Showing papers on "Circulant matrix published in 1971"

A matrix theory proof of the discrete convolution theorem

Abstract: In this paper we prove the discrete convolution theorem by means of matrix theory. The proof makes use of the diagonalization of a circulant matrix to show that a circular convolution is diagonalized by the discrete Fourier transform. The diagonalization of the circular convolution shows that the eigenvalues of a circular convolution operator are identical with the discrete Fourier frequency spectrum.

Biased estimation for nonparametric identification of linear systems

Abstract: A technique for solving convolution-type integral equations, previously investigated by the author for the inverse problem of radiography, is extended in this article to biological identification problems. The properties of solutions to convolution-type integral equations are formulated in terms of the eigenvalues of a circulant matrix, the eigenvalues being obtained by diagonalization by the discrete Fourier transform. Statistical properties of the solution, the bias and covariance matrix, are derived, and a discrete Fourier transform technique for computing the covariance matrix is also derived. A brief criticism of biological identification problems in the published literature is also given.

On Hadamard matrices constructible by circulant submatrices

Abstract: Let V2, be an H-matrix of order 2un constructible by using circulant n X n submatrices. A recursive method has been found to construct V4, by using circulant 2in X 2n submatrices which are derived from n X n submatrices of a given V-2. A similar method can be applied to a given W4w, an H-matrix of Williamson type with odd ni, to construct W8,. All V2, constructible by the standard type, for 1 I (see [3]), it is known that many H4,-matrices can be constructed by using circulant submatrices of order n or 2n. (For H-matrices of Williamson type, see [1], [2], [4].) Let V2, be an H2n-matrix constructible by using circulant n X n submatrices. Then V2, can be constructed by the following standard type: (*) M2n = L A B , where A, B are n X n circulant matrices --B TAT and CT means the transposed matrix of C. A recursive method has been found to construct V4, by circulant 2n X 2ii matrices which are derived by circulant n X n submatrices of a given V27.. (See Theorem 1, below.) Likewise, let W4, be an H47.-matrix of Williamson type with odd n; W8' can be constructed by using 2n X 2n symmetric circulant matrices which are derived from n X n symmetric circulant submatrices of a given W47.. (See Theorem 2.) Let Sn = ((ei)) be the n X n circulant matrix with the first row entries ei, (O < i ? n 1), all zero except for e, = 1. Then n X n circulant matrices A, B of (*) can be written as polynomials in S. (We shall omit the suffix n of Sn and others when there is no confusion.) n-1 n-I A = A(S) = E aiSt, B = B.(S) = i S, i=O i.=0 with coefficients ai, bi = 1 or 1; where S =In = the n X n identity matrix. A sufficient condition for the matrix M2n of type (*) being an H-matrix is that M-InM2'n= 2nI2, which is equivalent to (1) AAT + BBT = 2n In. Received March 31, 1970. AMS 1970 subject classifications. Primary 05B20, 62K05, 05A19; Secondary 15A36, 0504, 1504.

A Note on Fast Cyclic Convolution

Abstract: This note presents a new algorithm for computing the cyclic convolution of two vectors over a commutative ring. The algorithm requires n(n 1 +1)...(n k +1)/2kmultiplications for the convolution of two n-vectors, where n=n 1 ...n k is a factorization of n into factors which are pairwise relatively prime.

On Hadamard Matrices Constructible by Circulant Submatrices

Abstract: Let V2, be an H-matrix of order 2un constructible by using circulant n X n sub- matrices. A recursive method has been found to construct V4, by using circulant 2in X 2n