Browsing by Subject "Linear canonical transform (LCT)"

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Open Access
Digital computation of linear canonical transforms
(Institute of Electrical and Electronics Engineers, 2008) Koç A.; Özaktaş, Haldun M.; Candan, C.; Kutay, M. A.
We deal with the problem of efficient and accurate digital computation of the samples of the linear canonical transform (LCT) of a function, from the samples of the original function. Two approaches are presented and compared. The first is based on decomposition of the LCT into chirp multiplication, Fourier transformation, and scaling operations. The second is based on decomposition of the LCT into a fractional Fourier transform followed by scaling and chirp multiplication. Both algorithms take ∼ N log N time, where N is the time-bandwidth product of the signals. The only essential deviation from exactness arises from the approximation of a continuous Fourier transform with the discrete Fourier transform. Thus, the algorithms compute LCTs with a performance similar to that of the fast Fourier transform algorithm in computing the Fourier transform, both in terms of speed and accuracy.
Open Access
Discrete linear canonical transform based on hyperdifferential operators
(IEEE, 2019-05) Koç, Aykut; Bartan, Burak; Özaktaş, Haldun
Linear canonical transforms (LCTs) are of importance in many areas of science and engineering with many applications. Therefore, a satisfactory discrete implementation is of considerable interest. Although there are methods that link the samples of the input signal to the samples of the linear canonical transformed output signal, no widely-accepted definition of the discrete LCT has been established. We introduce a new approach to defining the discrete linear canonical transform (DLCT) by employing operator theory. Operators are abstract entities that can have both continuous and discrete concrete manifestations. Generating the continuous and discrete manifestations of LCTs from the same abstract operator framework allows us to define the continuous and discrete transforms in a structurally analogous manner. By utilizing hyperdifferential operators, we obtain a DLCT matrix, which is totally compatible with the theory of the discrete Fourier transform (DFT) and its dual and circulant structure, which makes further analytical manipulations and progress possible. The proposed DLCT is to the continuous LCT, what the DFT is to the continuous Fourier transform. The DLCT of the signal is obtained simply by multiplying the vector holding the samples of the input signal by the DLCT matrix.
Open Access
Operator theory-based computation of linear canonical transforms
(Elsevier, 2021-08-12) Koç, Aykut; Özaktaş, Haldun M.
Linear canonical transforms (LCTs) are extensively used in many areas of science and engineering with many applications, which requires a satisfactory discrete implementation. Recently, hyperdifferential operators have been proposed as a novel way of defining the discrete LCT (DLCT). Here we first focus on improving the accuracy of this approach by considering alternative discrete coordinate multiplication and differentiation operations. We also consider canonical decompositions of LCTs and compare them with the originally proposed Iwasawa decomposition. We show that accuracy of the approximation of the continuous LCT with the DLCT can be drastically improved. The advantage and elegance of this approach lie in the fact that it reduces the problem of defining sophisticated discrete transforms to merely defining discrete coordinate multiplication and differentiation operations, by reducing the transforms to these operations. As a result of systematic investigation of possible parameters and design choices, we achieve a DLCT that is both theoretically satisfying and highly accurate.