Browsing by Subject "Linear canonical transform"

Now showing 1 - 10 of 10

Open Access
Degrees of freedom of optical systems and signals with applications to sampling and system simulation
(Optical Society of America, 2013) Oktem F.S.; Özaktaş, Haldun M.
We study the degrees of freedom of optical systems and signals based on space-frequency (phase space) analysis. At the heart of this study is the relationship of the linear canonical transform domains to the space-frequency plane. Based on this relationship, we discuss how to explicitly quantify the degrees of freedom of first-order optical systems with multiple apertures, and give conditions for lossless transfer. Moreover, we focus on the degrees of freedom of signals in relation to the space-frequency support and provide a sub-Nyquist sampling approach to represent signals with arbitrary space-frequency support. Implications for simulating optical systems are also discussed. © 2013 Optical Society of America.
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
Fast and accurate algorithm for the computation of complex linear canonical transforms
(Optical Society of America, 2010-08-05) Koç A.; Özaktaş, Haldun M.; Hesselink, L.
A fast and accurate algorithm is developed for the numerical computation of the family of complex linear canonical transforms (CLCTs), which represent the input-output relationship of complex quadratic-phase systems. Allowing the linear canonical transform parameters to be complex numbers makes it possible to represent paraxial optical systems that involve complex parameters. These include lossy systems such as Gaussian apertures, Gaussian ducts, or complex graded-index media, as well as lossless thin lenses and sections of free space and any arbitrary combinations of them. Complex-ordered fractional Fourier transforms (CFRTs) are a special case of CLCTs, and therefore a fast and accurate algorithm to compute CFRTs is included as a special case of the presented algorithm. The algorithm is based on decomposition of an arbitrary CLCT matrix into real and complex chirp multiplications and Fourier transforms. The samples of the output are obtained from the samples of the input in ∼N log N time, where N is the number of input samples. A space-bandwidth product tracking formalism is developed to ensure that the number of samples is information-theoretically sufficient to reconstruct the continuous transform, but not unnecessarily redundant.
Open Access
Fast and accurate algorithms for quadratic phase integrals in optics and signal processing
(SPIE, 2011) Koç, A.; Özaktaş, Haldun M.; Hesselink L.
The class of two-dimensional non-separable linear canonical transforms is the most general family of linear canonical transforms, which are important in both signal/image processing and optics. Application areas include noise filtering, image encryption, design and analysis of ABCD systems, etc. To facilitate these applications, one need to obtain a digital computation method and a fast algorithm to calculate the input-output relationships of these transforms. We derive an algorithm of NlogN time, N being the space-bandwidth product. The algorithm controls the space-bandwidth products, to achieve information theoretically sufficient, but not redundant, sampling required for the reconstruction of the underlying continuous functions. © 2011 SPIE.
Open Access
Fast and accurate computation of two-dimensional non-separable quadratic-phase integrals
(Optical Society of America, 2010-05-12) Koç A.; Özaktaş, Haldun M.; Hesselink, L.
We report a fast and accurate algorithm for numerical computation of two-dimensional non-separable linear canonical transforms (2D-NS-LCTs). Also known as quadratic-phase integrals, this class of integral transforms represents a broad class of optical systems including Fresnel propagation in free space, propagation in gradedindex media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic/astigmatic/non- orthogonal cases. The general two-dimensional non-separable case poses several challenges which do not exist in the one-dimensional case and the separable two-dimensional case. The algorithm takes ∼ñ log ñ time, where ñ is the two-dimensional space-bandwidth product of the signal. Our method properly tracks and controls the space-bandwidth products in two dimensions, in order to achieve information theoretically sufficient, but not wastefully redundant, sampling required for the reconstruction of the underlying continuous functions at any stage of the algorithm. Additionally, we provide an alternative definition of general 2D-NS-LCTs that shows its kernel explicitly in terms of its ten parameters, and relate these parameters bidirectionally to conventional ABCD matrix parameters.
Open Access
Fast and accurate linear canonical transform algorithms
(IEEE, 2015) Özaktaş, Haldun M.; Koç, A.
Linear canonical transforms are encountered in many areas of science and engineering. Important transformations such as the fractional Fourier transform and the ordinary Fourier transform are special cases of this transform family. This family of transforms is especially important for the modelling of wave propagation. It has many applications such as noise removal, image encryption, and analysis of optical systems. Here we discuss algorithms for fast and accurate computation of these transforms. These algorithms can achieve the same accuracy and speed as fast Fourier transform algorithms, so that they can be viewed as optimal algorithms. Efficient sampling of signals plays an important part in the development of these algorithms.
Open Access
Linear canonical domains and degrees of freedom of signals and systems
(Springer Verlag, 2016) Öktem, F. S.; Özaktaş, Haldun; Healy, J. J.; Kutay, Mehmet Alper; Özaktaş, Haldun; Sheridan, J. T.
We discuss the relationships between linear canonical transform (LCT) domains, fractional Fourier transform (FRT) domains, and the space-frequency plane. In particular, 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 monotonically ordered by the corresponding fractional order parameter and provides a more transparent view of the evolution of light through an optical system modeled by LCTs. We then study the number of degrees of freedom of optical systems and signals based on these concepts. We first discuss the bicanonical width product (BWP), which is the number of degrees of freedom of LCT-limited signals. The BWP generalizes the space-bandwidth product and often provides a tighter measure of the actual number of degrees of freedom of signals. We illustrate the usefulness of the notion of BWP in two applications: efficient signal representation and efficient system simulation. In the first application we provide a sub-Nyquist sampling approach to represent and reconstruct signals with arbitrary space-frequency support. In the second application we provide a fast discrete LCT (DLCT) computation method which can accurately compute a (continuous) LCT with the minimum number of samples given by the BWP. Finally, we focus on the degrees of freedom of first-order optical systems with multiple apertures. We show how to explicitly quantify the degrees of freedom of such systems, state conditions for lossless transfer through the system and analyze the effects of lossy transfer.
Open Access
Linear canonical transforms, degrees of freedom, and sampling in optical signals and systems
(IEEE, 2014) Özaktaş, Haldun M.; Öktem, F. S.
We study the degrees of freedom of optical systems and signals based on space-frequency (phase-space) analysis. At the heart of this study is the relationship of the linear canonical transform domains to the space-frequency plane. Based on this relationship, we discuss how to explicitly quantify the degrees of freedom of first-order optical systems with multiple apertures, and give conditions for lossless transfer. Moreover, we focus on the degrees of freedom of signals in relation to the space-frequency support and provide a sub-Nyquist sampling approach to represent signals with arbitrary space-frequency support. Implications for simulating optical systems are also discussed.
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.
Open Access
Signal representation and recovery under partial information, redundancy, and generalized finite extent constraints
(2009) Öktem, Sevinç Figen
We study a number of fundamental issues and problems associated with linear canonical transforms (LCTs) and fractional Fourier transforms (FRTs). First, we study signal representation under generalized finite extent constraints. Then we turn our attention to signal recovery problems under partial and redundant information in multiple transform domains. In the signal representation part, we focus on sampling issues, the number of degrees of freedom, and the timefrequency support of the set of signals which are confined to finite intervals in two arbitrary linear canonical domains. We develop the notion of bicanonical width product, which is the generalization of the ordinary time-bandwidth product, to refer to the number of degrees of freedom of this set of signals. The bicanonical width product is shown to be the area of the time-frequency support of this set of signals, which is simply given by a parallelogram. Furthermore, these signals can be represented in these two LCT domains with the minimum number of samples given by the bicanonical width product. We prove that with these samples the discrete LCT provides a good approximation to the continuous LCT due to the underlying exact relation between them. In addition, the problem of finding the minimum number of samples to represent arbitrary signals is addressed based on the LCT sampling theorem. We show that this problem reduces to a simple geometrical problem, which aims to find the smallest parallelogram enclosing a given time-frequency support. By using this equivalence, we see that the bicanonical width product provides a better fit to the actual number of degrees of freedom of a signal as compared to the time-bandwidth product. We give theoretical bounds on the representational efficiency of this approach. In the process, we accomplish to relate LCT domains to the time-frequency plane. We show that each LCT domain is essentially a scaled FRT domain, and thus any LCT domain can be labeled by the associated fractional order, instead of its three parameters. We apply these concepts knowledge to the analysis of optical systems with arbitrary numbers of apertures. We propose a method to find the largest number of degrees of freedom that can pass through the system. Besides, we investigate the minimum number of samples to represent the wave at any plane in the system. In the signal recovery part of this thesis, we study a class of signal recovery problems where partial information in two or more fractional Fourier domains are available. We propose a novel linear algebraic approach to these problems 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, we explore the redundancy and information relations between the given data under different partial information conditions.