Showing papers on "Circulant matrix published in 1995"
TL;DR: The decorrelating power of the DCTs is studied, obtaining expressions that show the decor Relating behavior of each DCT with respect to any stationary processes, and it is proved that the eight types of D CTs are asymptotically optimal for all finite-order Markov processes.
Abstract: Since its introduction in 1974 by Ahmed et al., the discrete cosine transform (DCT) has become a significant tool in many areas of digital signal processing, especially in signal compression. There exist eight types of discrete cosine transforms (DCTs). We obtain the eight types of DCTs as the complete orthonormal set of eigenvectors generated by a general form of matrices in the same way as the discrete Fourier transform (DFT) can be obtained as the eigenvectors of an arbitrary circulant matrix. These matrices can be decomposed as the sum of a symmetric Toeplitz matrix plus a Hankel or close to Hankel matrix scaled by some constant factors. We also show that all the previously proposed generating matrices for the DCTs are simply particular cases of these general matrix forms. Using these matrices, we obtain, for each DCT, a class of stationary processes verifying certain conditions with respect to which the corresponding DCT has a good asymptotic behavior in the sense that it approaches Karhunen-Loeve transform performance as the block size N tends to infinity. As a particular result, we prove that the eight types of DCTs are asymptotically optimal for all finite-order Markov processes. We finally study the decorrelating power of the DCTs, obtaining expressions that show the decorrelating behavior of each DCT with respect to any stationary processes.
114 citations
TL;DR: In this article, the eigenvalue and singular-value distributions for matrices S −1 n A n and C − 1 n A N are examined, where A n, S n, and C n are Toeplitz matrices, simple circulants, and optimal circulant generated by the Fourier expansion of some function f.
69 citations
TL;DR: This work studies the application of $\tau $, circulant, and Hartley preconditioners to ill-conditioned Toeplitz matrices by proving that only the first class realizes a rate of convergence not depending on the dimension of the system.
Abstract: Several preconditioning techniques for solving Toeplitz systems are known in literature, but their convergence features are completely understood only in the well-conditioned case. We study the application of $\tau $, circulant, and Hartley preconditioners to ill-conditioned Toeplitz matrices by proving that only the first class realizes a rate of convergence not depending on the dimension of the system.
59 citations
01 Jan 1995
TL;DR: In this paper, a new class of finite unimodular sequences with unimodal Fourier transforms with complex entries is introduced, which is called Circulant Hadamard matrices with complex entry.
Abstract: New classes of finite unimodular sequences with unimodular Fourier transforms. Circulant Hadamard matrices with complex entries.
54 citations
TL;DR: A novel fast algorithm for computing the minimum MSE decision feedback equalizer settings is proposed, first by estimating the channel, and then by computing the coefficients in the frequency domain with the discrete Fourier transform.
Abstract: A novel fast algorithm for computing the minimum MSE decision feedback equalizer settings is proposed. The equalizer filters are computed indirectly, first by estimating the channel, and then by computing the coefficients in the frequency domain with the discrete Fourier transform (DFT). Approximating the correlation matrices by circulant matrices facilitates the whole computation with very small performance loss. The fractionally spaced equalizer settings are derived. The performance of the fast algorithm is evaluated through simulation. The effects of the channel estimation error and finite precision arithmetic are briefly analyzed. Results of simulation show the superiority of the proposed scheme.
52 citations
TL;DR: It is shown that in many instances a space domain implementation of TBM, with line realizations obtained from a circulant embedding method, has advantages over a spectral implementation in that without increase in computational costs (1) the linerealizations display exactly the required covariance structure, (2) knowledge of the spectral density of the process Y is not required, and (3) fine tuning of line process parameters is not needed.
Abstract: Given a correlation function c(x) with x in Rn, the turning bands method (TBM( for n = 2 and 3 constructs realizations of an n-dimensional and stationary process Y ∼ N(O, c(x)) from appropriately summed line processes. Therefore an implementation of TBM calls naturally for fast and accurate generations of line realizations. These have generally been generated by a spectral approach since the Fourier transform of the line covariances is linked in a very simple fashion to the n-dimensional Fourier transform of c(x). However, we show that in many instances a space domain implementation of TBM, with line realizations obtained from a circulant embedding method, has advantages over a spectral implementation in that without increase in computational costs (1) the line realizations display exactly the required covariance structure, (2) knowledge of the spectral density of the process Y is not required, and (3) fine tuning of line process parameters is not needed.
41 citations
TL;DR: In this article, it was shown that every finite group G has a character θ such that every other generalized character of G is a polynomial in θ with rational coefficients.
38 citations
TL;DR: A new preconditioner for symmetric, positive definite Toeplitz systems is presented, an element of the n--dimensional vector space of matrices that are diagonalized by the discrete sine transform that is comparable, and in some cases superior, to the standard circulants of Tony Chan.
Abstract: We present a new preconditioner for $n\times n$ symmetric, positive definite Toeplitz systems. This preconditioner is an element of the $n$--dimensional vector space of matrices that are diagonalized by the discrete sine transform. Conditions are given for which the preconditioner is positive definite and for which the preconditioned system has asymptotically clustered eigenvalues. The diagonal form of the preconditioner can be calculated in $O(n\log(n))$ operations if $n=2^k-1.$ Thus only $n$ additional parameters need be stored. Moreover, complex arithmetic is not needed. To use the preconditioner effectively, we develop a new technique for computing a fast convolution using the discrete sine transform (also requiring only real arithmetic). The results of numerical experimentation with this preconditioner are presented. Our preconditioner is comparable, and in some cases superior, to the standard circulant preconditioner of Tony Chan. Possible generalizations for other fast transforms are also indicated.
34 citations
TL;DR: This paper is mainly devoted to show the relationship between the Smith normal form for integral matrices and the dimensions of such (di)graphs, that is the minimum ranks of the groups they can arise from.
27 citations
TL;DR: A class of highly regular fast cyclic convolution algorithms, based on block pseudocirculant matrices, is obtained.
Abstract: Pseudocirculant matrices have been studied in the past in the context of FIR filtering, block filtering, polyphase networks and others. For completeness, their relation to cyclic convolution, stride permutations, circulant matrices, and to certain permutations of the Fourier matrix is explicitly established in this work. Within this process, a class of highly regular fast cyclic convolution algorithms, based on block pseudocirculant matrices, is obtained. >
22 citations
TL;DR: Under the assumptions in the above-referenced paper, it is proved that if the generating function of $f(x,y)$ of $T_{mn} $ is positive, then $T_mn} is posit...
Abstract: In [SIAM J. Sci. Statist. Comput., 13 (1992), pp. 948–966], Ku and Kuo proposed and analysed a block circulant preconditioner $R_{mn} $ for solving a family of block Toeplitz systems $T_{mn} v = b$. For a special class of block matrices called the quadrantally symmetric Toeplitz matrices, they proved that the eigenvalues of $R_{mn}^{ - 1} T_{mn} $ are clustered around one except at most $O(m + n)$ outliers with $T_{mn} $ generated by a two-dimensional rational function. The superior convergence rate of the preconditioned conjugate gradient (PCG) method is explained by the clustering property of the spectrum of $R_{mn}^{ - 1} T_{mn} $. However, in their analysis, there is no discussion on the positive definiteness of the matrix $T_{mn} $, and the preconditioner $R_{mn} $ is assumed to be invertible. In this paper, we give some results on these two aspects. Under the assumptions in the above-referenced paper, we prove that if the generating function $f(x,y)$ of $T_{mn} $ is positive, then $T_{mn} $ is posit...
23 Oct 1995
TL;DR: This paper shows that a particular combined diagonal/Fourier preconditioner yields a more accurate approximation to the Hessian and gives significantly faster convergence rates than does either preconditionser used alone.
Abstract: Iterative methods for tomographic image reconstruction often converge slowly. Preconditioning methods can often accelerate gradient-based iterations. Previous preconditioning methods for PET reconstruction have used either diagonal or Fourier-based preconditioners. Fourier-based preconditioners are well suited to problems with near-circulant Hessian matrices. However, due to the nonuniform Poisson noise variance in PET, the circulant approximation to the Hessian is suboptimal. This paper shows that a particular combined diagonal/Fourier preconditioner yields a more accurate approximation to the Hessian and gives significantly faster convergence rates than does either preconditioner used alone.
TL;DR: In this article, a complete characterization of banded block circulant matrices that have banded inverse is derived by factorizations similar to those used for orthogonal matrices of this kind.
TL;DR: It is proved that if F AF∗ has Property A, where F is the Fourier matrix, then Cb minimizes ∥C − A∥F over all circulant matrices C, where ∥ · ∥F denotes the Frobenius norm.
TL;DR: Three fast algorithms for solving Toeplitz systems of equations are presented, which improve over FFT-based algorithms in avoiding roundoff error in an ill-conditioned problem.
TL;DR: A parallel method for solving the circulant block-tridiagonal systems by generalizing Muller and Scheerer's method which is used to parallelize the tridiagonal solvers is presented.
Abstract: Generalizing Muller and Scheerer's method which is used to parallelize the tridiagonal solvers, this paper presents a parallel method for solving the circulant block-tridiagonal systems. The applications of our result to solve the block-tridiagonal systems, the banded systems, and the circulant tridiagonal systems (for example, solving the closed B-spline curve fitting) are also addressed.
TL;DR: This work proposes an aperiodic model of deconvolution, which holds without regard to the size of the kernel and the image, and in the general case can lead to a nonsingular system of equations that has a lower condition number than the circulant one.
Abstract: It is known that discretization of a continuous deconvolution problem can alleviate the ill-posedness of the problem. The currently used circulant matrix model, however, does not play such a role. Moreover, the approximation of deconvolution problems by circulant matrix model is rational only if the size of the kernel function is very small. We propose an aperiodic model of deconvolution. For discrete and finite deconvolution problems the new model is an exact one. In the general case, the new model can lead to a nonsingular system of equations that has a lower condition number than the circulant one, and the related computations in the deconvolution can be done efficiently by means of the DFT technique, as in the ease for circulant matrices. The rationality of the new model holds without regard to the size of the kernel and the image. The use of the aperiodic model is illustrated by gradient-based algorithms. >
TL;DR: In this article, the circulant preconditioned conjugate gradient method was applied to solving the Wiener-Hopf equation and it was shown that it converges superlinearly.
Abstract: In this paper, we study the solutions of finite-section Wiener-Hopf equations by the preconditioned conjugate gradient method. Our main aim is to give an easy and general scheme of constructing good circulant integral operators as preconditioners for such equations. The circulant integral operators are constructed from sequences of conjugate symmetric functions {Cτ}τ. Letk(t) denote the kernel function of the Wiener-Hopf equation and\(\hat k(t)\) be its Fourier transform. We prove that for sufficiently large τ if {Cτ}τ is uniformly bounded on the real lineR and the convolution product of the Fourier transform ofCτ with\(\hat k(t)\) uniformly onR, then the circulant preconditioned Wiener-Hopf operator will have a clustered spectrum. It follows that the conjugate gradient method, when applied to solving the preconditioned operator equation, converges superlinearly. Several circulant integral operators possessing the clustering and fast convergence properties are constructed explicitly. Numerical examples are also given to demonstrate the performance of different circulant integral operators as preconditioners for Wiener-Hopf operators.
TL;DR: It is proved that it is possible to compute p elements of  in time O(p + N log N) with only O(N) auxiliary storage, and even if all elements have to be computed, the algorithm is faster than traditional methods.
TL;DR: Fourier harmonics provide universal eigenvectors, and they are applied to several homogeneous examples: k-wta, k-cluster, on/center off/surround, and the assignment problem.
Abstract: We introduce a generalization of mutually inhibitory networks called homogeneous networks. Such networks have symmetric connection strength matrices that are circulant (one-dimensional case) or block circulant with circulant blocks (two-dimensional case). Fourier harmonics provide universal eigenvectors, and we apply them to several homogeneous examples: k-wta, k-cluster, on/center off/surround, and the assignment problem. We also analyze one nonhomogeneous case: the subset-sum problem. We present the results of 10000 trials on a 50-node k-cluster problem and 100 trials on a 25-node subset-sum problem. >
TL;DR: It is proved that the spectra of the resulting preconditioned operators ( 1 u )∑ v (αI+P τ (u,v) ) −1 are clustered around 1 and thus the algorithm converges sufficiently fast, and the methods converges faster than those preconditionsed by using circulant integral operators.
TL;DR: In this paper, the authors consider the infinite analogues of circulant and random infinite circulants, and their connectivities and hamiltonian properties are discussed, and answer a question of [4] in the case of infinite (undirected) Circulants and some results on random infinite Circulant are also obtained.
Abstract: In recent years diverse literatures have been published on circulants (cf. [2] and the references cited therein). In this paper we consider the infinite analogues of circulant and random infinite circulant, and their connectivities and hamiltonian properties are discussed. Especially we answer a question of [4] in the case of infinite (undirected) circulants, and some results on random infinite circulants are also obtained.
TL;DR: In this paper, it was shown that the eigenvectors of circulant matrices of prime dimension can be expressed succinctly in the notation of classical cyclotomy, and that the cyclotomy can be used to express the eigvector of a circular matrices.
15 Dec 1995
TL;DR: The structured approach to the structured factorisation of banded circulant matrices is competitive to the deconvolution one based on the FFT and in many practical applications where large matrices with narrow bandwidths occur the structured approach is the most efficient.
Abstract: The solution of banded circulant systems based on structured factorisation is presented. The structured factorisation of banded circulant matrices is obtained by solving a set of non-linear equations. When the matrix is strictly diagonally dominant, the corresponding non-linear system can be solved by an iterative algorithm, e.g., a non-linear SOR or Newton's iteration. Both approaches are presented. The computational complexity of algorithms for computing the structured factorisation depends on the matrix bandwidth and not on its dimensions. A coupled system obtained by the factorisation is solved using a rank-annihilation formula. The structured approach is competitive to the deconvolution one based on the FFT. Moreover, in many practical applications where large matrices with narrow bandwidths occur the structured approach is the most efficient.
TL;DR: With a fast algorithm, it is shown that C is a good preconditioner for solving block Toeplitz systems with tensor structure and only O ( mn m log n ) operations are required for the solution of the preconditionsed systems.
01 Jan 1995
TL;DR: It is proved that if the number of servers s is fixed independent of the queue size n, then for sufficiently large n, the preconditioners are invertible and the precONDitioned systems have singular values clustered around 1.
Abstract: We consider finding the stationary probability distribution vectors of Markovian queueing models having batch arrivals by using the preconditioned conjugate gradient (PCG) method. The preconditioners are constructed by exploiting the near-Toeplitz structure of the generator matrix of the model and are products of circulant matrices and band-Toeplitz matrices. We prove that if the number of servers s is fixed independent of the queue size n, then for sufficiently large n, the preconditioners are invertible and the preconditioned systems have singular values clustered around 1. Hence if the systems are solved by the preconditioned conjugate gradient method, we expect superlinear convergence. Our numerical results show that the PCG method with our preconditioner converges in finite number of steps independent of n and s whereas the numbers required by the Jacobi method or the PCG method without any preconditioner increase like O(n).
09 Jun 1995
TL;DR: It is illustrated that the restoration algorithms in a weighted space using the block circulant matrices can be modified into the aperiodic model based algorithms and there are more possibilities to reduce the computational cost for the modified algorithms.
Abstract: The aperiodic matrix deconvolution model has been shown to have a better condition number than the popularly used circulant matrix model. In this paper, we illustrate that the restoration algorithms in a weighted space using the block circulant matrices can be modified into the aperiodic model based algorithms and we have more possibilities to reduce the computational cost for the modified algorithms. Examples are used to show the modified algorithm. We address that more interesting is the use of the model to develop some new algorithms, for example, for the unknown blur case. >
Journal Article•
TL;DR: In this paper, a comparison of the two models including their ill-posedness, the rationality of the approximation by the models, and their computational efficiency is made. And the comparison shows that the aperiodic model is promising in the development of new restoration algorithms.
28 Apr 1995
TL;DR: The real parts of the eigenvalues of the feedback matrix associated with any CNN dynamical system provide useful information on the behaviour of its trajectories and this paper shows how circulant and block circulants arise as feedback matrices of ring CNNs.
Abstract: The real parts of the eigenvalues of the feedback matrix associated with any CNN dynamical system provide useful information on the behaviour of its trajectories. In this paper we show how circulant and block circulant matrices arise as feedback matrices of ring CNNs. Such matrices possess many pleasant properties and we are able to give formulae for their eigenvalues and thus completely describe the spectrum of any feedback matrix associated with a ring CNN. Armed with this knowledge we are able to present a stability theorem for ring CNNs with a (specific) two-dimensional cloning template. This theorem provides a parameter range for which convergence of the CNN dynamical system is assured. Interestingly, this condition differs from the well-known diagonal dominance condition; moreover, it is of practical import since CNNs with feedback matrices which are diagonally dominant will not 'transform' bipolar images (such images lie in the basin of attraction of stable equilibria) and this must be considered a practical drawback to such a condition. The condition on the parameters presented here results in CNNs which will process bipolar images.