The orthogonal projection matrices on the eigenspaces of the DFT-IV matrix

Abstract

Since having orthonormal Hermite-Gaussian-like eigenvectors of the DFT-IV matrix G is essential for developing a fractional discrete Fourier transform of type IV (FDFTIV), some methods for the generation of those eigenvectors are analyzed in a detailed simulation study involving evaluating the execution time, orthonormality error and approximation error. Since six of the nine methods included in the study necessitate knowledge of the orthogonal projection matrices on the eigenspaces of the DFT-IV matrix, explicit expressions are derived for those matrices. Based on this contribution it is no longer essential to generate the eigenvectors of a nearly tridiagonal matrix S which commutes with matrix G as a way for obtaining eigenvectors of the latter. The simulation results show the tradeoff between the speed of execution and the numerical robustness of the computation of the various techniques.