Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
V.D.Sharma Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.1463-1...
V.D.Sharma Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05
...
V.D.Sharma Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05
...
V.D.Sharma Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05
...
V.D.Sharma Int. Journal of Engineering Research and Application
ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05
...
Upcoming SlideShare
Loading in …5
×

Ij3514631467

177 views

Published on

International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.

Published in: Technology
  • Be the first to comment

  • Be the first to like this

Ij3514631467

  1. 1. V.D.Sharma Int. Journal of Engineering Research and Application ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.1463-1467 RESEARCH ARTICLE www.ijera.com OPEN ACCESS Modulation and Parsvels Theorem For Generalized Fractional Fourier Transform V. D. Sharma Mathematics Department, Arts, Commerce and Science College, Amravati, 444606 (M.S.) India Abstract The fractional Fourier transform (FrFT) was first introduced as a way to solve certain classes of ordinary and partial differential equations arising in quantum mechanics. FrFt has established itself as a powerful tool for the analysis of time-varying signals, especially in optics. FrFt has found applications in areas of signal processing such as repeated filtering, fractional convolution and correlation, beam forming, optimal filter, convolution, filtering and wavelet transforms, time frequency representations. In this paper operational calculus for Generalized fractional Fourier transform are obtained. Keywords: Generalized function, Fractional Fourier transform, Fourier transform, signal processing, Test function space. I. Introduction The Fourier transform is one of the most important mathematical tools used in Physical optics, linear system theory, signal processing and so on [1]. The conventional Fourier transform can be regarded as a rotation in the time-frequency plane and the FrFT performs a rotation of signal to any angle. Moreover, fractional Fourier transform serves as an orthonormal signal representation for chirp signal. The Fractional Fourier transform is also called rotational Fourier transform or angular Fourier transform in some documents. The FrFT is also related to other timevarying signal processing tools such as Wigner distribution, Short time Fourier transform, Wavelet transform [2]. In our previews work we have defined in [3, 4] as, 1.1. Conventional fractional Fourier transform: The FrFT with parameter of denoted by performs a linear operation given by Where 1.2 The test function space E: An infinitely differentiable complex valued smooth function on belongs to if for each compact set , where 0, I Rn. , where p=1, 2, 3..... Thus will denote the space of all with suppose contained in . Note that the space E is complete and therefore a Frechet space. Moreover we say that is a fractional Fourier transformable if it is a member of , the dual space of . FrFT has many applications in solution of differential equations, optical beam propagtioan and spherical mirror resonators, optical diffraction theory, quantum mechanics, statistical optics, signal detectors, pattern recognition, space image recovery etc. [5, 6]. In the present work Generalization of the FrFT is presented by kernel method. Scaling property, Modulation theorem, and Parsvels theorem is proved for Generalized Fractional Fourier transform. II. Distributional Fractional Fourier Transform The transforms of Distributional fractional Fourier can be defined by where www.ijera.com 1463 | P a g e
  2. 2. V.D.Sharma Int. Journal of Engineering Research and Application ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05 The right hand side of (3.1) has a meaning as the application of III. www.ijera.com to . Scaling Property where Proof: putting where Parseval’s Identity for Fractional Fourier transform IV. If and then i. ii. Proof: By definition of using inversion formula of Taking complex conjugate we get Now www.ijera.com 1464 | P a g e
  3. 3. V.D.Sharma Int. Journal of Engineering Research and Application ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05 Hence proved ii) Let eqn (4.2) becomes then If denotes generalized fractional Fourier transform of and V. www.ijera.com then Modulation property for FrFT then Proof: 6. If www.ijera.com denotes generalized fractional Fourier transform of then 1465 | P a g e
  4. 4. V.D.Sharma Int. Journal of Engineering Research and Application ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05 www.ijera.com Proof: Generalized Fractional Fourier Transform Sr. No 1. 2. 3. 4. www.ijera.com 1466 | P a g e
  5. 5. V.D.Sharma Int. Journal of Engineering Research and Application ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.01-05 www.ijera.com 5. 6. 7. 8. VI. Conclusion I n this paper Modulation theorem, Scaling property, Parsvels theorem is proved for Generalized Fractional Fourier transform. References [1]. [2]. [3]. [4]. [5]. [6]. [7]. [8]. [9]. R. N. Bracewll, The Fourier Transform and its Application, 2nd ed., McGraw-Hill, New York, 1986 H.M. Ozaktes, A. Zalevsky, M.A. Katay, the FRFT with Application in optics and signal processing, John Wiky, Chichester New York, USA, 2001. V.D.Sharma, Generalization of Twodimensional Fractional Fourier transforms, Asian Journal of current Engineering and Maths, 46-48, 2012. V.D.Sharma, operation transform Formulae on Generalized FrFT, processing of Int. conf. in Recent trends in Inf. Tech. and Comp. Science published in Int. J. of computer applications 15-18, 2012. Bultheel A. et.al., Recent developments in the theory of the fractional Fourier & linear canonical transform, Preprint (2004), 1-33. Felipe Salazon, An Introduction to the Fractional Fourier transform and its applications, Dec. 7, (2011) 1-8 Zemanian A.H., Generalized Integral transform, Interscience Publication, New York, 1968. B.N.Bhosale, Integral transformation of Generalized functions. Discovery publishing hour, 2005. Ervin S., Djurovic I. et.al., fractional Fourier transform as a signal processing tool; an review of recent developments, signal processing (2011), 1351-1369. www.ijera.com 1467 | P a g e

×