Matrices generating eigenvectors for constructing fractional trigonometric transforms

In this paper, we propose a method for obtaining eigenvectors of discrete cosine and sine transforms of types I and IV. The approach is based on constructing an initial eigenvector of one of such trigonometric transforms and a generating matrix. Multiplying powers of the matrix by the initial eigenvector, new eigenvectors are obtained. It is shown how the generated eigenvectors can be used in the fractionalization of the respective transform. Finally, we illustrate the applicability of the developed theory in the scenario of image encryption.

[1]  Hua Yu,et al.  Parameter Estimation of Wideband Underwater Acoustic Multipath Channels based on Fractional Fourier Transform , 2016, IEEE Transactions on Signal Processing.

[2]  Liyun Xu,et al.  Reconstruction of uniformly sampled signals from non-uniform short samples in fractional Fourier domain , 2016, IET Signal Process..

[3]  Utkarsh Singh,et al.  Application of fractional Fourier transform for classification of power quality disturbances , 2017 .

[4]  Wei Zhang,et al.  A generalized convolution theorem for the special affine Fourier transform and its application to filtering , 2016 .

[5]  Soo-Chang Pei,et al.  Generating matrix of discrete Fourier transform eigenvectors , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[6]  Di Xiao,et al.  Edge-based lightweight image encryption using chaos-based reversible hidden transform and multiple-order discrete fractional cosine transform , 2013 .

[7]  Stephen A. Martucci,et al.  Symmetric convolution and the discrete sine and cosine transforms , 1993, IEEE Trans. Signal Process..

[8]  Cagatay Candan,et al.  The discrete fractional Fourier transform , 2000, IEEE Trans. Signal Process..

[9]  Ye Liu,et al.  Single-channel color image encryption algorithm based on fractional Hartley transform and vector operation , 2013, Multimedia Tools and Applications.

[10]  J. McClellan,et al.  Eigenvalue and eigenvector decomposition of the discrete Fourier transform , 1972 .

[11]  Ran Tao,et al.  Multiple-Parameter Discrete Fractional Transform and its Applications , 2016, IEEE Transactions on Signal Processing.

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

[13]  Ljupco Kocarev,et al.  Chaos-Based Cryptography - Theory, Algorithms and Applications , 2011, Chaos-Based Cryptography.