# Stationary points of a real-valued function of a complex variable

### David G. Messerschmitt

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2006-93

June 27, 2006

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-93.pdf

The optimization problem of maximizing or minimizing some real-valued objective function of a complex variable (or vector of complex variables) arises often in signal processing. For example, the mean-square error is such a function. A challenge that arises is that such a function is often not analytic, and thus not differentiable using the ordinary tools of complex variable theory. This tutorial report shows how this challenge can be bypassed by reformulationg the problem as a function of two real variables (the real and imaginary parts), finding the solution, and then relating this back to complex variables.

BibTeX citation:

@techreport{Messerschmitt:EECS-2006-93, Author = {Messerschmitt, David G.}, Title = {Stationary points of a real-valued function of a complex variable}, Institution = {EECS Department, University of California, Berkeley}, Year = {2006}, Month = {Jun}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-93.html}, Number = {UCB/EECS-2006-93}, Abstract = {The optimization problem of maximizing or minimizing some real-valued objective function of a complex variable (or vector of complex variables) arises often in signal processing. For example, the mean-square error is such a function. A challenge that arises is that such a function is often not analytic, and thus not differentiable using the ordinary tools of complex variable theory. This tutorial report shows how this challenge can be bypassed by reformulationg the problem as a function of two real variables (the real and imaginary parts), finding the solution, and then relating this back to complex variables.} }

EndNote citation:

%0 Report %A Messerschmitt, David G. %T Stationary points of a real-valued function of a complex variable %I EECS Department, University of California, Berkeley %D 2006 %8 June 27 %@ UCB/EECS-2006-93 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-93.html %F Messerschmitt:EECS-2006-93