=left><FONT size=2 face=Tahoma><B>From:</B> Bernard Arthur
Hutchins, Jr [mailto:bah13@cornell.edu] <BR><B>Sent:</B> Tuesday, January 12,
2021 11:08 AM<BR><B>To:</B> synth-diy@synth-diy.org; David G Dixon; Ian
Fritz<BR><B>Subject:</B> [sdiy] 90-degree phase displacement network
calculations <BR></FONT><BR></DIV>
<DIV></DIV><SPAN
style="BACKGROUND-COLOR: #ffecb3; COLOR: #000000; FONT-SIZE: 12px"><SPAN
style="BACKGROUND-COLOR: #ffecb3; COLOR: #000000; FONT-SIZE: 12px"><SPAN
style="PADDING-BOTTOM: 3px; LINE-HEIGHT: 1.6; BACKGROUND-COLOR: #ffecb3; FONT-STYLE: normal; PADDING-LEFT: 3px; PADDING-RIGHT: 3px; COLOR: #000000; FONT-SIZE: 12px; FONT-WEIGHT: normal; PADDING-TOP: 3px">[<STRONG>CAUTION:</STRONG>
Non-UBC Email]</SPAN></SPAN></SPAN>
<DIV>
<P style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt" class=MsoNormal><FONT
color=#ff0000 face="Arial, sans-serif"><SPAN
style="BACKGROUND-COLOR: rgb(250,250,250)"></SPAN></FONT> </P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; COLOR: rgb(0,0,0); FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt"><A
href="#_Hlk61346225"><O:P></O:P></A></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; COLOR: rgb(0,0,0); FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt">The proper terms to
contrast/compare here are perhaps CLOSED-FORM and
ALGORITHMIC?<O:P> </O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; COLOR: rgb(0,0,0); FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt"><O:P></O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; COLOR: rgb(0,0,0); FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt">Equiripple FILTER
designs, of necessity, employ well-studied ALGORITHMS (Remez Exchange,
Parks-McClellan) which converge TOWARD the UNIQUE-BEST solution (Alternation
Theorem).<O:P> </O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; COLOR: rgb(0,0,0); FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt"><O:P></O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; COLOR: rgb(0,0,0); FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt">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.<O:P> </O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; COLOR: rgb(0,0,0); FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt"><O:P></O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; COLOR: rgb(0,0,0); FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt">-Bernie<O:P> </O:P></SPAN></P><BR></DIV></BODY></HTML>