Electrical Engineering
      and Computer Sciences

Electrical Engineering and Computer Sciences

COLLEGE OF ENGINEERING

UC Berkeley

The Fixed-Point Theory of Strictly Causal Functions

Eleftherios Matsikoudis and Edward A. Lee

EECS Department
University of California, Berkeley
Technical Report No. UCB/EECS-2013-122
June 9, 2013

http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-122.pdf

We ask whether strictly causal components form well defined systems when arranged in feedback configurations. The standard interpretation for such configurations induces a fixed-point constraint on the function modelling the component involved. We define strictly causal functions formally, and show that the corresponding fixed-point problem does not always have a well defined solution. We examine the relationship between these functions and the functions that are strictly contracting with respect to a generalized distance function on signals, and argue that these strictly contracting functions are actually the functions that one ought to be interested in. We prove a constructive fixed-point theorem for these functions, introduce a corresponding induction principle, and study the related convergence process.


BibTeX citation:

@techreport{Matsikoudis:EECS-2013-122,
    Author = {Matsikoudis, Eleftherios and Lee, Edward A.},
    Title = {The Fixed-Point Theory of Strictly Causal Functions},
    Institution = {EECS Department, University of California, Berkeley},
    Year = {2013},
    Month = {Jun},
    URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-122.html},
    Number = {UCB/EECS-2013-122},
    Abstract = {We ask whether strictly causal components form well defined systems when arranged in feedback configurations. The standard interpretation for such configurations induces a fixed-point constraint on the function modelling the component involved. We define strictly causal functions formally, and show that the corresponding fixed-point problem does not always have a well defined solution. We examine the relationship between these functions and the functions that are strictly contracting with respect to a generalized distance function on signals, and argue that these strictly contracting functions are actually the functions that one ought to be interested in. We prove a constructive fixed-point theorem for these functions, introduce a corresponding induction principle, and study the related convergence process.}
}

EndNote citation:

%0 Report
%A Matsikoudis, Eleftherios
%A Lee, Edward A.
%T The Fixed-Point Theory of Strictly Causal Functions
%I EECS Department, University of California, Berkeley
%D 2013
%8 June 9
%@ UCB/EECS-2013-122
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-122.html
%F Matsikoudis:EECS-2013-122