Showing papers on "Circulant matrix published in 2006"
Book•
01 Jan 2006
414 citations
TL;DR: It is conjectured that there are exactly 2^@t^(^n^-^1 non-isomorphic integral circulant graphs on n vertices, where @t(n) is the number of divisors of n.
134 citations
TL;DR: This work considers a suggestion by Michael Stein, who proposed nonnegative definite periodic embeddings based on suitably modified, compactly supported covariance functions, andoretical support is given, and software for its implementation is provided.
Abstract: The circulant embedding technique allows for the fast and exact simulation of stationary and intrinsically stationary Gaussian random fields. The method uses periodic embeddings and relies on the fast Fourier transform. However, exact simulations require that the periodic embedding is nonnegative definite, which is frequently not the case for two-dimensional simulations. This work considers a suggestion by Michael Stein, who proposed nonnegative definite periodic embeddings based on suitably modified, compactly supported covariance functions. Theoretical support is given to this proposal, and software for its implementation is provided. The method yields exact simulations of planar Gaussian lattice systems with 106 and more lattice points for wide classes of processes, including those with powered exponential, Matern, and Cauchy covariances.
76 citations
TL;DR: In this article, an explicit construction of circulant graphs of very high energy is given, based on Gauss sums, and the Littlewood conjecture can be used to establish new result for a certain class of graphs.
57 citations
TL;DR: The authors propose the use of "cheap methods" to find an approximation for the inverse Gabor frame matrix, based on (double) preconditioning, and obtain good approximations of the true dual Gabor atom at low computational costs.
Abstract: The authors present an application of the general idea of preconditioning in the context of Gabor frames. While most (iterative) algorithms aim at a more or less costly exact numerical calculation of the inverse Gabor frame matrix, we propose here the use of "cheap methods" to find an approximation for it, based on (double) preconditioning. We thereby obtain good approximations of the true dual Gabor atom at low computational costs. Since the Gabor frame matrix commutes with certain time-frequency shifts, it is natural to make use of diagonal and circulant preconditioners sharing this property. Part of the efficiency of the proposed scheme results from the fact that all the matrices involved share a well-known block matrix structure. At least, for the smooth Gabor atoms typically used, the combination of these two preconditioners leads consistently to good results. These claims are supported by numerical experiments in this paper. For numerical evaluations we introduce two new matrix norms, which can be calculated efficiently by exploiting the structure of the frame matrix
57 citations
TL;DR: In this article, the degree condition is used to test the separability of density matrices of graphs and it is shown that circulant and other matrices constructed from groups always satisfy the condition and indeed are separable with respect to any split.
Abstract: We reconsider density matrices of graphs as defined in quant-ph/0406165. The density matrix of a graph is the combinatorial Laplacian of the graph normalized to have unit trace. We describe a simple combinatorial condition (the ``degree condition'') to test the separability of density matrices of graphs. The condition is directly related to the Peres-Horodecki partial transposition condition. We prove that the degree condition is necessary for separability, and we conjecture that it is also sufficient. We prove special cases of the conjecture involving nearest-point graphs and perfect matchings. We observe that the degree condition appears to have a value beyond the density matrices of graphs. In fact, we point out that circulant density matrices and other matrices constructed from groups always satisfy the condition and indeed are separable with respect to any split. We isolate a number of problems and delineate further generalizations.
45 citations
TL;DR: The proposed computation method for the discrete Fourier transform is based on factorizing the transform matrix into a product of a binary block circulant matrix and a diagonal block circULant matrix.
Abstract: The discrete Fourier transform over a finite field finds applications in algebraic coding theory. The proposed computation method for the discrete Fourier transform is based on factorizing the transform matrix into a product of a binary block circulant matrix and a diagonal block circulant matrix.
32 citations
TL;DR: In this article, a self-contained modern linear stability analysis of a system of n equal mass bodies in circular orbit about a single more massive body is given, based on the result that J.C. Maxwell found for large n in his seminal paper on the nature and stability of Saturn's rings.
Abstract: We give a self-contained modern linear stability analysis of a system of n equal mass bodies in circular orbit about a single more massive body. Starting with the mathematical description of the dynamics of the system, we form the linear approximation, compute all of the eigenvalues of the linear stability matrix, and finally derive inequalities that guarantee that none of these eigenvalues have positive real part. In the end, we rederive the result that J.C. Maxwell found for large n in his seminal paper on the nature and stability of Saturn's rings, which was published 150 years ago. In addition, we identify the exact matrix that defines the linearized system even when n is not large. This matrix is then investigated numerically (by computer) to find stability inequalities. Furthermore, using properties of circulant matrices, the eigenvalues of the large 4nx4n matrix can be computed by solving n quartic equations, which further facilitates the investigation of stability. Finally, we have implemented an n-body simulator and we verify that the threshold mass ratios that we derived mathematically or numerically do indeed identify the threshold between stability and instability. Throughout the paper we consider only the planar n-body problem so that the analysis can be carried out purely in complex notation, which makes the equations and derivations more compact, more elegant and therefore, we hope, more transparent. The result is a fresh analysis that shows that these systems are always unstable for 2 6 they are stable provided that the central mass is massive enough. We give an explicit formula for this mass-ratio threshold.
26 citations
TL;DR: This paper shows how, when used with a standard 'powers of two' FFT algorithm, circulant embedding can be readily adapted to handle complex-valued Gaussian stationary processes.
26 citations
TL;DR: In this article, it was shown that the structured pseudospectrum of matrix polynomials equals the unstructured pseudo-pectrum for Toeplitz, circulant, Hankel and symmetric structures.
25 citations
TL;DR: It is shown that there is a Hamilton cycle decomposition of every 6-regular circulant graph 〈S〉n in which S has an element of order n.
Abstract: The circulant graph 〈S〉n, where S⊆Zn∖{0}, has vertex set Zn and edge set {{x,x+s}|x ∈ Zn,s ∈ S}. It is shown that there is a Hamilton cycle decomposition of every 6-regular circulant graph 〈S〉n in which S has an element of order n.
TL;DR: It is shown that the set of linear equations relating the autocorrelation matrix of the block received signal and the product channel coefficients can be rearranged into one with a distinctive block circulant structure.
Abstract: This paper proposes a periodic-modulation-based blind channel identification scheme for single-carrier (SC) block transmission with frequency-domain equalization (FDE). The proposed approach relies on the block system model and exploits the circulant channel matrix structure after the cyclic prefix is removed. It is shown that the set of linear equations relating the autocorrelation matrix of the block received signal and the product channel coefficients can be rearranged into one with a distinctive block circulant structure. The identification equations thus obtained lead to a very simple identifiability condition, as well as a natural formulation of the optimal modulating sequence design problem which, based on the block circulant signal structure, can be cast as a constrained quadratic problem that allows for a simple closed-form solution. The impact of the optimal modulating sequence on the peak-to-average power ratio (PAPR) is investigated. Also, it is shown that the optimal sequence results in a consistent channel estimate irrespective of white noise perturbation. Pairwise error probability analysis is used to examine the equalization performance and based on which several design tradeoffs are discussed. Simulation results are used for illustrating the performance of the proposed method.
TL;DR: The main result is the “black dot algorithm” and its fast version for the construction of a new circulant preconditioner for Toeplitz matrices that is likely to be the best of all possible circulants.
TL;DR: In this article, the degenerate scale for plate problems is studied for both continuous and discrete models, where the circulant is employed to determine the rank deficiency of the influence matrix.
Abstract: In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen. For the discrete model, the circulant is employed to determine the rank deficiency of the influence matrix. Both approaches, continuous and discrete models, lead to the same result of degenerate scale. We study the nonunique solution analytically for the circular plate and find degenerate scales. The similar properties of solvability condition between the membrane (Laplace) and plate (biharmonic) problems are also examined. The number of degenerate scales for the six boundary integral formulations is also determined.
TL;DR: It is shown that a circulant weighing matrix of order n and weight 16 exists if and only if n>=21 and n is a multiple of 14,21 or 31.
TL;DR: In this article, Pocklington's integral equation is formulated for the circular loop geometry, and the method of moments with point matching is applied to cast the equation's discrete counterpart.
Abstract: This paper is presenting a very efficient method for computing the currents on a circular loop antenna with arbitrarily large size. Pocklington's integral equation is formulated for the circular loop geometry, and the method of moments with point matching is applied to cast the equation's discrete counterpart. The basis functions are chosen in such a way that the relevant square matrix is circulant, and therefore amenable to exact eigenvalue analysis. Subsequently, the matrix is diagonalized and inverted analytically, yielding simple, analytical expressions for the weights of the basis functions, and hence the current and the input admittance of the antenna. The algorithm utilizes mainly elementary mathematical functions, as opposed to already known expressions in the literature, and yields results in the form of a rapidly convergent, single series, applicable to extremely large loops
TL;DR: A complete computer search is made for all circulant weighing matrices of order 16 such that MMT = kIn for some positive integer t and new structural results are obtained.
Abstract: A weighing matrix of weight k is a square matrix M with entries 0, ± 1 such that MM T = kI n . We study the case that M is a circulant and k = 22t for some positive integer t. New structural results are obtained. Based on these results, we make a complete computer search for all circulant weighing matrices of order 16.
Patent•
20 Dec 2006TL;DR: In this article, a method of encoding a binary source message u, by calculating x:=Au, calculating y:=B′x, resolving the equation Dp=y for p, and incorporating u and p to produce an encoded binary message v, was presented.
Abstract: A method of encoding a binary source message u, by calculating x:=Au, calculating y:=B′x, resolving the equation Dp=y for p, and incorporating u and p to produce an encoded binary message v, where A is a matrix formed only of permutation sub matrices, B′ is a matrix formed only of circulant permutation sub matrices, and D is a matrix of the form D = ( T 0 … 0 0 0 T … 0 0 … … … … … 0 0 … T 0 I I … I I )
where T is a two-diagonal, circulant sub matrix, and I is an identity sub matrix.
TL;DR: The general expression of the l -th power for one type of symmetric circulant matrices of order n = 2 p ( p ∈ N, p ⩾ 2) is derived.
TL;DR: In this article, the set of all kronecker products of unitary Fourier matrices is split into permutation equivalence classes, which can be used to construct inequivalent, in a certain sense, maximally entangled bases.
TL;DR: The complete derivation of the general expression of the l th power for one type of even order symmetric circulant matrices is given and expressions of eigenvectors and of the transforming matrix and its inverse are given.
07 May 2006
TL;DR: The theoretical and practicable capacity efficiencies for known-channel MIMO are compared for two idealized channels, with the circulant channel having a higher capacity efficiency than that of the random i.i.d. channel, for practical SNR values.
Abstract: The impact of communications signal processing such as QAM modulations (instead of gaussian signals), finite block lengths (instead of infinitely long codes), and using simpler algorithms (instead of expensive-to-implement ones), etc., is a lower practicable capacity efficiency than that of the Shannon limit. In this paper, the theoretical and practicable capacity efficiencies for known-channel MIMO are compared for two idealized channels. The motivation is to identify worthwhile trade-offs between capacity reduction and complexity reduction. The channels are the usual complex gaussian random i.i.d., and also the complex gaussian circulant. The comparison reveals new and interesting capacity behaviour, with the circulant channel having a higher capacity efficiency than that of the random i.i.d. channel, for practical SNR values. A circulant channel would also suggest implementation advantages owing to its fixed eigenvectors. Because of the implementation complexity of water filling, the simpler but sub-optimum solution of equal power allocation is investigated and shown to be worthwhile.
TL;DR: By exploiting the block circulant structure of the coefficient matrices appearing when the MFS is applied to certain axisymmetric harmonic and biharmonic problems, efficient matrix decomposition algorithms for their solution are developed.
Abstract: In this work we apply the Method of Fundamental Solutions (MFS) with fixed singularities and boundary collocation to certain axisymmetric harmonic and biharmonic problems. By exploiting the block circulant structure of the coefficient matrices appearing when the MFS is applied to such problems, we develop efficient matrix decomposition algorithms for their solution. The algorithms are tested on several examples
TL;DR: In this paper, the authors proposed a method that offers the fast and accurate analysis of large-scale periodic array antennas by conjugate-gradient fast Fourier transform (CG-FFT) combined with an equivalent sub-array preconditioner.
Abstract: SUMMARY This paper presents method that offers the fast and accurate analysis of large-scale periodic array antennas by conjugate-gradient fast Fourier transform (CG-FFT) combined with an equivalent sub-array preconditioner. Method of moments (MoM) is used to discretize the electric field integral equation (EFIE) and form the impedance matrix equation. By properly dividing a large array into equivalent sub-blocks level by level, the impedance matrix becomes a structure of Three-level Block Toeplitz Matrices. The Three-level Block Toeplitz Matrices are further transformed to Circulant Matrix, whose multiplication with a vector can be rapidly implemented by one-dimension (1-D) fast Fourier transform (FFT). Thus, the conjugate-gradient fast Fourier transform (CG-FFT) is successfully applied to the analysis of a large-scale periodic dipole array by speeding up the matrix-vector multiplication in the iterative solver. Furthermore, an equivalent sub-array preconditioner is proposed to combine with the CG-FFT analysis to reduce iterative steps and the whole CPU-time of the iteration. Some numerical results are given to illustrate the high efficiency and accuracy of the present method.
TL;DR: Multicarrier multiple access with channel knowledge and prescribed power at the transmitters is shown to maximize the sum-rate for circulant intersymbol-interference (ISI) channels.
Abstract: Multicarrier multiple access with channel knowledge and prescribed power at the transmitters is shown to maximize the sum-rate for circulant intersymbol-interference (ISI) channels. A low-complexity iterative algorithm is derived for optimal subcarrier allocation to multiple users, while power is loaded per user by specializing an existing iterative algorithm to circulant ISI channels. It is analytically shown that each subcarrier should be allocated to the user having relatively better subcarrier gain and that different users may share certain subcarriers.
Posted Content•
TL;DR: In this article, the problem of constructing dense lattices of R^n with a given automorphism group was considered and a family of lattices with density at least cn/2^n was constructed.
Abstract: We consider the problem of constructing dense lattices of R^n with a given automorphism group. We exhibit a family of such lattices of density at least cn/2^n, which matches, up to a multiplicative constant, the best known density of a lattice packing. For an infinite sequence of dimensions n, we exhibit a finite set of lattices that come with an automorphism group of size n, and a constant proportion of which achieves the aforementioned lower bound on the largest packing density. The algorithmic complexity for exhibiting a basis of such a lattice is of order exp(nlogn), which improves upon previous theorems that yield an equivalent lattice packing density. The method developed here involves applying Leech and Sloane's construction A to a special class of codes with a given automorphism group, namely the class of double circulant codes.
TL;DR: The spectrum of large benchmark (circulant) matrices is obtained by combining the Sylvester–Hadamard transform, the Singular Value Decomposition (SVD) and the theory of Hamiltonian symplectic matrices.
Abstract: The Huckel theory is reviewed and improved. The usefulness of several Hadamard fast transforms when preconditioning binary Huckel matrices is compared and analysed. The spectrum of large benchmark (circulant) matrices is obtained by combining the Sylvester–Hadamard transform, the Singular Value Decomposition (SVD) and the theory of Hamiltonian symplectic matrices. The developed methodology is used to calculate the spectrum of Huckel matrices representing nanotubes of 16 000 atoms.