<!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=017592903-09012021><FONT color=#0000ff
size=2 face=Arial>Thank you, Bernie.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=017592903-09012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=017592903-09012021><FONT color=#0000ff
size=2 face=Arial>In fact, the Orchard result is "exactly" equiripple, as this
is built into the derivation.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=017592903-09012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=017592903-09012021><FONT color=#0000ff
size=2 face=Arial>As far as the band edges, it looks as if the Weaver result
shoots up within the band itself. Indeed, at the very band edge in the
simulation I showed before, the phase in the Weaver scenario is 88 degrees -- an
error of 2 degrees. Of course, it's probably not a big
deal.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=017592903-09012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=017592903-09012021><FONT color=#0000ff
size=2 face=Arial>I guess that I'm still not sure why Weaver and others bothered
with anything other than Orchard's solution, given that it is trivially easy to
calculate once you know how (although maybe a little tedious without a
computer).</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=017592903-09012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr align=left><SPAN class=017592903-09012021><FONT color=#0000ff
size=2 face=Arial>In any case, I have learned a lot about elliptic functions and
Chebyshev approximation. I'm going to study these topics a bit more just
for fun. I may even be able to use the Chebyshev stuff in my
day-job.</FONT></SPAN></DIV>
<DIV dir=ltr align=left><SPAN class=017592903-09012021><FONT color=#0000ff
size=2 face=Arial></FONT></SPAN> </DIV>
<DIV dir=ltr lang=en-us class=OutlookMessageHeader align=left>
<HR tabIndex=-1>
<FONT size=2 face=Tahoma><B>From:</B> Bernard Arthur Hutchins, Jr
[mailto:bah13@cornell.edu] <BR><B>Sent:</B> Friday, January 08, 2021 6:26
PM<BR><B>To:</B> synth-diy@synth-diy.org<BR><B>Cc:</B> David G
Dixon<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 style="FONT-FAMILY: Arial, Helvetica, sans-serif; COLOR: rgb(0,0,0)">
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif"
class=MsoNormal><SPAN
style="FONT-FAMILY: Arial, sans-serif; FONT-SIZE: 12pt"></SPAN><BR> </P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt">Thanks David-
Outstanding<O:P> </O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; 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; FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt">The Orchard curve
(yellow) looks nearly “Equiripple” (Chebyshev error) and would be that (by the
alternation theorem) if it has the right number of extremal frequencies: one
more than the number of degrees of freedom in the design.<SPAN
style="mso-spacerun: yes"> </SPAN>It has 16 pole choices and 17 extremals
(both ends count) – so Winner!<SPAN style="mso-spacerun: yes">
</SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><O:P> </O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; 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; FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt">We likely have a
prejudice (looks “intentionally esthetic”) for a pretty-looking equiripple
response, so keep in mind that the Orchard (like the Weaver shown) in fact
shoots rapidly upward beyond the band edges.<SPAN
style="mso-spacerun: yes"> </SPAN>In addition this min-max-error is
achieved at the expense of increased error in other
regions.<O:P> </O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; 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; FONT-SIZE: 11pt"
class=MsoNormal><SPAN style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt">I
suspect that with the tiny max error of under 0.2 degrees, this all makes little
difference. Readers seeking more info on equiripple procedures for background
and as applied to digital filter design might look at my “A New Simplified
Equiripple Design Algorithm for FIR Filters” Electronotes 175 August 1989, pp
3-50. <O:P> </O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><SPAN
style="FONT-FAMILY: 'Arial',sans-serif; FONT-SIZE: 12pt">-<SPAN
style="mso-spacerun: yes"> </SPAN>Bernie<O:P> </O:P></SPAN></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><O:P></O:P></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><O:P></O:P></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><O:P></O:P></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><O:P></O:P></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><O:P></O:P></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><O:P></O:P></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><O:P></O:P></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><O:P></O:P></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><O:P></O:P></P>
<P
style="LINE-HEIGHT: 107%; MARGIN: 0in 0in 8pt; FONT-FAMILY: Calibri, sans-serif; FONT-SIZE: 11pt"
class=MsoNormal><O:P></O:P></P><BR></DIV></BODY></HTML>