Discrete Pseudo-Fractional Hadamard Transform and its Fast Algorithm

In this paper a new discrete fractional transform for data vectors whose size <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula> is a power of two is proposed. The basic operation of the introduced transform is a discrete fractional Hadamard transform. Since the described transform is not a classical discrete fractional Hadamard transform in its pure form, we called it a pseudo-fractional discrete Hadamard transform. A generalization of the new transform is also defined so that it depends on many parameters. Fast algorithm for calculating the proposed transform is developed.

[1]  M. Alper Kutay,et al.  Image representation and compression with the fractional Fourier transform , 2001 .

[2]  S. Pei,et al.  Improved discrete fractional Fourier transform. , 1997, Optics letters.

[3]  Soo-Chang Pei,et al.  The discrete fractional cosine and sine transforms , 2001, IEEE Trans. Signal Process..

[4]  Granino A. Korn,et al.  Mathematical handbook for scientists and engineers. Definitions, theorems, and formulas for reference and review , 1968 .

[5]  Ioannis Pitas,et al.  Digital watermarking in the fractional Fourier transformation domain , 2001, J. Netw. Comput. Appl..

[6]  Cagatay Candan,et al.  The discrete fractional Fourier transform , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[7]  R. W. Johnson,et al.  A methodology for designing, modifying, and implementing Fourier transform algorithms on various architectures , 1990 .

[8]  Naitong Zhang,et al.  A novel fractional wavelet transform and its applications , 2011, Science China Information Sciences.

[9]  Soo-Chang Pei,et al.  Tridiagonal Commuting Matrices and Fractionalizations of DCT and DST Matrices of Types I, IV, V, and VIII , 2008, IEEE Transactions on Signal Processing.

[10]  Ran Tao,et al.  The multiple-parameter discrete fractional Hadamard transform , 2009 .

[11]  Deyun Wei,et al.  Novel Tridiagonal Commuting Matrices for Types I, IV, V, VIII DCT and DST Matrices , 2014, IEEE Signal Processing Letters.

[12]  Ran Tao,et al.  Analysis and comparison of discrete fractional fourier transforms , 2019, Signal Process..

[13]  Kehar Singh,et al.  Fully phase encryption using fractional Fourier transform , 2003 .

[14]  Alexander Graham,et al.  Kronecker Products and Matrix Calculus: With Applications , 1981 .

[15]  R. Yarlagadda,et al.  A note on the eigenvectors of Hadamard matrices of order 2n , 1982 .

[16]  J. Sylvester.  LX. Thoughts on inverse orthogonal matrices, simultaneous signsuccessions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers , 1867 .

[17]  John T. Sheridan,et al.  Fractional Fourier transform-based image encryption: phase retrieval algorithm , 2003 .

[18]  Dorota Majorkowska-Mech,et al.  Fast algorithm for discrete fractional Hadamard transform , 2014, Numerical Algorithms.

[19]  S. Pei,et al.  Discrete fractional Hartley and Fourier transforms , 1998 .

[20]  S Liu,et al.  Optical image encryption with multistage and multichannel fractional Fourier-domain filtering. , 2001, Optics letters.

[21]  Dorota Majorkowska-Mech,et al.  A New Fast Algorithm for Discrete Fractional Hadamard Transform , 2019, IEEE Transactions on Circuits and Systems I: Regular Papers.

[22]  Zhang Naitong,et al.  A novel fractional wavelet transform and its applications , 2012 .

[23]  Soo-Chang Pei,et al.  Discrete fractional Hadamard transform , 1999, ISCAS'99. Proceedings of the 1999 IEEE International Symposium on Circuits and Systems VLSI (Cat. No.99CH36349).