Revised direct batch evaluation of optimal orthonormal eigenvectors of the DFT-IV matrix using the notion of matrix pseudoinverse

Abstract

Batch algorithms for the generation of optimal orthonormal eigenvectors of a square unitary matrix F are presented. Since the unitarity of a matrix implies the orthogonality of its eigenspaces pertaining to its distinct eigenvalues, the problem is decoupled in the sense that optimal eigenvectors are individually generated for the eigenspaces. In one algorithm, the singular value decomposition (SVD) technique is applied to a rectangular matrix whose columns are nonorthogonal eigenvectors of matrix F pertaining to a distinct eigenvalue. In another algorithm, the SVD technique is applied to a square matrix obtained by pre multiplying this rectangular matrix by its complex conjugate transpose. For the sake of generality, this rectangular matrix is allowed to be rank-deficient and the notion of matrix pseudoinverse is resorted to in order to find optimal orthonormal eigenvectors. The general results are applied to the discrete Fourier transform of type IV (DFT-IV) kernel matrix G where the objective is the generation of Hermite-Gaussian-like (HGL) orthonormal eigenvectors as an essential step towards the development of the fractional discrete Fourier transform of type IV (FDFTIV). Since the contributed algorithms necessitate knowledge of the orthogonal projection matrices on the eigenspaces of the unitary matrix G, explicit expressions are derived for those projection matrices. The extensive simulation results show the relative merits of the various algorithms especially for large values of the order N of the matrix.