An FPT algorithm with a modularized structure for computing two-dimensional discrete Fourier transforms

With the change of variables U = U*q, substitution into (21) results in UCU*q = Ud. Consequently, the solution vector U of (20) can be obtained from the solution vector q of (22) or q can be obtained from U. Note that if a system of real equations is Toeplitz-plus-Hankel (T + H) , where T i s symmetric Toeplitz and H i s skew-centrosym-metric Hankel, then the equations may be transformed into Her-mitian Toeplitz and solved with 1 .25n2 + O(n) complex multiplies or 3.752, + O (n) real multiplies. This is a significant improvement in complexity over the approach of [8] which requires 12n2 + O (n) real multiplies, and is slightly lower in complexity than the approach found in [9] which uses an entirely different development and requires 6n2 + O (n) real multiplies. 111. CONCLUSION In this correspondence, we have shown that constant unitary matrices exist which transform a Hermitian Toeplitz matrix into a real Toeplitz-plus-Hankel structure. As a consequence of this property, some real symmetric Toeplitz-plus-Hankel matrices may be converted to Hermitian Toeplitz matrices and vice versa. The unitary matrices presented are different from the one given in [2] and have two advantages: i) they transform Hermitian Toeplitz matrices into real matrices also possessing a special form, i.e., Toeplitz-plus-Hankel, (the result of [ 2 ] does not) and ii) the unitary matrices are simple to express mathematically, thus making it easier to manipulate and interpret results based on them for analytical purposes. A consequence of the unitary transformation U , presented in this correspondence is a relationship between the real and imaginary parts of eigenvectors of Hermitian Toeplitz matrices. This relationship also differs from the eigenvector symmetry property often given in the literature, e. g. , [I]. As a practical observation on eigenspace decomposition consider the following. If all eigenvalues and eigenvectors are required, it is more efficient to transform a Hermitian Toeplitz matrix to a real matrix (using either the result of this paper o r that of [2]) and use a standard algorithm such as the QR algorithm o r Jacobi rotations [ 5 ]. If only a few eigenvaludeigenvector pairs are needed, however , it will be more efficient to work on with the Hermitian Toe-plitz matrix using a fast algorithm such as the Toeplitz eigensystem solver (TESS) [IO] that exploits the special structure and can find specific eigenpairs. Abstract-In this correspondence, …