Eigenvalues and eigenvectors of generalized DFT, generalized DHT, DCT-IV and DST-IV matrices

In this paper, the eigenvalues and eigenvectors of the generalized discrete Fourier transform (GDFT), the generalized discrete Hartley transform (GDHT), the type-IV discrete cosine transform (DCT-IV), and the type-IV discrete sine transform (DST-IV) matrices are investigated in a unified framework. First, the eigenvalues and their multiplicities of the GDFT matrix are determined, and the theory of commuting matrices is applied to find the real, symmetric, orthogonal eigenvectors set that constitutes the discrete counterpart of Hermite Gaussian function. Then, the results of the GDFT matrix and the relationships among these four unitary transforms are used to find the eigenproperties of the GDHT, DCT-IV, and DST-IV matrices. Finally, the fractional versions of these four transforms are defined, and an image watermarking scheme is proposed to demonstrate the effectiveness of fractional transforms