In this paper, we have &nbsp;established the N- dimentional fractional Fourier transform and its mathematical expression in a easier manner and discus the&nbsp; eigenvalues and eigenfunctions of &nbsp; -dimensional fractional Fourier transform

Iqbal, Saleem

Rehman, Abdul

Sarwar, Farhana

Ullah, Inayat

English

American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS)

  210  American Scientific Research Journal for Engineering, Technology,  and Sciences  (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers  http://asrjetsjournal.org/   N-dimensional  Fractional  Fourier Transform and its Eigenvalues and Eigenfunctions Inayat Ullaha, Saleem Iqbalb*, Farhana Sarwarc, Abdul  Rehmand a,b,dDepartment of Mathematics, University of Balochistan, Quetta 87300, Pakistan cDepartment of Mathematics F.G.Girls Degree College, Madrissa Road , Quetta, Cantt, 87300, Pakistan aEmail: inayat1611@yahoo.com bEmail: saleemiqbal81@yahoo.com cEmail: f_saleem10@yahoo.com dEmail: abdul_maths@yahoo.com Abstract In this paper, we have  established the N- dimentional fractional Fourier transform and its mathematical expression in a easier manner and discus the  eigenvalues and eigenfunctions of   -dimensional fractional Fourier transform. Keywords: Fourier transform; Fractional Fourier Transform; Eigenvalues;Eigenfunctions . 1. Introduction  In 1929 Norbert Wiener introduce the idea of the fractional power of the Fourier operator[1], later  in  1in 1937 and 1939 Condon and   Kober  extended  this  idea of fractional order [2,3] but it is  established in more precise manner by  Victor Namias in 1980 and  introduced  the fractional Fourier transform as a generalized form of classical Fourier transform  and successfully established the mathematical properties of this very  important transform, he also applied the FRFT to solve the Schrodinger wave equations [4]. Later on McBride and Kerr extended this 1idea of fractional powers of the Fourier operator in terms of mathematical formulation [ 5] and in 1994 Almaida used this transform in time frequency plane [6]. FRFT has been used quantum mechanics , optics  and signal processing. Since 1990 the mathematician, physicist  and  scientist  highly attracted towards this famous transforms a series of boom publication was published and it is still going on. We are  in this article focus with the mathematical formulation  of N-dimensional fractional Fourier transform and investigating their eigenvalues eigenfunctions in a similar manner as for classical Fourier transform. ------------------------------------------------------------------------ * Corresponding author American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (202019) Volume 62, No  1, pp 210-214  211  1.1. The  N-dimensional Fourier transform (FT) In this section we  recall  definition and some of the properties and eigenvalues and eigenfunction of the classical continuous Fourier transform so that it will motivate to dicuss the eigen values for N-dimensional  fractional Fourier transform.  The Fourier and its inverse Fourier transform is defined by the following pair of equations  ( )   √  ∫  ( )                                      ( )  ( )   √  ∫  ( )                                     ( ) The one of the  main application of Fourier transform (FT) find  in designing of filters real filters, reconstruction of signal from digital to analog conversion, sinusoidal modulation and pulse amplitude modulation schemes in facial recognition the predefined eigenfaces are used to construct the images from eigenvalues [7]. One of the important role of eigenfunctions  as a role in linear time invariant  systems where eigenfunctions represent as  arbitrary signals .The eigenvalues and eigenfunctions of the ordinary Fourier transform are well known and discussed in [8]. There are Hermite-Gaussian functions  ( )commonly known as the eigensolutions of the harmonic oscillator in quantum mechanics or the modes of propagation of quadratic graded-index media in optics. The eigenvalues may be expressed as    (      ) and the eigen value equation for the ordinary Fourier transform may be written as     ( )           ( )        ( ) A detailed discussions is found in [9] on the analysis of eigenvalues eigenfunctions of   N- dimensional Fourier transform  and are given by  (√  )   (√  )   (√  )    (√  )   There are number of applications of eigenvalues and eigenfunctions are found   in various areas like  of eigenfunctions is also find in image processing, voice recognition, medical imaging analysis, handwriting analysis, lip reading and sign language/hand gestures interpretation[9]. In case a signal is represented in linear combination of eigenfunctions then it will reduce the evalution of spectrum and also useful in developing . This representation is also useful in developing  the new signal compression techniques.The eigenfunction which correspond the eigenvalues  are also useful in study of signal spaces spanned and  can revolutionize the transmission of data such as   higher dimensional signals like images, video etc.  2. Results and Discussions 2.1. One  dimensional Fractional Fourier transform  The  one dimensional linear operator version of fractional Fourier transform is given  as follows, American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (202019) Volume 62, No  1, pp 210-214  212    , ( )-    , ( )-  ∫  ( )  (   )                                 ( ) The   (   ) is the kernel of transform defined in [6]   (   )  { ( )     .       /                                                                           (   )                                                                                                                (   )                                                                                              (    )         (5) where  ( )  √         ,     ,    , and   The inverse-FRFT with respect to angle   is the FRFT with angle    given by from eqn (4) is  ( )  ∫  ( )   (   )                                                                     (6) Since FRFT is  the generalized  form of  Fourier transform. In  classical Fourier transform   the  concept of rotating  angle is      where in  FrFT the  concept of rotating  angleis generalized to over an arbitrary angle    =       with    . Where   is the order Fractional Fourier transform  This FRFT operator shall be denoted as     where        corresponds to the classical FT operator over rotational angle     , therefore it is  similar to the Fourier Transform whenever the rotation of angle is      , for     then      represent the axis as the reverese time axix  known as  the parity operator , i.e., the time axis rotated over an angle  , for       the rotation of the axis over an angle      or      which represent the inversion Fourier transform  ,i.e.,          . It will now be clear that by another rotation of angle     ,i.e.,       and    , which brings us back to the original time axis, hence        [6].  2.2. Two dimensional Fractional Fourier transform  TIn two-dimensional (2D)-fractional Fourier transform that is extension of equation (4) with two variable and with two angles it is denoted as     * (     )+          * (     )+(     )          (     )   ∫ ∫  (     )       (           )                  ( )         where        (           )      (     )    (     )  and defined        (           )   ( )    { .         /        .         /                            }                                                             ( ) American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (202019) Volume 62, No  1, pp 210-214  213  Since two dimensional fractional Fourier transform is the generalized form of two dimensional (2-D) classical Fourier transform if  two angles of rotation replaced by                  n                        it will reduce the(2-D)FT with same rotational axis properties . If one of these angle is zero, then  (2-D) FRFT reduced to (1-D) FRFT.  2.3. N- dimensional Fractional Fourier transform  In a similar manner  equation (6) can be extended to 3-D and so on up to n-dimensional fractional Fourier transform. Let us consider a function with n variables and with n rotational angles the FRFTcan be denoted and defined defined as follows     * (         )+             * (         )+(          )             (          )   ∫  ∫  (     )       (           )                   ( )         Where            (                    )  ( )    { (         )        (         )          (         )                                    }                                                             ( ) The N- dimensional fractional Fourier transform is the generalized form of N- dimensional classical Fourier transform if  N-angles of rotation replaced by                          furthe if each rotational angle is rotated over an angle     it will posses the same roatational properties of N-D Fourier transform. If one of these angle is zero, then  N-D FRFT reduced to (n-1) dimensional  FRFT.  2.4. Eigenvalues and Eigenfunctions The eigenfunctions of the fractional Fourier transform are the Hermite functions which is same as for classical N-dimensional Fourier transform also hold eqn (3) and the corresponding eigenvalues are    ( )                                                 If we put       leads to identity operator  and we take      the result leads to the parity operator. If we define the order n of the fractional Fourier transform by       (   )  we see that the regular Fourier transform is of order 1. Order 0 is the identity operation and order 2 is the parity transformation. Negative orders correspond to inverse transforms. The whole range of transforms is covered by the orders —        . Transforms with non-integral values of n may be called Fourier transforms of fractional order. Thus we can define the eigen values for the N-D fractional Fourier transform as ( ( )    ) ,  where             If we consider       then  it will be eigenvalues  of I-D fractional Fourier transform where         For       then it will eigenvalues of  2-D fractional Fourier transform ain this case            American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (202019) Volume 62, No  1, pp 210-214  214      Similarly if     then it will eigenvalues for N-D fractional Fourier transform and                   For if                 then it will easily obtain the same eigenvalues as for N-dimensional Fourier transform (√  )   (√  )   (√  )    (√  )  3.  Conclusion  We have  eatablished the mathematical formulation of N-dimesional Fractional Fourier transform in very easier manner and discuss their eigenvalues and eigen functions in a simila manner as for we have classical Fourier transform it will help to apply these concepet in more meaning full manner in various applications specially in case of face deduction from images.  References [1] N. Wiener. Hermitian polynomials and Fourier analysis. J. Math. Phys., 8:70{73, 1929. [2] H. Kober, Wurzeln aus der Hankel-, Fourier- und aus anderen stetigen transfornationen, Q. J Math.(Oxford),   10, 45–59 (1939).  [3]  E.U. Condon. Immersion of the Fourier transform in a continuous group of functional transformations. Proc.National Academy Sciences, 23:158{164, 1937. [4] V. Namias, “The fr ction l or er Fourier tr nsform  n  its  pplic tion to qu ntum  mech nics”, IMA Journal    of Applied Mathematics, 25(3) 241-265(1980) [5]  A.C. McBride and F.H. Kerr. On Namias's fractional Fourier transforms. IMA J. Appl. Math., 39:159-    175, 1987. [6]  L.B. Almeida. The fractional Fourier transform and time-frequency representation.IEEE Trans. Sig. Proc.,   42:3084-3091, 1994 [7]  http://en.wikipedia.org/wiki/Eigenface  [8]  Y.K run   n  R. Dhuli  et.el “ Complete  n lysis of the eigen functions of Fourier tr nsform”    International Conference on Computer Communication and Informatics , India  (ICCCI -2013)  [9 ] R.C. Gonzalez, R.E. Woods, and S.L. Eddins. Digital image processing using MATLAB. Prentice Hall    Upper Saddle River, NJ, 2004.  

N-dimensional  Fractional  Fourier Transform and its Eigenvalues and Eigenfunctions

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N-dimensional Fractional Fourier Transform and its Eigenvalues and Eigenfunctions

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