Properly Handling Complex Differentiation in Optimization and Approximation Problems

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


On higher order approximations for hermite-gaussian functions and discrete fractional Fourier transforms
Candan, Çağatay (Institute of Electrical and Electronics Engineers (IEEE), 2007-10-01)
Discrete equivalents of Hermite-Gaussian functions play a critical role in the definition of a discrete fractional Fourier transform. The discrete equivalents are typically calculated through the eigendecomposition of a commutator matrix. In this letter, we first characterize the space of DFT-commuting matrices and then construct matrices approximating the Hermite-Gaussian generating differential equation and use the matrices to accurately generate the discrete equivalents of Hermite-Gaussians.
Exact Relation Between Continuous and Discrete Linear Canonical Transforms
Öktem, Sevinç Figen (Institute of Electrical and Electronics Engineers (IEEE), 2009-08-01)
Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. The Fourier and fractional Fourier transforms are special cases of LCTs. We present the exact relation between continuous and discrete LCTs (which generalizes the corresponding relation for Fourier transforms), and also express it in terms of a new definition of the discrete LCT (DLCT), which is independent of the sampling interval. This p...
On the Eigenstructure of DFT Matrices
Candan, Çağatay (Institute of Electrical and Electronics Engineers (IEEE), 2011-03-01)
The discrete Fourier transform (DFT) not only enables fast implementation of the discrete convolution operation, which is critical for the efficient processing of analog signals through digital means, but it also represents a rich and beautiful analytical structure that is interesting on its own. A typical senior-level digital signal processing (DSP) course involves a fairly detailed treatment of DFT and a list of related topics, such as circular shift, correlation, convolution operations, and the connectio...
Spherical wave expansion of the time-domain free-space Dyadic Green's function
Azizoglu, SA; Koç, Seyit Sencer; Buyukdura, OM (Institute of Electrical and Electronics Engineers (IEEE), 2004-03-01)
The importance of expanding Green's functions, particularly free-space Green's functions in terms of orthogonal wave functions is practically self-evident when frequency domain scattering problems are of interest. With the relatively recent and widespread interest in time-domain scattering problems, similar expansions of Green's functions are expected to be useful in the time-domain. In this paper, an expression, expanded in terms of orthogonal spherical vector wave functions, for the time-domain free-space...
Factor Graph Based LMMSE Filtering for Colored Gaussian Processes
Sen, Pinar; Yılmaz, Ali Özgür (Institute of Electrical and Electronics Engineers (IEEE), 2014-10-01)
We propose a reduced complexity, graph based linear minimum mean square error (LMMSE) filter in which the non-white statistics of a random noise process are taken into account. Our method corresponds to block LMMSE filtering, and has the advantage of complexity linearly increasing with the block length and the ease of incorporating the a priori information of the input signals whenever possible. The proposed method can be used with any random process with a known autocorrelation function by use of an approx...
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: