Persymmetric matrix

Abstract: This letter presents a method to estimate the directions of arrival (DOAs) by taking spatial averaging algorithm for circular array. By utilizing the symmetric configuration of the circular array to form a centrosymmetric array, the covariance matrix of the received data can be constructed in the form of the Hermitian persymmetric matrix, and then the spatial averaging algorithm can be applied to estimate the DOAs. Compared with the conventional MUSIC method, the method proposed in this letter has a better performance at low SNR and few snapshots. Simulations are performed to demonstrate the effectiveness of our method.

Abstract: In this study, the authors deal with the problem of adaptive detection of point-like targets in Gaussian disturbance with unknown but persymmetric structured covariance matrix induced by the space and/or time symmetry of the sensing system. In this framework, they devise and assess two selective receivers exploiting the Rao test and the generalised likelihood ratio test design criteria. The performance assessment, conducted by Monte Carlo simulation, has shown that the proposed receivers can significantly outperform their unstructured counterparts and guarantee enhanced rejection performance of unwanted signals with respect to their natural competitors.

Abstract: Both computer time and storage can be saved in the calculation of the eigensystem of Hermitian persymmetric matrices if advantage is taken of their special structure. For any vector v, let vt denote its transpose and let J denote the permutation matrix obtained by reversing the rows of the identity matrix I; Ji ,n+1i = Iii for n X n matrices. Note that J = J' = J-' Definition. M is persymmetric if JMJ = Mt. Note that all Toeplitz matrices (ti, = t. + , I+ 1) are persymmetric. Any complex persymmetric matrix P of even order can be written in partitioned form as (1) P = [JF c] with C = C', F= F'. JF JB tJ If, in addition, P is Hermitian (Pt = 1) then we will split it into real and imaginary parts: (2) P A K J] + { J 1JH JA J, JN JSJ where A, H, N are real symmetric and S is real and skew (St = -S). The form of (2) suggests applying P to vectors of special form. Let x, y be any real vectors conformable with A and write (3) ~ ~ ~ ~ v V x + i . ix -Y. Then, we find that (4) Pv r (A + H)x + (NS)y 1 +F (N + S)x + (AH)y 1 J(A + H)x + J(NS)yi -J(N + S)xJ(AH)y Consequently, v will be an eigenvector of P corresponding to an eigenvalue X if and only if Received February 22, 1973. AMS (MOS) subject classifications (1970). Primary 15A18, 65F15.

Abstract: The problem of generating a matrix A with specified eigenpair, where A is an anti-symmetric and persymmetric matrix, is presented. The solvability conditions are studied. A general expression of such a matrix is provided. We denote the set of such matrices by En. The best approximation problem associated with En is discussed, that is: to find the nearest matrix to a given matrix A* by A En. The existence and uniqueness of the best approximation problem is proved and the expression of this nearest matrix is provided.