On higher order approximations for hermite-gaussian functions and discrete fractional Fourier transforms

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.


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...
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...
Properly Handling Complex Differentiation in Optimization and Approximation Problems
Candan, Çağatay (Institute of Electrical and Electronics Engineers (IEEE), 2019-03-01)
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 lect...
Maximum likelihood estimation of transition probabilities of jump Markov linear systems
Orguner, Umut (Institute of Electrical and Electronics Engineers (IEEE), 2008-10-01)
This paper describes an online maximum likelihood estimator for the transition probabilities associated with a jump Markov linear system (JMLS). The maximum likelihood estimator is derived using the reference probability method, which exploits an hypothetical probability measure to find recursions for complex expectations. Expectation maximization (EM) procedure is utilized for maximizing the likelihood function. In order to avoid the exponential increase in the number of statistics of the optimal EM algori...
Derivation of length extension formulas for complementary sets of sequences using orthogonal filterbanks
Candan, Çağatay (Institution of Engineering and Technology (IET), 2006-11-23)
A method for the construction of complementary sets of sequences using polyphase representation of orthogonal filterbanks is presented. It is shown that the case of two-channel filterbanks unifies individually derived length extension formulas for complementary sequences into a common framework and the general M-channel case produces novel formulas for the extension of complementary sets of sequences. The presented technique can also be used to generate polyphase and multilevel sequences.
Citation Formats
Ç. Candan, “On higher order approximations for hermite-gaussian functions and discrete fractional Fourier transforms,” IEEE SIGNAL PROCESSING LETTERS, pp. 699–702, 2007, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/47829.