Showing papers on "Circulant matrix published in 1977"

Toeplitz and circulant matrices

Abstract: The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

Circulant weighing matrices

Abstract: CHAPTER I HISTORY AND APPLICATIONS ........................... 1 CHAPTER II BASIC PROPERTIES ................................... 8 A Geometric Visualisation .......................... 14 Equivalence of Circulant Matrices .................. 14 Extending Circulant Matrices ...................... 16 CHAPTER III EXISTENCE ............................................ 20 Projective Planes .............................. 20 An Inequality ....................................... 23 Circulant Weighing Matrices of Weight 4 26 Reduction Theorems ............................... 32 CHAPTER IV OVALS IN CYCLIC PROJECTIVE PLANES ................... 36 Ovals in Finite Projective Planes .................. 36 The Wallis-Whiteman Theorem. .......................... 42 Equations in Finite Cyclic Projective Planes . . . . 46 Postscript........................................... 48 REFERENCES ........................................................ 49 1 CHAPTER I HISTORY AND APPLICATIONS When I first learned that some mathematicians spend their time studying matrices with entries from (-1, 0,1} , my reaction was like that of a young upperclass lady to the local sanitary can collector! Aren't matrices passe? What kind of mathematicians are these that haven’t heard of linear transformations? Why study matrices with entries in (-1, 0, 1} ? How does such a study relate to the total intrastructure of mathematics? Specifically we are interested in weighing matrices; orthogonal integer matrices with entries 0,1 or -1 which have the same number of zeros in each row. Historically the study of such matrices began with the work of James Sylvester in 1876 and Jacques Hadamard in 1893. In 1893 Hadamard [12] showed that the absolute value of an n x n determinant, all of whose entries were complex and lay within the unit disc, is no greater than n . Hadamard then showed that if such a determinant attained the bound, then all the entries lie on the perimeter of the unit disc (that is, the unit circle) and the rows of the determinant are pairwise orthogonal. Thus in the case when all the entries of the determinant are real, the entries must all be either -1 or 1 and in this case Hadamard showed that either n 2 or n is divisible by four. Sylvester [26] had already constructed a family of real n x n determinants with these properties when n was a power of two. Real n x n matrices whose determinants have the above properties are called Hadamard matrices and are examples of weighing matrices. Weighing matrices also arise naturally in a practical context. Suppose we have a balance which records the difference in weight between the right and the left pans. How can we determine the weight of n objects as accurately as possible in n weighings? As an example, suppose that we have two objects of weights and x2 . Let e be the error made each time the balance is used. We suppose 2 that e is a random variable with zero mean and variance G . Make two measurements, the first with both the objects in the left pan and the second with one object in each pan. If and e2 are the errors made in the first and second weighings respectively and if y and y ̂ are the first and second measurements respectively, then (1.1) y± = x± + x2 + e1 , (1.2) y2 = x± x2 + e2 . Equations (1.1) and (1.2) can be solved easily to give us estimates of and x2 . The distributions of the estimates obtained for and x2 2 using this procedure have variance ö /2 while the variance of the distributions of the estimates of x ̂ and x2 obtained by weighing each object separately have variance G Equations (1.1) and (1.2) may be written

Constrained least squares filtering

Abstract: In the following we: 1) present a frequency domain derivation of the constrained least squares filter, which is much simpler than previous approaches relying on the diagonalization of block circulant matrices; 2) derive a lower bound on the Lagrange multiplier appearing in the filter and obtain conditions under which the Lagrange multiplier may assume negative values; and 3) indicate how the frequency domain expression for the error energy can be used to reduce computation time in the iterative determination of the Lagrange multiplier.

Fast Computational Techniques for Pseudoinverse and Wiener Image Restoration

Abstract: New computational techniques for pseudoinverse and Wiener image restoration are presented. These techniques are based upon the modification of an observed image array so that the numerical model for the image blur contains a circulant matrix blur operator. Utilization of Fourier transform processing techniques then permits scalar computation, as contrasted to vector computation required for standard processing. Several image restoration examples are given.

Orthogonal designs in order 24

Abstract: This paper uses circulant matrices, computer techniques and product designs to construct orthogonal designs in order 24.

On the convolution operator on a finite interval

Abstract: In this paper a convolution operator whose kernel is the Fourier transform of a rational function is studied on a finite interval. Criteria are found for completeness and basicity of the root system of such an operator.Bibliography: 9 titles.

An algorithm for the permanent of circulant matrices

Abstract: The permanent of an n x n matrix A = (a;j) is the matrix function ( 1) per A = ∑ al1r(1)••• a .. ",( .. )".C~" where the summation is over all permutations in the symmetric group, S ... An n x n matrix A is a circulant if there are scalars ab ... ,a,. such that (2) A= ∑ a;pi-l where P is the n x n permutation matrix corresponding to the cycle (12• .. n) in s". In general the computation of the permanent function is quite difficult chiefly because it is not invariant under addition of a multiple of one row to another. Using the principle of "inclusion and exclusion", Ryser [6, p. 27J gave an expansion for the permanent. Also the Laplace expansion is available for the permanent [2, p. 20]. Neither of these methods are particularly efficient. In [4J Mine considered the permanents of matrices with entires either 0 or 1. Mine also studied tridiagonal circulants in [5J]. Metropolis, Stein, and Stein [3] have given recurrence relations for evaluating the permanents of circulant matrices (2) where the first k scalars are 1 and the remaining ones are O. Permanents of circulant matrices were also studied by Tinsley [7]. Disciplines Physical Sciences and Mathematics Publication Details Cummings, LJ and Seberry, J, An algorithm for the permanent of circulant matrices, Bull. Canada. Math. Soc, 20, 1977, 67-70. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/981 Canad. Math. Bull. Vol. 20 (1), 1977 AN ALGORITHM FOR THE PERMANENT OF CIRCULANT MATRICES BY LARRY J. CUMMINGS AND JENNIFER SEBERRY WALLIS

Number Theoretic Transforms

Abstract: The theory of Number Theoretic Transforms having circular convolution properties is developed from the definition of circular convolution. The application of these transforms to digital signal processing is discussed. The lectures include the following sections. 1. Circular convolution; 2. Circulant diagonalisation; 3. Finite arithmetic structures; 4. Application of Number Theoretic Transforms.

The infinity norm of a certain type of symmetric circulant matrix

Abstract: An attainable bound for the infinity norm of the inverse of a whole class of symmetric circulant Toeplitz matrices is found. The class of matrices includes those arising from interpolation with both odd and even degree periodic polynomial splines on a uniform mesh.

Binary matrices with specified properties

Abstract: The binary matrices A which are circulant with one or more of the following properties: ( 1 ) A is symmetric, i.e. A = A T , ( 2 ) A is orthogonal, i.e. AA T = I ( mod 2 ), ( 3 ) A has low multiplicative order, i.e. A m = I , occur often in communication, control and network theory problems. In this paper we construct a number of such matrices. The results are based on the theory of power-residues modulo an odd prime p, and the fact that the algebra of all p × p circulant matrices is isomorphic to the algebra of polynomials modulo (x p − 1 ).

A linear-filtering transform algorithm based on permutations

Abstract: An algorithm is described which computes the discrete Fourier transform using only the following operations: multiplication by a term of constant frequency; multiplication by a sequence containing only the four numbers ± 1, ± j; permutation of the sequence; convolution with a fixed impulse response; permutation of the convolved sequence, multiplication by a sequence containing only ± 1, ± j, and multiplication by a term of constant frequency. Using charge-coupled devices, the fixed convolution and the multiplications by ± 1, ± j are relatively easy operations. The permutations are also believed to be reasonably easy. The multiplications by constant frequency terms can be ignored in many applications and the resulting "transform" still represents the spectrum and still has a straightforward convolution property.

On circulant codes with prescribed distances

Abstract: Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.