Generalized eigenvectors and fractionalization of offset DFTs and DCTs

The offset discrete Fourier transform (DFT) is a discrete transform with kernel exp[-j2π(m-a)(n-b)/N]. It is more generalized and flexible than the original DFT and has very close relations with the discrete cosine transform (DCT) of type 4 (DCT-IV), DCT-VIII, discrete sine transform (DST)-IV, DST-VIII, and discrete Hartley transform (DHT)-IV. In this paper, we derive the eigenvectors/eigenvalues of the offset DFT, especially for the case where a+b is an integer. By convolution theorem, we can derive the close form eigenvector sets of the offset DFT when a+b is an integer. We also show the general form of the eigenvectors in this case. Then, we use the eigenvectors/eigenvalues of the offset DFT to derive the eigenvectors/eigenvalues of the DCT-IV, DCT-VIII, DST-IV, DST-VIII, and DHT-IV. After the eigenvectors/eigenvalues are derived, we can use the eigenvectors-decomposition method to derive the fractional operations of the offset DFT, DCT-IV, DCT-VIII, DST-IV, DST-VIII, and DHT-IV. These fractional operations are more flexible than the original ones and can be used for filter design, data compression, encryption, and watermarking, etc.