Block Circulant Matrix Research Articles

Algorithm for the Characterization of the Cross-Correlation Structure in Multivariate Time Series

In this paper, it is shown that the block circulant matrix decomposition technique makes the multivariate singular spectrum analysis (M-SSA) a well suited tool for detecting of changes of the correlation structure in non-stationary multivariate time series in the presence of high observational noise levels. The major drawback of M-SSA, that it operates on a large covariance matrix and becomes computationally expensive, can be avoided by reordering the Toeplitz-block covariance matrix into a block Toeplitz matrix, embedding this into a block circulant matrix and efficiently block-diagonalizing this by the means of the Fast Fourier Transform (FFT) using the well known algorithm. The overall degree of synchronization among multiple-channel signals is defined by the synchronization index (the S-estimator) of the rearranged and truncated eigenvalue spectrum. Throughout the experiment, the high capability of the proposed algorithm to detect the lag-synchronized state under the influence of strong noise is validated with simulated data—a network of time series generated by autoregressive models (AR) and a network of coupled chaotic Roessler oscillators.

Sparse block circulant matrices for compressed sensing

An undetermined measurement matrix can capture sparse signals losslessly if the matrix satisfies the restricted isometry property (RIP) in compressed sensing (CS) framework. However, existing measurement matrices suffer from high computational burden because of their completely unstructured nature. In this study, the authors propose to construct a novel measurement matrix with a specific structure, called sparse block circulant matrix (SBCM), to reduce the computational burden. The RIP of the proposed SBCM is also guaranteed with overwhelming probability. The simulation results validate that SBCM reduces the computational burden significantly whereas keeps similar signal recovery accuracy as Gaussian random matrices.

Characteristic Polynomials and Spectra of Some Block Circulant Graphs

A block matrix A=(Aij ) n×n is called a block circulant matrix, denoted by C(A 11, A 12, …, A 1n ) if Aij =A 1,j−i+1 for all 1⩽i, j⩽n, where Aij is a matrix and j−i+1 is taken modulo n. Especially when all Aij are of order 1, A is called a circulant matrix. A digraph is called a (block) circulant digraph if its adjacency matrix is congruent to a (block) circulant matrix. In this article we will calculate the characteristic polynomials and spectra of several special block circulant graphs having some physical and chemical backgrounds.

Counting Short Cycles of Quasi Cyclic Protograph LDPC Codes

An efficient method for counting short cycles in the Tanner graphs of quasi cyclic (QC) protograph low-density parity-check (LDPC) codes is presented. The method is based on the relationship between the number of short cycles in the graph and the eigenvalues of the directed edge matrix of the graph. We demonstrate that for a QC protograph LDPC code, the complexity of computing such eigenvalues can be reduced significantly by representing the directed edge matrix as a block circulant matrix. Numerical results are presented to show the lower complexity of the proposed method compared to the existing algorithms for counting short cycles. These results also reveal that QC LDPC codes on average have a superior short cycle and girth distribution compared to similar randomly constructed codes.

Efficiently Encodable Irregular QC-LDPC Codes Constructed from Self-Reciprocal Generator Polynomials of MDS Codes

This letter presents a class of quasi-cyclic (QC) low-density parity-check (LDPC) codes which are efficiently encodable and have better memory efficiency than other randomly constructed QC-LDPC codes. The parity-check matrix of the codes is obtained from a block circulant matrix that is generated from a self-reciprocal generator polynomial of a maximum distance separable (MDS) code by replacing some of its block entries by all zero matrices. We show by simulations that the codes can achieve excellent error performance.

A direct method to solve block banded block Toeplitz systems with non-banded Toeplitz blocks

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman–Morrison–Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diagonal matrix with Toeplitz blocks, and the resulting Toeplitz systems are solved by means of a fast Toeplitz solver. The computational complexity in the case one uses fast Toeplitz solvers is equal to ξ ( m , n , k ) = O ( m n 3 ) + O ( k 3 n 3 ) flops, there are m block rows and m block columns in the matrix, n is the order of blocks, 2 k + 1 is the bandwidth. The validity of the approach is illustrated by numerical experiments.

On the Construction of Quasi-Systematic Block-Circulant LDPC Codes

A class of quasi-systematic block-circulant LDPC (QSBC-LDPC) Codes is proposed. The parity-check matrix of a QSBC-LDPC code is a sparse block-circulant matrix with a quasi-systematic structure. Due to the special structure of the parity-check matrix, only limited memories, XOR computations and cyclic-shifting operations are needed in the recursive encoding process of the QSBC-LDPC codes. Simulations show that the QSBC-LDPC codes provide remarkable performance improvement with low encoding complexity.

CT-perfusion imaging of the human brain: Advanced deconvolution analysis using circulant singular value decomposition

According to indicator dilution theory tissue time–concentration curves have to be deconvolved with arterial input curves in order to get valid perfusion results. Our aim was to adapt and validate a deconvolution method originating from magnetic resonance techniques and apply it to the calculation of dynamic contrast enhanced computed tomography perfusion imaging. The application of a block-circulant matrix approach for singular value decomposition renders the analysis independent of tracer arrival time to improve the results.

Mean-Squared Error Estimation for Linear Systems with Block Circulant Uncertainty

We consider the problem of estimating a vector ${\bf x}$ in the linear model ${\bf A}{\bf x} \approx {\bf y}$, where ${\bf A}$ is a block circulant (BC) matrix with $N$ blocks and ${\bf x}$ is assumed to have a weighted norm bound. In the case where both ${\bf A}$ and ${\bf y}$ are subjected to noise, we propose a minimax mean-squared error (MSE) approach in which we seek the linear estimator that minimizes the worst-case MSE over a BC structured uncertainty region. For an arbitrary choice of weighting, we show that the minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved efficiently. For a Euclidean norm bound on ${\bf x}$, the SDP is reduced to a simple convex program with $N+1$ unknowns. Finally, we demonstrate through an image deblurring example the potential of the minimax MSE approach in comparison with other conventional methods.

A Double-Stage Multiuser Detector with FFT-Based Equalization for Asynchronous CDMA Ultra-Wideband Communication Systems

A novel multiuser transmission and detection scheme for high-speed asynchronous ultra-wideband (UWB) communication systems is proposed. Firstly, a block-spreading code-division multiple-access (BS-CDMA) scheme with zero correlation window (ZCW) is designed for application in asynchronous UWB communication systems with dense frequency selective fading propagation. As a result, multiple access interference can be completely removed by maintaining the spreading code orthogonality. Secondly, the UWB channel model is converted from a block Toeplitz matrix into block circulant matrix. Consequently, the inter-symbol interference is eliminated through the use of a fast Fourier transform (FFT) based minimum mean square error equalization. It is shown that the proposed multiuser transmission and detection scheme for UWB systems can deliver superior performance relative to the existing schemes, with low computation complexity

A method for computation of the discrete Fourier transform over a finite field

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.

Robust Mean-Squared Error Estimation of Multiple Signals in Linear Systems Affected by Model and Noise Uncertainties

This paper is a continuation of the work in [11] and [2] on the problem of estimating by a linear estimator, N unobservable input vectors, undergoing the same linear transformation, from noise-corrupted observable output vectors. Whereas in the aforementioned papers, only the matrix representing the linear transformation was assumed uncertain, here we are concerned with the case in which the second order statistics of the noise vectors (i.e., their covariance matrices) are also subjected to uncertainty. We seek a robust mean-squared error estimator immuned against both sources of uncertainty. We show that the optimal robust mean-squared error estimator has a special form represented by an elementary block circulant matrix, and moreover when the uncertainty sets are ellipsoidal-like, the problem of finding the optimal estimator matrix can be reduced to solving an explicit semidefinite programming problem, whose size is independent of N.

Right division in groups, dedekind-frobenius group matrices, and ward quasigroups

The variety of quasigroups satisfying the identity (xy)(zy) = xz mirrors the variety of groups, and offers a new look at groups and their multiplication tables. Such quasigroups are constructed from a group using right division instead of multiplication. Their multiplication tables consist of circulant blocks which have additional symmetries and have a concise presentation. These tables are a reincarnation of the group matrices which Frobenius used to give the first account of group representation theory. Our results imply that every group matrix may be written as a block circulant matrix and that this result leads to partial diagonalization of group matrices, which are present in modern applied mathematics. We also discuss right division in loops with the antiautomorphic inverse property.

An application of the modified Leverrier–Faddeev algorithm to the spectral decomposition of symmetric block-circulant matrices

The Leverrier–Faddeev algorithm is little-known but, in a modified form, is useful for deriving the algebraic, rather than numerical, spectral structure of matrices occurring in statistical methodology. An example is given of deriving the spectral decomposition of any symmetric block-circulant matrix, which in turn provides the singular value decomposition of any block-circulant matrix. Such problems arise as short-cuts to certain computations that arise in special forms of principal components analysis and correspondence analysis.

A Preconditioned Finite Element Method for the p‐Laplacian Parabolic Equation

AbstractIn this paper we propose a method for the discretization of the parabolic p‐Laplacian equation. In particular we use alternately either the backward Euler scheme or the Crank‐Nicolson scheme for the time‐discretization and the first order Finite Element Method for space‐discretization as in [7]. To obtain the numerical solution we have to invert a block Toeplitz matrix with Toeplitz blocks. To this aim we use a Conjugate Gradient (CG) algorithm preconditioned by a block circulant matrix with circulant blocks. A Two‐Dimensional Discrete Fast Sine‐Cosine Transform (2D‐DFSCT) is applied to invert the block circulant matrix with circulant blocks. The experimental results show how the application of the preconditioner reduces the iterations of the CG algorithm of about the 56% –75%. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Focal plane array infrared camera transfer function calculation and image restoration

Infrared images often present distortions induced by the measurement system. Image processing is thus an essential part of infrared measurements. A distortion model based on a convolution product is presented. The analytical form of the convolution kernel has been obtained from an image formation theory, along with an analysis of the sampling of the focal plane array camera detector's matrix. Image restoration is an ill-posed problem, and its solution can be obtained using regularization methods. In this work, image restoration is performed using a variation of Tikhonov regularization that makes use of the particular form of the convolution kernel matrix, which is built as a block-circulant matrix that admits a diagonal form in the 2-D Fourier space. The restoration procedure is used to restore a knife-edge infrared source image.

Regularization Method in Infrared Image Processing

Infrared images often present distortions induced by the measurement system. Thus, image processing is a vital part of infrared measurements. A distortion model based on a convolution product is presented. Image restoration is an ill-posed problem and its solution can be obtained using regularization methods. In this paper, image restoration is performed using a variation of Tikhonov regularization that makes use of the particular form of the convolution kernel matrix, which is built as a block-circulant matrix that admits a diagonal form in the two-dimensional Fourier space. The restoration procedure is used to restore a knife-edge infrared source image.

A superfast method for solving Toeplitz linear least squares problems

In this paper we develop a superfast O((m+n) log 2(m+n)) complexity algorithm to solve a linear least squares problem with an m× n Toeplitz coefficient matrix. The algorithm is based on the augmented matrix approach. The augmented matrix is further extended to a block circulant matrix and DFT is applied. This leads to an equivalent tangential interpolation problem where the nodes are roots of unity. This interpolation problem can be solved by a divide and conquer strategy in a superfast way. To avoid breakdowns and to stabilize the algorithm pivoting is used and a technique is applied that selects “difficult” points and treats them separately. The effectiveness of the approach is demonstrated by several numerical examples.

An application of the Gröbner basis in computation for the minimal polynomials and inverses of block circulant matrices

Algorithms for the minimal polynomial and the inverse of a level- n( r 1, r 2,…, r n )-block circulant matrix over any field are presented by means of the algorithm for the Gröbner basis for the ideal of the polynomial ring over the field, and two algorithms for the inverse of a level- n( r 1, r 2,…, r n )-block circulant matrix over a quaternion division algebra are given, which can be realized by CoCoA 4.0, an algebraic system, over the field of rational numbers or the field of residue classes of modulo a prime number.

Generation of “Similar” Images from a Given Discrete Image

A discrete image of several colors is viewed as a discrete random field obtained by clipping or quantizing a Gaussian random field at several levels. Given a discrete image, parameters of the unobserved original Gaussian random field are estimated. Discrete images, statistically similar to the original image, are then obtained by generating different realizations of the Gaussian field and clipping them. To overcome the computational difficulties, the block Toeplitz covariance matrix of the Gaussian field is embedded into a block circulant matrix which is diagonalized by the fast Fourier transform. The Gibbs sampler is used to apply the stochastic EM algorithm for the estimation of the field's parameters.