Browsing by Subject "Fractional Fourier transform (FRT)"

Now showing 1 - 3 of 3

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 discrete fractional Fourier transform
(Springer, 2019) Koç, Aykut
The fractional Fourier transform is of importance in several areas of signal processing with many applications including optical signal processing. Deploying it in practical applications requires discrete implementations, and therefore defining a discrete fractional Fourier transform (DFRT) is of considerable interest. We propose an operator theory-based approach to defining the DFRT. By deploying hyperdifferential operators, a DFRT matrix can be defined compatible with the theory of the discrete Fourier transform. The proposed DFRT only uses the ordinary Fourier transform and the coordinate multiplication and differentiation operations. We also propose and compare several alternative discrete definitions of coordinate multiplication and differentiation operations, each of which leads to an alternative DFRT definition. Unitarity and approximation to the continuous transform properties are also investigated in detail. The proposed DFRT is highly accurate in approximating the continuous transform.