Re: Bilinear Transforms-> Möbius transforms

• To: mathgroup at smc.vnet.net
• Subject: [mg50807] Re: Bilinear Transforms-> Möbius transforms
• From: Roger Bagula <tftn at earthlink.net>
• Date: Wed, 22 Sep 2004 00:11:10 -0400 (EDT)
• References: <cijen5\$hmb\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```It appear to be Fuchsian with determinant in lamda squared.
I usually similfy these by hand:
u'=(z-1/Lamda)/(z/Lamda+1)=z/(z/Lamda+1)-(1/Lamda)/(z/Lamda+1)
I think that's what you want.
These are called Möbius transforms and
are associated with the Poincare
nonEuclidean plane.
Fuchsian groups are a big well studied area.
A tip is to multiply by (z/z) and (-1/Lamda)/(-1/Lamda)
to get a form where it is:
u'=(a*z+b)/(c*z+b)
By bilinear they mean that the 2by2:
M={{a,b},{c,d}}
multiplies when the the functions are taken like f(g(z)).
These matrices are also associated as well with Klein groups and
Teichmuller space
in Mathematics.
Chris Williams wrote:

>Hi everyone,
>
>I have an warped FIR filter with order around 1400 - however to unwarp
>this filter configuration for use on my data I need to apply the
>bilinear transform:
>
>u' ->  (z^-1 - lambda)/(1-lambda*z^-1)
>
>So essentially each unit delay becomes an all-pass element.
>I've substituted in the transform into a transfer function expression
>for my filter, and now I want to get the transfer function back into the
>normal form of a fraction of polynomials of z.
>
>I've tried Together[] and Simplify[] without much success. Has anyone
>accomplished something similar - any help would be greatly appreciated ;)
>
>Cheers and thanks,
>
>Chris.
>
>
>

--
Respectfully, Roger L. Bagula