Direct sequential evaluation of optimal orthonormal eigenvectors of the discrete Fourier transform matrix by constrained optimization

The recent emergence of the discrete fractional Fourier transform has spurred research activity aiming at generating Hermite-Gaussian-like (HGL) orthonormal eigenvectors of the discrete Fourier transform (DFT) matrix F. By exploiting the unitarity of matrix F – resulting in the orthogonality of its eigenspaces pertaining to the distinct eigenvalues – the problem decouples into finding orthonormal eigenvectors for each eigenspace separately. A Direct Sequential Evaluation by constrained Optimization Algorithm (DSEOA) is contributed for the generation of optimal orthonormal eigenvectors for each eigenspace separately. This technique is direct in the sense that it does not require the generation of initial orthonormal eigenvectors as a prerequisite for obtaining the final optimal ones. The resulting eigenvectors are optimal in the sense of being as close as possible to samples of the Hermite-Gaussian functions. The technique is found to be numerically robust because total pivoting is allowed in performing the QR matrix decomposition step. The DSEOA is proved to be theoretically equivalent to each of the Gram-Schmidt algorithm (GSA) and the sequential orthogonal procrustes algorithm (SOPA). However the three techniques are algorithmically quite distinct. An extensive comparative simulation study shows that the DSEOA is by far the most numerically robust technique among all sequential algorithms thus paying off for its relatively long computation time.