Browsing by Subject "Mathematical operators"
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Item Open Access Digital computation of the fractional Fourier transform(Institute of Electrical and Electronics Engineers, 1996-09) Özaktaş, Haldun M.; Arıkan, Orhan; Kutay, M. A.; Bozdağı, G.An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.Item Open Access Effects of linear filter on stability and performance of human-in-the-loop model reference adaptive control architectures(ASME, 2017) Yousefi, Ehsan; Demir, Didem Fatma; Sipahi, R.; Yücelen, T.; Yıldız, YıldırayModel reference adaptive control (MRAC) can effectively handle various challenges of the real world control problems including exogenous disturbances, system uncertainties, and degraded modes of operations. In human-in-the-loop settings, MRAC may cause unstable system trajectories. Basing on our recent work on the stability of MRAC-human dynamics, here we follow an optimization based computations to design a linear filter and study whether or not this filter inserted between the human model and MRAC could help remove such instabilities, and potentially improve performance. To this end, we present a mathematical approach to study how the error dynamics of MRAC could favorably or detrimentally influence human operator's error dynamics in performing a certain task. An illustrative numerical example concludes the study.Item Open Access EFIE and MFIE, why the difference?(IEEE, 2008-07) Chew W.C.; Davis, C. P.; Warnick, K. F.; Nie, Z. P.; Hu, J.; Yan, S.; Gürel, LeventEFIE (electric field integral equation) suffers from internal resonance, and the remedy is to use MFIE (magnetic field integral equation) to come up with a CFIE (combined field integral equation) to remove the internal resonance problem. However, MFIE is fundamentally a very different integral equation from EFIE. Many questions have been raised about the differences.Item 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.Item Open Access Improving the accuracy of the MFIE with the choice of basis functions(IEEE, 2004-06) Ergül, Özgür; Gürel, LeventThe accuracy of the magnetic-field integral equation (MFIE) for the Method of Moments (MOM) and fast-multiple-method (FMM) solutions of the electromagnetic scattering problems modeled by abitrary planar triangulations is improved by the choice of basis functions. The four different basis functions, Rao-Wilton-Glisson (RWG), TL, n × RWG and n × TL are used to define the planar triangles. The current distribution and the radar cross section (RCS) results are found to be more accurate for the TL and n × TL functions. TL and n × TL functions are used to improve the efficiency inspite of the increased computational cost.Item 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.Item Open Access Morphological subband decomposition structure using GF(N) arithmetic(IEEE, 1996-09) Gürcan, Metin Nafi; Çetin, A. Enis; Gerek, Ömer, N.Linear filter banks with critical subsampling and perfect reconstruction (PR) property have received much interest and found numerous applications in signal and image processing. Recently, nonlinear filter bank structures with PR and critical subsampling have been proposed and used in image coding. In this paper, it is shown that PR nonlinear subband decomposition can be performed using the Gallois Field (GF) arithmetic. The result of the decomposition of an n-ary (e.g. 256-ary) input signal is still n-ary at different resolutions. This decomposition structure can be utilized for binary and 2k (k is an integer) level signal decompositions. Simulation studies are presented.Item Open Access New characterizations of ℓ1 solutions to overdetermined systems of linear equations(Elsevier, 1994) Madsen, K.; Nielsen H. B.; Pınar, M. Ç.New characterizations of the ℓ1 solutions to overdetermined system of linear equations are given. The first is a polyhedral characterization of the solution set in terms of a special sign vector using a simple property of the ℓ1 solutions. The second characterization is based on a smooth approximation of the ℓ1 function using a "Huber" function. This allows a description of the solution set of the ℓ1 problem from any solution to the approximating problem for sufficiently small positive values of an approximation parameter. A sign approximation property of the Huber problem is also considered and a characterization of this property is given. © 1994.Item Open Access Object-oriented query language facilitating construction of new objects(Elsevier, 1993) Alhajj, R.; Arkun, M. E.In object-oriented database systems, messages can be used to manipulate the database; however, a query language is still a required component of any kind of database system. In the paper, we describe a query language for object-oriented databases where both objects as well as behaviour defined in them are handled. Not only existing objects are manipulated; the introduction of new relationships and new objects constructed out of existing ones is also facilitated. The operations supported in the described query language subsumes those of the relational algebra aiming at a more powerful query language than the relational algebra. Among the additional operators, there is an operator that handles the application of an aggregate function on objects in an operand while still having the result possessing the characteristics of an operand. The result of a query as well as the operands are considered to have a pair of sets, a set of objects and a set of message expressions; where a message expression is a sequence of messages. A message expression handles both stored and derived values and hence provides a full computational power without having an embedded query language with impedance mismatch. Therefore the closure property is maintained by having the result of a query possessing the characteristics of an operand. Furthermore, we define a set of objects and derive a set of message expressions for every class; hence any class can be an operand. Moreover, the result of a query has the characteristics of a class and its superclass/subclass relationships with the operands are established to make it persistent. © 1993.Item Open Access On the analyticity of functions approximated by their q-Bernstein polynomials when q > 1(2010) Ostrovskii I.; Ostrovska, S.Since in the case q > 1 the q-Bernstein polynomials Bn,q are not positive linear operators on C[0, 1], the investigation of their convergence properties for q > 1 turns out to be much harder than the one for 0 < q < 1. What is more, the fast increase of the norms ∥Bn,q∥ as n → ∞, along with the sign oscillations of the q-Bernstein basic polynomials when q > 1, create a serious obstacle for the numerical experiments with the q-Bernstein polynomials. Despite the intensive research conducted in the area lately, the class of functions which are uniformly approximated by their q-Bernstein polynomials on [0, 1] is yet to be described. In this paper, we prove that if f:[0,1]→C is analytic at 0 and can be uniformly approximated by its q-Bernstein polynomials (q > 1) on [0, 1], then f admits an analytic continuation from [0, 1] into {z: z < 1}. © 2010 Elsevier Inc. All rights reserved.Item Open Access On the classification of Darboux integrable chains(American Institute of Physics, 2008) Habibullin, I.; Zheltukhina, N.; Pekcan, A.We study differential-difference equation (d/dx) t (n+1,x) =f (t (n,x),t (n+1,x), (d/dx) t (n,x)) with unknown t (n,x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, { t (n+k,x) } k=-∞ ∞, {(dk /d xk) t (n,x) } k=1 ∞, such that Dx F=0 and DI=I, where D x is the operator of total differentiation with respect to x and D is the shift operator: Dp (n) =p (n+1). Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function f is of the special form f (u,v,w) =w+g (u,v). © 2009 American Institute of Physics.Item Open Access Proper orthogonal decomposition for reduced order modeling: 2D heat flow(IEEE, 2003-06) Efe, M. Ö.; Özbay, HitayModeling issues of infinite dimensional systems is studied in this paper. Although the modeling problem has been solved to some extent, use of decomposition techniques still pose several difficulties. A prime one of this is the amount of data to be processed. Method of snapshots integrated with POD is a remedy. The second difficulty is the fact that the decomposition followed by a projection yields an autonomous set of finite dimensional ODEs that is not useful for developing a concise understanding of the input operator of the system. A numerical approach to handle this issue is presented in this paper. As the example, we study 2D heat flow problem. The results obtained confirm the theoretical claims of the paper and emphasize that the technique presented here is not only applicable to infinite dimensional linear systems but also to nonlinear ones.Item Open Access Quantum stereographic projection and the homographic oscillator(American Physical Society, 1996) Hakioǧlu T.; Arik, M.The quantum deformation created by the stenographic mapping from S2 to C is studied. It is shown that the resulting algebra is locally isomorphic to su(2) and is an unconventional deformation of which the undeformed limit is a contraction onto the harmonic oscillator algebra. The deformation parameter is given naturally by the central invariant of the embedding su(2). The deformed algebra is identified as a member of a larger class of quartic q oscillators. We next study the deformations in the corresponding Jordan-Schwinger representation of two independent deformed oscillators and solve for the deforming transformation. The invertibility of this transformation guarantees an implicit coproduct law which is also discussed. Finally we discuss the analogy between Poincaré's geometric interpretation of the quantum Stokes parameters of polarization and the stereographic projection as an important physical application of the latter.Item Open Access Query model for object-oriented databases(IEEE, 1993-04) Alhajj, Reda; Arkun, M. ErolA query language should be a part of any database system. While the relational model has a well defined underlying query model, the object-oriented database systems have been criticized for not having such a query model. One of the most challenging steps in the development of a theory for object-oriented databases is the definition of an object algebra. A formal object-oriented query model is described here in terms of an object algebra, at least as powerful as the relational algebra, by extending the latter in a consistent manner. Both the structure and the behavior of objects are handled. An operand and the output from a query in the object algebra are defined to have a pair of sets, a set of objects and a set of message expressions where a message expression is a valid sequence of messages. Hence the closure property is maintained in a natural way. In addition, it is proved that the output from a query has the characteristics of a class; hence the inheritance (sub/superclass) relationship between the operand(s) and the output from a query is derived. This way, the result of a query can be persistently placed in its proper place in the lattice.Item Open Access Radar cross-section study of cylindrical cavity-backed apertures with outer or inner material coating: the case of E-polarization(IEEE, 1993) Çolak, D.; Nosich, A. I.; Altintaş, A.A dual-series-based solution is obtained for the scattering of an E-polarized plane wave from a cavity-backed aperture which is formed by a slitted infinite circular cylinder coated with absorbing material. The material coating can be done on the inner or outer surface of the cylinder. For both cases, numerical results are presented for the radar cross section and comparisons are given for two different realistic absorbing materials. The radar cross-section results are also given for the aspect angle of the screen. Finally, the dependence of radar cross section on the thickness of the absorbing layer is presented. © 1993 IEEEItem Open Access Signal recovery from partial fractional fourier transform information(IEEE, 2004) Çetin, A. Enis; Özaktaş, H.; Özaktaş, Haldun M.The fractional Fourier transform has found many applications in signal and image processing and optics. An iterative algorithm for signal recovery from partial fractional Fourier transform information is presented. The signal recovery algorithm is constructed by using the method of projections onto convex sets and convergence of the algorithm is assured.Item Open Access Teager energy based feature parameters for robust speech recognition in car noise(IEEE, Piscataway, NJ, United States, 1999) Jabloun, F.; Çetin, A. EnisIn this paper, a new set of speech feature parameters based on multirate signal processing and the Teager Energy Operator is developed. The speech signal is first divided into nonuniform subbands in mel-scale using a multirate filter-bank, then the Teager energies of the subsignals are estimated. Finally, the feature vector is constructed by log-compression and inverse DCT computation. The new feature parameters have a robust speech recognition performance in car engine noise which is low pass in nature.Item Open Access Vortex lattice of a Bose-Einstein condensate in a rotating anisotropic trap(American Physical Society, 2004) Oktel, M. Ö.The behavior of Bose-Einstein condensates (BEC) with a vortex lattice in a rotating anisotropic trap was studied. The single-particle wave functions for a rotating anisotropy trap were found by exactly diagonalizing the Hamiltonian. The wave functions in the lowest Landau level (LLL) were shown to be written as simple analytic functions in the small anisotropy limit. The results show that the regularity of the vortex configurations observed in an axisymmetric trap is disturbed by the anisotropy of the confinement.