Properly Handling Complex Differentiation in Optimization and Approximation Problems

Date

2019-03-01

Author

Candan, Çağatay

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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

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Department of Electrical and Electronics Engineering, Article

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Citation Formats

Ç. Candan, “Properly Handling Complex Differentiation in Optimization and Approximation Problems,” IEEE SIGNAL PROCESSING MAGAZINE, pp. 117–124, 2019, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/37965.