[sdiy] 90-degree phase displacement network calculations

[David G Dixon]

Thank you, Bernie.
 
In fact, the Orchard result is "exactly" equiripple, as this is built into
the derivation.
 
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.
 
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).
 
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.
 
  _____  

From: Bernard Arthur Hutchins, Jr [mailto:bah13 at cornell.edu] 
Sent: Friday, January 08, 2021 6:26 PM
To: synth-diy at synth-diy.org
Cc: David G Dixon
Subject: [sdiy] 90-degree phase displacement network calculations


[CAUTION: Non-UBC Email] 


 

Thanks David- Outstanding 

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.  It has 16 pole choices and 17 extremals (both ends count) - so
Winner!     

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.  In
addition this min-max-error is achieved at the expense of increased error in
other regions. 

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.  

-  Bernie 


-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://synth-diy.org/pipermail/synth-diy/attachments/20210108/f00b5aa4/attachment.htm>