Browsing by Subject "Fractional Fourier domains"

Now showing 1 - 10 of 10

Open Access
Condition number in recovery of signals from partial fractional fourier domain information
(Optical Society of America, 2013-06) Oktem F. S.; Özaktaş, Haldun M.
The problem of estimating unknown signal samples from partial measurements in fractional Fourier domains arises in wave propagation. By using the condition number of the inverse problem as a measure of redundant information, we analyze the effect of the number of known samples and their distributions.
Open Access
Cost-efficient approximation of linear systems with repeated and multi-channel filtering configurations
(IEEE, 1998-05) Kutay, Mehmet Alper; Erden, M. F.; Özaktaş, Haldun M.; Arıkan, Orhan; Candan, Ç.; Güleryüz, Ö.
It is possible to obtain either exact realizations or useful approximations of linear systems or matrix-vector products arising in many different applications, by synthesizing them in the form of repeated or multi-channel filtering operations in fractional Fourier domains, resulting in much more efficient implementations with acceptable decreases in accuracy. By varying the number and configuration of filter blocks, which may take the form of arbitrary flow graphs, it is possible to trade off between accuracy and efficiency in the desired manner. The proposed scheme constitutes a systematic way of exploiting the information inherent in the regularity or structure of a given linear system or matrix, even when that structure is not readily apparent.
Open Access
Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space-bandwidth product
(Optical Society of America, 2010-07-30) Oktem, F. S.; Özaktaş, Haldun M.
Linear canonical transforms (LCTs) form a three-parameter family of integral transforms with wide application in optics. We show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space-frequency plane. This allows LCT domains to be labeled and ordered by the corresponding fractional order parameter and provides insight into the evolution of light through an optical system modeled by LCTs. If a set of signals is highly confined to finite intervals in two arbitrary LCT domains, the space-frequency (phase space) support is a parallelogram. The number of degrees of freedom of this set of signals is given by the area of this parallelogram, which is equal to the bicanonical width product but usually smaller than the conventional space-bandwidth product. The bicanonical width product, which is a generalization of the space-bandwidth product, can provide a tighter measure of the actual number of degrees of freedom, and allows us to represent and process signals with fewer samples.
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
Filtering in fractional Fourier domains and their relation to chirp transforms
(IEEE, 1994-04) Özaktaş, Haldun M.; Barshan, Billur; Onural, Levent; Mendlovic, D.
Fractional Fourier transforms, which are related to chirp and wavelet transforms, lead to the notion of fractional Fourier domains. The concept of filtering of signals in fractional domains is developed, revealing that under certain conditions one can improve upon the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.
Open Access
The fractional fourier transform
(IEEE, 2001) Özaktas, Haldun M.; Kutay, M. A.
A brief introduction to the fractional Fourier transform and its properties is given. Its relation to phase-space representations (time- or space-frequency representations) and the concept of fractional Fourier domains are discussed. An overview of applications which have so far received interest are given and some potential application areas remaining to be explored are noted.
Open Access
Generalization of time-frequency signal representations to joint fractional Fourier domains
(IEEE, 2005-09) Durak, L.; Özdemir, A. K.; Arıkan, Orhan; Song, I.
The 2-D signal representations of variables rather than time and frequency have been proposed based on either Hermitian or unitary operators. As an alternative to the theoretical derivations based on operators, we propose a joint fractional domain signal representation (JFSR) based on an intuitive understanding from a time-frequency distribution constructing a 2-D function which designates the joint time and frequency content of signals. The JFSR of a signal is so designed that its projections on to the defining joint fractional Fourier domains give the modulus square of the fractional Fourier transform of the signal at the corresponding orders. We derive properties of the JFSR including its relations to quadratic time-frequency representations and fractional Fourier transformations. We present a fast algorithm to compute radial slices of the JFSR.
Open Access
Linear algebraic analysis of fractional Fourier domain interpolation
(IEEE, 2009) Öktem, Figen S.; Özaktaş, Haldun M.
In this work, we present a novel linear algebraic approach to certain signal interpolation problems involving the fractional Fourier transform. These problems arise in wave propagation, but the proposed approach to these can also be applicable to other areas. We see this interpolation problem as the problem of determining the unknown signal values from the given samples within some tolerable error. We formulate the problem as a linear system of equations and use the condition number as a measure of redundant information in given samples. By analyzing the effect of the number of known samples and their distributions on the condition number with simulation examples, we aim to investigate the redundancy and information relations between the given data.
Open Access
Optimal filtering in fractional Fourier domains
(IEEE, 1995) Kutay, M. Alper; Onural, Levent; Özaktaş Haldun M.; Arıkan, Orhan
The ordinary Fourier transform is suited best for analysis and processing of time-invariant signals and systems. When we are dealing with time-varying signals and systems, filtering in fractional Fourier domains might allow us to estimate signals with smaller minimum-mean-square error (MSE). We derive the optimal fractional Fourier domain filter that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel. We present an example for which the MSE is reduced by a factor of 50 as a result of filtering in the fractional Fourier domain, as compared to filtering in the conventional Fourier or time domains. We also discuss how the fractional Fourier transformation can be computed in O(N log N) time, so that the improvement in performance is achieved with little or no increase in computational complexity.
Open Access
Signal processing with repeated filtering in fractional Fourier domains
(Board of Optronics Lasers, Tian-Jin City, China, 1998) Ozaktas, Haldun M.; Erden, M. F.; Kutay, M. A.