The direct batch generation of Hermite-Gaussian-like eigenvectors of the DFT matrix using the notion of matrix pseudoinverse

The discrete fractional Fourier transform (DFRFT) depends heavily on the availability of Hermite-Gaussian-like (HGL) orthonormal eigenvectors of the DFT matrix F. The direct batch evaluation by constrained optimization algorithm (DBEOA) was developed under the assumption that the orthogonal projection of the approximate eigenvectors on the corresponding eigenspaces results in linearly independent vectors. The present paper handles the case when those vectors are not linearly independent which happens when the order N of matrix F is large. A more general treatment of the batch generation of HGL eigenvectors is presented and the notion of matrix pseudoinverse is used for solving the linear system which arises in the solution of the constrained optimization problem. The contributed technique is termed the Direct Batch Evaluation by constrained Optimization Algorithm using the notion of matrix Pseudoinverse (DBEOAP). The simulation results show that the DBEOAP results in smaller values of the norms of the approximation error vectors than those obtained when applying the DBEOA.