SOO-CHANG PEIJIAN-JIUN DING2018-09-102018-09-102004-071053587Xhttp://scholars.lib.ntu.edu.tw/handle/123456789/310460https://www.scopus.com/inward/record.uri?eid=2-s2.0-3142715345&doi=10.1109%2fTSP.2004.828904&partnerID=40&md5=011cadeab94ea196477579f093ee9191The 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 water-marking, etc. © 2004 IEEE.application/pdfapplication/pdfConvolution; Cosine transforms; Cryptography; Data compression; Digital filters; Digital watermarking; Discrete Fourier transforms; Eigenvalues and eigenfunctions; Frequency domain analysis; Matrix algebra; Polynomials; Theorem proving; Commutative matrix method; Discrete cosine transforms; Discrete fractional Fourier transform; Hermite polynomial; Offset discrete Fourier transform; Digital signal processingjournal article10.1109/TSP.2004.8289042-s2.0-3142715345WOS:000222136200017