Showing papers on "Circulant matrix published in 2002"

Method and decoding apparatus using linear code with parity check matrices composed from circulants

Abstract: The present invention provides a novel method and apparatus for decoding digital information transmitted through the communication channel or recorded on a recording medium. The method and apparatus are preferably applied in the systems where data is encoded using regular LDPC codes with parity check matrices composed from circulants (a matrix is called a circulant if all its column or row are cyclic shifts each other).

Fast and exact synthesis for 1-D fractional Brownian motion and fractional Gaussian noises

Abstract: In this letter, it is shown that fast and exact fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) signals can be synthesized by the circulant embedding method (CEM). CEM consists in embedding the N/spl times/N covariance matrix of the stationary fGn process in a larger 2M/spl times/2M circulant matrix such that M /spl ges/N-1. CEM is exact, since second-order statistics of the generated data are those of the Gaussian fGn. CEM is fast, since the optimal case M=N-1 can be reached. Fast and exact fBm sequences can be easily recovered from fGn ones.

Circulant Matrices and Time-Series Analysis

Abstract: This paper sets forth some salient results in the algebra of circulant matrices which can be used in time-series analysis. It provides easy derivations of some results that are central to the analysis of statistical periodograms and empirical spectral density functions. A statistical test for the stationarity or homogeneity of empirical processes is also presented.

Speech encryption using circulant transformations

Abstract: A new analog encryption technique using unitary circulant transformations of the sampled analog signal is proposed. This technique is less vulnerable to attack than other known schemes based on transposition (scrambling) of the signal components in the time or frequency domain. The new technique introduces primarily a phase distortion. A new distortion measure capable of quantifying phase distortion is proposed. An optimal speaker-specific key generation scheme is then developed for maximizing an objective function based on the new distortion measure.

Fast computation of multi-input-multi-output decision feedback equalizer coefficients

Abstract: Multi-Input-Multi-Output (MIMO) Optimal Decision Feedback Equalizer (DFE) coefficients are determined from a channel estimate h by casting the MIMO DFE coefficient problem as a standard recursive least squares (RLS) problem and solving the RLS problem. In one embodiment, a fast recursive method, e.g., fast transversal filter (FTF) technique, then used to compute the Kalman gain of the RLS problem, which is then directly used to compute MIMO Feed Forward Equalizer (FFE) coefficients gopt. The complexity of a conventional FTF algorithm is reduced to one third of its original complexity by choosing the length of a MIMO Feed Back Equalizer (FBE) coefficients bopt (of the DFE) to force the FTF algorithm to use a lower triangular matrix. The MIMO FBE coefficients bop, are computed by convolving the MIMO FFE coefficients gopt with the channel impulse response h. In performing this operation, a convolution matrix that characterizes the channel impulse response h extended to a bigger circulant matrix. With the extended circulant matrix structure, the convolution of the MIMO FFE coefficients gopt with the channel impulse response h may be performed easily performed in the frequency domain.

Periodic solutions for planar 2N-body problems

Abstract: In this paper we study some necessary conditions and sufficient conditions for the nested polygonal solutions of planar 2N-body problems.

Fast computation of mimo decision feedback equalizer coefficients

Abstract: Multi-Input-Multi-Output (MIMO) Decision Feedback Equalizer (DFE) coefficients are determined from a channel estimate h by casting the MIMO DFE coefficient problem as a standard recursive least squares (RLS) problem and solving the RLS problem In one embodiment, a fast recursive method, eg, fast transversal filter (FTF) technique, then used to compute the Kalman gain of the RLS problem, which is then directly used to compute MIMO Feed Forward Equalizer (FFE) coefficients gopt The complexity of a conventional FTF algorithm is reduced to one third of its original complexity by choosing the length of a MIMO Feed Back Equalizer (FBE) coefficients bopt (of the DFE) to force the FTF algorithm to use a lower triangular matrix The MIMO FBE coefficients bopt are computed by convolving the MIMO FFE coefficients gopt with the channel impulse response h In performing this operation, a convolution matrix that characterizes the channel impulse response h is extended to a bigger circulant matrix With the extended circulant matrix structure, the convolution of the MIMO FFE coefficients gopt with the channel impulse response h may be performed easily performed in the frequency domain

A Note on the Superoptimal Matrix Algebra Operators

Abstract: We study the superoptimal Frobenius operators in several matrix vector spaces and in particular in the circulant algebra, by emphasizing both the algebraic and geometric properties. More specifically we prove a series of "negative" results that explain why this approximation procedure is not competitive with the optimal Frobenius approximation, although it could be used for regularization purposes.

Multi-carrier multiple access is sum-rate optimal for block transmissions over circulant ISI channels

Abstract: We establish that practical multiple access based on finite size information blocks transmitted with prescribed power and with loaded multicarrier modulation, is optimal with respect to maximizing the sum-rate of circulant intersymbol interference (ISI) channels, that are assumed available at the transmitter. Circulant ISI channels are ensured either with cyclic prefixed block transmissions and an overlap-save reception, or, with zero-padded block transmissions and an overlap-add reception. Analysis asserts that sum-rate optimal multicarrier users could share one or more subcarriers depending on the underlying channels. Optimal loading is performed by specializing an existing iterative low-complexity algorithm to circulant ISI channels.

Real-Valued, Low Rank, Circulant Approximation

Abstract: Partially due to the fact that the empirical data collected by devices with finite bandwidth often neither preserves the specified structure nor induces a certain desired rank, retrieving the nearest structured low rank approximation from a given data matrix becomes an imperative task in many applications. This paper investigates the case of approximating a given target matrix by a real-valued circulant matrix of a specified, fixed, and low rank. A fast Fourier transform (FFT)-based numerical procedure is proposed to speed up the computation. However, since a conjugate-even set of eigenvalues must be maintained to guarantee a real-valued matrix, it is shown by numerical examples that the nearest real-valued, low rank, and circulant approximation is sometimes surprisingly counterintuitive.

Low complexity data detection using fast fourier transform of channel correlation matrix

Abstract: A combined signal is received over a shared spectrum in a time slot in a time division duplex communication system using code division multiple access. Each data signal experiences a similar channel response. The similar channel response is estimated. A matrix representing a channel of the data signals based on in part the estimated channel response is constructed. A spread data vector is determined based on in part a fast fourier transform (FFT) decomposition of a circulant version of the channel matrix. The spread data vector is despread to recover data from the received combined signal.

On Circulant Complex Hadamard Matrices

Abstract: We show that a circulant complex Hadamard matrix of order n is equivalent to a relative difference set in the group C_4\times C_n where the forbidden subgroup is the unique subgroup of order two which is contained in the C_4 component. We obtain non-existence results for these relative difference sets. Our results are sufficient to prove there are no circulant complex Hadamard matrices for many orders.

Spectral approximation of banded Laurent matrices with localized random perturbations

Abstract: This paper explores the relationship between the spectra of perturbed infinite banded Laurent matricesL(a)+K and their approximations by perturbed circulant matricesC n (a)+P n KP n for largen. The entriesK jk of the perturbation matrices assume values in prescribed sets Ω jk at the sites (j, k) of a fixed finite setE, and are zero at the sites (j, k) outsideE. WithK Ω denoting the ensemble of these perturbation matrices, it is shown that $$\mathop {\lim }\limits_{n \to \infty } \bigcup\limits_{K \in K_\Omega ^E } {{\text{sp}}} {\text{ }}(C_n (a) + P_n KP_n ) = \bigcup\limits_{K \in K_\Omega ^E } {{\text{sp}}} {\text{ (}}L(a) + K)$$ under several fairly general assumptions onE and Ω.

The best circulant preconditioners for Hermitian Toeplitz systems II: The multiple-zero case

Abstract: In [10,14], circulant-type preconditioners have been proposed for ill-conditioned Hermitian Toeplitz systems that are generated by nonnegative continuous functions with a zero of even order. The proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions. It was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero. Therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems. In [10,14], it was claimed that this result can be the case where the generating functions have multiple zeros. The main aim of this paper is to give a complete convergence proof of the method in [10,14] for this class of generating functions.

Acceleration of free-space discrete body of revolution codes by exploiting circulant submatrices

Abstract: Discrete body of revolution (DBOR) codes can accelerate moment method (MoM) solutions for scattering geometries possessing discrete circular symmetry. Traditional DBOR formulations require modifications to the incident field, such as decomposition into circular modes. While appropriate for methods employing direct solvers; approaches employing iterative solvers are disadvantaged by this dramatic increase in the number of right-hand sides that must be solved. In this correspondence, we show how the block circulant property of the impedance matrix can be exploited to avoid incident field modification while compressing matrix storage and accelerating computation of the iterative solver's associated matrix-vector product above that of the typical iterative MoM approach.

Some Bounds on \(\ell_p\) Matrix and \(\ell_p\) Operator Norms of Almost Circulant, Cauchy-Toeplitz and Cauchy-Hankel Matrices

Abstract: Let Cn, Tn and Hn denote almost circulant, Cauchy-ToeplitZ and Cauchy-Hankel matrices, respectively. We find some upper bounds for \(\ell_p\) matrix norm and \(\ell_p\) operator norm of this matrices. Moreover, we give some results for Kronecker products Cn\(\otimes\)Tn and Cn\(\otimes\)Hn.

A Search for Hadamard Matrices constructed from Williamson Matrices

Abstract: We describe the implementation of a distributed computer search that uses Williamson’s construction for Hadamard matrices. The search program is used to perform a complete search for matrices of orders 100 through 148. No new results are found, confirming existing results. We are convinced that no further matrices of any order less than 156 may be constructed. For reference purposes, we present tables of Hadamard matrices of orders 100 through 180 constructed using four circulant symmetric (1,−1) matrices in the Williamson array.

The Moore-Penrose Pseudoinverse of an Arbitrary, Square, k -circulant Matrix

Abstract: In 1974 Cline, Plemmons and Worm showed that A † is a k -circulant matrix if and only if A is k -circulant with | k | = 1. However they left open the nature of A † when |k| p 1. We present a simple derivation of the Moore-Penrose pseudoinverse of an arbitrary, square, k -circulant matrix.

Lattice low-density parity check codes and their application in partial response systems

Abstract: This paper introduces a combinatorial construction of high rate low-density parity check codes based on two-dimensional integer lattices. A class of codes with a wide range of lengths, column weights and minimum distances is obtained. The resulting codes are doubly circulant, i.e. the matrix of parity checks is an array of circulant matrices. The bounds on the minimum distance are established and the performance of these codes in partial response channels is investigated.

The Spectrum of Circulant-Like Preconditioners for Some General Linear Multistep Formulas for Linear Boundary Value Problems

Abstract: The spectrum of the eigenvalues, the conditioning, and other related properties of circulant-like matrices used to build up block preconditioners for the nonsymmetric algebraic linear equations of time-step integrators for linear boundary value problems are analyzed. Moreover, results concerning the entries of a class of Toeplitz matrices related to the latter are proposed. Generalizations of implicit linear multistep formulas in boundary value form are considered in more detail. It is proven that there exists a new class of approximations which are well conditioned and whose eigenvalues have positive and bounded real and bounded imaginary part. Moreover, it is observed that preconditioners based on other circulant-like approximations, which are well suited for Hermitian linear systems, can be severely ill conditioned even if the matrices of the nonpreconditioned system are well conditioned.

Iterative fast fourier transform error correction

Abstract: Data is to be estimated from a received plurality of data signals in a code division multiple access communication system. The data signals are transmitted in a shared spectrum at substantially a same time. A combined signal of the transmitted data signals are received over the shared spectrum (42) and sampled (43). A channel response for the transmitted data signals is estimated. Data of the data signals is estimated (44) using the samples and the estimated channel response. The data estimation uses a fourier transform based data estimating approach. An error in the data estimation introduced from a circulant approximation used in the fourier transform based approach is iteratively reduced.

Lower bounds on the bounded coefficient complexity of bilinear maps

Abstract: We prove lower bounds of order n log n for both the problem to multiply polynomials of degree n, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the linear problem to multiply a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp. 305-306, 1973] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix.