# Exact Relation Between Continuous and Discrete Linear Canonical Transforms

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 provides the foundation for approximately computing the samples of the LCT of a continuous signal with the DLCT. The DLCT in this letter is analogous to the DFT and approximates the continuous LCT in the same sense that the DFT approximates the continuous Fourier transform. We also define the bicanonical width product which is a generalization of the time-bandwidth product.
IEEE SIGNAL PROCESSING LETTERS

# Suggestions

 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.
 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...
 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...
 Hybrid connection of RF MEMS and SMT components in an impedance tuner Unlu, Mehmet; Topalli, Kagan; Atasoy, Halil Ibrahim; Demir, Şimşek; Aydın Çivi, Hatice Özlem; Akın, Tayfun (Elsevier BV, 2010-01-01) This paper presents a systematic construction of a model for a hybrid connected RF MEMS and SMT components in a reconfigurable impedance tuner. The double stub hybrid impedance tuner which employs a high number of MEMS switches is selected to demonstrate the feasibility of the connections. In the hybrid tuner, MEMS switches are actuated with DC bias signals, where SMT resistors de-couple RF from the DC lines. The hybrid tuner is realized in two steps, where the MEMS impedance tuner is fabricated on a glass ...
 3D object recognition from range images using transform invariant object representation AKAGÜNDÜZ, erdem; Ulusoy, İlkay (Institution of Engineering and Technology (IET), 2010-10-28) 3D object recognition is performed using a scale and orientation invariant feature extraction method and a scale and orientation invariant topological representation. 3D surfaces are represented by sparse, repeatable, informative and semantically meaningful 3D surface structures, which are called multiscale features. These features are extracted with their scale (metric size and resolution) using the classified scale-space of 3D surface curvatures. Triplets of these features are used to represent the surfac...
Citation Formats
S. F. Öktem, “Exact Relation Between Continuous and Discrete Linear Canonical Transforms,” IEEE SIGNAL PROCESSING LETTERS, pp. 727–730, 2009, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/49166. 