Browsing by Subject "Fractional Fourier transform"

Now showing 1 - 20 of 27

Open Access
About the Wigner distribution of a graded index medium and the fractional fourier transform operation
(SPIE, 1993-08) Özaktaş, Haldun M.; Mendlovic, D.; Lohmann, A. W.
Upon propagation through quadratic graded index media, the Wigner distribution of the wavefunction of light rotates uniformly. As a consequence, a definition of fractional Fourier transforms based on rotating the functions Wigner distribution, and another based on propagation through graded index media, are equivalent.
Open Access
Applications of the fractional Fourier transform in optics and signal processing-a review
(SPIE, 1996) Özaktaş, Haldun M.; Mendlovic, D.
The fractional Fourier transform The fractional Fourier transform is a generalization of the common Fourier transform with an order parameter a. Mathematically, the ath order fractional Fourier transform is the ath power of the fractional Fourier transform operator. The a = 1st order fractional transform is the common Fourier transform. The a = 0th transform is the function itself. With the development of the fractional Fourier transform and related concepts, we see that the common frequency domain is merely a special case of a continuum of fractional domains, and arrive at a richer and more general theory of alternate signal representations, all of which are elegantly related to the notion of space-frequency distributions. Every property and application of the common Fourier transform becomes a special case of that for the fractional transform. In every area in which Fourier transforms and frequency domain concepts are used, there exists the potential for generalization and improvement by using the fractional transform.
Open Access
Complex signal recovery from two fractional Fourier transform intensities: order and noise dependence
(Elsevier, 2005-01) Ertosun, M. G.; Atlı, H.; Özaktaş, Haldun M.; Barshan, B.
The problem of recovering a complex signal from the magnitudes of two of its fractional Fourier transforms is addressed. This corresponds to phase retrieval from the transverse intensity profiles of an optical field at two arbitrary locations along the optical axis. The convergence of the iterative algorithm, the effects of noise or measurement errors, and their dependence on the fractional transform order are investigated. It is observed that in general, better results are obtained when the fractional transform order is close to unity and poorer results are obtained when the order is close to zero. It follows that to the extent that conditions allow, the fractional order between the two measurement planes should be chosen as close to unity (or other odd integer) as possible for best results.
Open Access
Diffraction and holography from a signal processing perspective
(SPIE, 2006) Onural, Levent; Özaktaş, Haldun M.
The fact that plane waves are solutions of the Helmholtz equation in free space allows us to write the exact solution to the diffraction problem as a superposition of plane waves. The solution of other related problems can also be expressed in similar forms. These forms are very well suited for directly importing various signal processing tools to diffraction related problems. Another signal processing-diffraction link is the application of novel sampling theorems and procedures in signal processing to diffraction for the purpose of more convenient and efficient discrete representation and the use of associated computational algorithms. Another noteworthy link between optics and signal processing is the fractional Fourier transform. Revisiting diffraction from a modern signal processing perspectiv is likely to yield both interesting viewpoints and improved techniques.
Open Access
The discrete fractional Fourier transformation
(IEEE, 1996) Arıkan, Orhan; Kutay, M. Alper; Özaktaş, Haldun M.; Akdemir, Özer Koray
Based on the fractional Fourier transformation of sampled periodic functions, the discrete form of the fractional Fourier transformation is obtained. It is found that only for a certain dense set of fractional orders such a discrete transformation is possible to define. Also, for its efficient computation a fast algorithm, which has the same complexity as the FFT, is given.
Open Access
The effect of distribution of information on recovery of propagating signals
(2015-09) Karabulut, Özgecan
Interpolation is one of the fundamental concepts in signal processing. The analysis of the di fficulty of interpolation of propagating waves is the subject of this thesis. It is known that the information contained in a propagating wave fi eld can be fully described by its uniform samples taken on a planar surface transversal to the propagation direction, so the eld can be found anywhere in space by using the wave propagation equations. However in some cases, the sample locations may be irregular and/or nonuniform. We are concerned with interpolation from such samples. To be able to reduce the problem to a pure mathematical form, the fractional Fourier transform is used thanks to the direct analogy between wave propagation and fractional Fourier transformation. The linear relationship between each sample and the unknown field distribution is established this way. These relationships, which constitute a signal recovery problem based on multiple partial fractional Fourier transform information, are analyzed. Recoverability of the fi eld is examined by comparing the condition numbers of the constructed matrices corresponding to di fferent distributions of the available samples.
Open Access
An efficient algorithm to extract components of a composite signal
(IEEE, 2000) Özdemir, A. Kemal; Arıkan, Orhan
An efficient algorithm is proposed to extract components of a composite signal. The proposed approach has two stages of processing in which the time-frequency supports of the individual signal components are identified and then the individual components are estimated by performing a simple time-frequency domain incision on the identified support of the component. The use of a recently proposed time-frequency representation [1] significantly improves the performance of the proposed approach by providing very accurate description on the auto-Wigner terms of the composite signal. Then, simple fractional Fourier domain incision provides reliable estimates for each of the signal components in O(N log N) complexity for a composite signal of duration N.
Open Access
Every Fourier optical system is equivalent to consecutive fractional-Fourier-domain filtering
(Optical Society of America, 1996-06-10) Özaktaş, Haldun M.; Mendlovic, D.
We consider optical systems composed of an arbitrary number of lenses and filters, separated by arbitrary distances, under the standard approximations of Fourier optics. We show that every such system is equivalent to (i) consecutive filtering operations in several fractional Fourier domains and (ii) consecutive filtering operations alternately in the space and the frequency domains.
Open Access
Feature extraction with the fractional Fourier transform
(1998) Güleryüz, Özgür
In this work, alternative design and implementation techniques for feature extraction applications are proposed. The proposed techniques amount to decomposing the overall feature extraction problem into a global linear system followed by a local nonlinear system. Different output representations for representation of input features are also allowed and used in these techniques. These different output representations bring cui additional degree of freedom to the feature extraction problems. The systems provide multi-outputs consisting of different features of the input signal or image. Efficient implementation of the linear part of the .system is obtained by using fractional Fourier filtering circuits. Expressions for the proposed techniques are derived and several illustrative examples cxre given in which efficient implementations for feature extraction applications are obtained.
Open Access
The Fractional Fourier transform and harmonic oscillation
(Springer, 2002) Kutay, M. A.; Özaktaş, Haldun M.
The ath-order fractional Fourier transform is a generalization of the ordinary Fourier transform such that the zeroth-order fractional Fourier transform operation is equal to the identity operation and the first-order fractional Fourier transform is equal to the ordinary Fourier transform. This paper discusses the relationship of the fractional Fourier transform to harmonic oscillation; both correspond to rotation in phase space. Various important properties of the transform are discussed along with examples of common transforms. Some of the applications of the transform are briefly reviewed.
Open Access
Generalized time-bandwidth product optimal short-time fourier transformation
(IEEE, 2002-05) Durak, Lütfiye; Arıkan, Orhan
By extending the time-bandwidth product concept to fractional Fourier domains, a generalized time-bandwidth product (GTBP) is introduced. The GTBP provides a rotation independent measure for the support of the signals in time-frequency domain. A close form expression for the adaptive kernel of STFT that provides the minimum increase on the GTBP of a signal is derived. Also, a linear canonical decomposition of the obtained GTBP optimal STFT is presented to identify its relation to the rotationally invariant STFT analysis.
Open Access
High resolution time frequency representation with significantly reduced cross-terms
(IEEE, 2000-06) Özdemir, A. Kemal; Arıkan, Orhan
A novel algorithm is proposed for efficiently smoothing the slices of the Wigner distribution by exploiting the recently developed relation between the Radon transform of the ambiguity function and the fractional Fourier transformation. The main advantage of the new algorithm is its ability to suppress cross-term interference on chirp-like auto-components without any detrimental effect to the auto-components. For a signal with N samples, the computational complexity of the algorithm is O(N log N) flops for each smoothed slice of the Wigner distribution.
Open Access
Improved acoustics signals discrimination using fractional Fourier transform based phase-space representations
(Elsevier, 2001-04-01) Zalevsky, Z.; Mendlovic, D.; Kutay, M. A.; Özaktaş, Haldun M.; Solomon, J.
In this communication we propose performing two-dimensional correlation operation between phase-space representations based on the fractional Fourier transform, instead of correlating the signals themselves. A numerical examples clearly indicates superior discrimination performance. (C) 2001 Published by Elsevier Science B.V.
Open Access
Interpolating between periodicity and discreteness through the fractional Fourier transform
(IEEE, 2006) Özaktaş, H. M.; Sümbül, U.
Periodicity and discreteness are Fourier duals in the same sense as operators such as coordinate multiplication and differentiation, and translation and phase shift. The fractional Fourier transform allows interpolation between such operators which gradually evolve from one member of the dual pair to the other as the fractional order goes from zero to one. Here, we similarly discuss the interpolation between the dual properties of periodicity and discreteness, showing how one evolves into the other as the order goes from zero to one. We also discuss the concepts of partial discreteness and partial periodicity and relate them to fractional discreteness and periodicity. © 2006 IEEE.
Open Access
Maximally selective fractional fourier pooling
(IEEE, 2024-06-23) Koç, Emirhan; Ekiz, Yunus Emre; Özaktaş, Haldun; Koç, Aykut
In traditional image classification models, global average pooling is typically employed in the final layer to mitigate model complexity. However, this approach is prone to loss of information while reducing the complexity. Recent studies have proposed alternatives, replacing this layer by propagating information in various domains. In our work, we propose replacing this conventional pooling layer with a fractional Fourier transform (FrFT) based pooling layer. We first transform the feature of the last convolutional layer to the FrFT domain and transfer only the k-largest coefficients to the following layer in each channel, thereby enhancing efficiency by preserving only the essential information. To support our proposal, we conducted experiments on two datasets using various image classification models. Our results show that the integration of the FrFT as a pooling layer not only improves model performances but also does not add significant computational burden to model complexity.
Open Access
Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions
(Elsevier BV * North-Holland, 1995-10-15) Sahin, A.; Özaktaş, Haldun M.; Mendlovic, D.
Previous optical implementations of the two-dimensional fractional Fourier transform have assumed identical transform orders in both dimensions. We let the orders in the two orthogonal dimensions to be different and present general design formulae for optically implementing such transforms. This design formulae allows us to specify the two orders and the input, output scale parameters simultaneously.
Open Access
Optimal fractional fourier filtering for graph signals
(IEEE, 2021-05-19) Öztürk, Cüneyd; Özaktaş, Haldun M.; Gezici, Sinan; Koç, Aykut
Graph signal processing has recently received considerable attention. Several concepts, tools, and applications in signal processing such as filtering, transforming, and sampling have been extended to graph signal processing. One such extension is the optimal filtering problem. The minimum mean-squared error estimate of an original graph signal can be obtained from its distorted and noisy version. However, the best separation of signal and noise, and thus the least error, is not always achieved in the ordinary Fourier domain, but rather a fractional Fourier domain. In this work, the optimal filtering problem for graph signals is extended to fractional Fourier domains, and theoretical analysis and solution of the proposed problem are provided along with computational cost considerations. Numerical results are presented to illustrate the benefits of filtering in fractional Fourier domains.
Open Access
Optimal image restoration with the fractional Fourier transform
(OSA - The Optical Society, 1998-04) Kutay, M. A.; Özaktaş, Haldun M.
The classical Wiener filter, which can be implemented in O(N log N) time, is suited best for space-invariant degradation models and space-invariant signal and noise characteristics. For space-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N2) time for implementation. Optimal filtering in fractional Fourier domains permits reduction of the error compared with ordinary Fourier domain Wiener filtering for certain types of degradation and noise while requiring only O(N log N) implementation time. The amount of reduction in error depends on the signal and noise statistics as well as on the degradation model. The largest improvements are typically obtained for chirplike degradations and noise, but other types of degradation and noise may also benefit substantially from the method (e.g., nonconstant velocity motion blur and degradation by inhomegeneous atmospheric turbulence). In any event, these reductions are achieved at no additional cost. © 1998 Optical Society of America.
Open Access
Optimal representation and processing of optical signals in quadratic-phase systems
(Elsevier, 2016) Arik, S. Ö.; Özaktaş, Haldun M.
Optical fields propagating through quadratic-phase systems (QPSs) can be modeled as magnified fractional Fourier transforms (FRTs) of the input field, provided we observe them on suitably defined spherical reference surfaces. Non-redundant representation of the fields with the minimum number of samples becomes possible by appropriate choice of sample points on these surfaces. Longitudinally, these surfaces should not be spaced equally with the distance of propagation, but with respect to the FRT order. The non-uniform sampling grid that emerges mirrors the fundamental structure of propagation through QPSs. By providing a means to effectively handle the sampling of chirp functions, it allows for accurate and efficient computation of optical fields propagating in QPSs.
Embargo
Relationships between two definitions of the discrete Wigner distribution and the continuous Wigner distribution
(Elsevier, 2025-03) Korkmaz, Sayit; Özaktaş, Haldun M.
We present a very simple relationship between two widely used discrete-time discrete-frequency Wigner distributions. The first one is obtained through sampling and the second one is obtained from the representation theory of the finite Heisenberg group. This relation shows that the values of one can simply be obtained by permuting the values of the other along the frequency axis, which in turn implies a relationship of the second definition to the samples of the continuous Wigner distribution, and the first definition to group representation theory. In the process, we derive a simplified form for the second definition which is completely analogous to the continuous Wigner distribution, and develop a set of relationships relating this definition to a discrete ambiguity function and auxiliary functions.