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Properly Handling Complex Differentiation in Optimization and Approximation Problems
Date
2019-03-01
Author
Candan, Çağatay
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Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License
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Functions of complex variables arise frequently in the formulation of signal processing problems. The basic calculus rules on differentiation and integration for functions of complex variables resemble, but are not identical to, the rules of their real variable counterparts. On the contrary, the standard calculus rules on differentiation, integration, series expansion, and so on are the special cases of the complex analysis with the restriction of the complex variable to the real line. The goal of this lecture note is to review the fundamentals of the functions of complex variables, highlight the differences and similarities with their real variable counterparts, and study the complex differentiation operation with the optimization and approximation applications in mind. More specifically, the take-home result of this lecture note is to understand the differentiation with respect to the conjugate variable (∂/∂z̅)f(z, z̅), which is known as Wirtinger calculus, and its application in optimization and approximation problems
Subject Keywords
Signal Processing
,
Electrical and Electronic Engineering
,
Applied Mathematics
URI
https://hdl.handle.net/11511/37965
Journal
IEEE SIGNAL PROCESSING MAGAZINE
DOI
https://doi.org/10.1109/msp.2018.2876761
Collections
Department of Electrical and Electronics Engineering, Article