Subspace Matrix Research Articles

Performance Analysis of Blind Multiuser Detectors for Cdma

This paper deals with the performance of blind multiuser detectors for CDMA is analyzed. The blind multi- user detectors are Direct Matrix Inversion (DMI) blind detector and subspace blind detector. The performance analysis is performed by means of the Signal to Interference Noise Ration (SINR) and Bit Error Rate (BER). The numerical results are plotted as variation of SINR Vs SNR and ρ , SINR with respect to correlation coefficient ( ρ ) and BER Vs signal samples (M) using MAT LAB software. It is observed that the SINR Vs SNR and ρ for both blind detectors. The performance of subspace blind detector deteriorates in the high cross correlation and low SNR region whereas the performance of the DMI blind detector is less sensitive to cross correlation and SNR in this region. The performance gain offered by the subspace detector is significant for smaller value of K, and the gain diminishes as K increases. For large number of signal samples (M), both detectors converge to the true linear MMSE detector, with the subspace blind detector converging much faster than the DMI blind detector and the performance gain offered by the subspace detector is quite significant for small values of M. The BER performance of subspace approach is better than the DMI approach with increasing SNR.

Computational geometry of positive definiteness

In matrix computations, such as in factoring matrices, Hermitian and, preferably, positive definite elements are occasionally required. Related problems can often be cast as those of existence of respective elements in a matrix subspace. For two dimensional matrix subspaces, first results in this regard are due to Finsler. To assess positive definiteness in larger dimensional cases, the task becomes computational geometric for the joint numerical range in a natural way. The Hermitian element of the Frobenius norm one with the maximal least eigenvalue is found. To this end, extreme eigenvalue computations are combined with ellipsoid and perceptron algorithms.

Dynamic Multi-Document Summarization Model

This paper introduces two models to describe dynamic evolution of network information: identify and analysis the document collection on the same topic in different stages.In order to construct dynamic of evolution content differences,two dynamic multi-document summarization models are presented,which are matrix subspace analysis model,text similarity cumulative model.Based on these models,some efficient dynamic sentence weighting algorithms are implemented.Experiments on the test data of Update Summarization in TAC 2008 and comparative results between new models and TAC 2008 evaluation,shows the effectiveness of the models.

A New Compressive Feedback Scheme Based on Distributed Compressed Sensing for Time-Correlated MIMO Channel

In this paper, a new compressive feedback (CF) scheme based on distributed compressed sensing (DCS) for time-corrected MIMO channel is proposed. First, the channel state information (CSI) is approximated by using a subspace matrix, then, the approximated CSI is compressed using a compressive matrix. At the base station, the approximated CSI can be robust recovered with simultaneous orthogonal matching pursuit (SOMP) algorithm by using forgone CSIs. Simulation results show our proposed DCS-CF method can improve the reliability of system without creating a large performance loss.

Distributed Principal Subspace Estimation in Wireless Sensor Networks

Motivated by applications in multi-sensor array detection and estimation, this paper studies the problem of tracking the principal eigenvector and the principal subspace of a signal's covariance matrix adaptively in a fully decentralized wireless sensor network (WSN). Sensor networks are traditionally designed to simply gather raw data at a fusion center, where all the processing occurs. In large deployments, this model entails high networking cost and creates a computational and storage bottleneck for the system. By leveraging both sensors' abilities to communicate and their local computational power, our objective is to propose distributed algorithms for principal eigenvector and principal subspace tracking. We show that it is possible to have each sensor estimate only the corresponding entry of the principal eigenvector or corresponding row of the p-dimensional principal subspace matrix and do so by iterating a simple computation that combines data from its network neighbors only. This paper also examines the convergence properties of the proposed principal eigenvector and principal subspace tracking algorithms analytically and by simulations.

Differential geometry of matrix inversion

Essentially, there exists just the dimension segregating (square) matrix subspaces. In view of algebraic operations, this quantity is not particularly descriptive. For differential geometric information on matrix inversion, the second fundamental form is found for the set of inverses of the invertible elements of a matrix subspace. Several conditions for this form to vanish are given, such as being equivalent to a Jordan subalgebra. Global measures of curvature are introduced in terms of an analogy of the Nash fiber.

Subspace Approach to Identification of Step-Response Model from Closed-Loop Data

We investigate direct estimation of step-response models from closed-loop data using subspace identification. Necessary information concerning impulse-response coefficients is embedded in subspace matrices. Therefore, the step-response coefficients can be directly obtained from this matrix by integration without the need of extracting state space models first, as the conventional subspace identification does. Since the estimated subspace matrix contains more than one set of impulse-response coefficients, a question arises about how to best synthesize them to obtain an optimal estimate of the impulse-response coefficients and subsequently the step-response coefficients. For this purpose, a reformulation of the subspace identification problem is required for which the variance of all elements in the related subspace matrix can be evaluated. The calculated variances are then used to perform a weighted averaging on the estimated impulse-response coefficients to attenuate the noise influence on the final step-response model estimation. Monte Carlo simulations and pilot-scale experiments are provided to illustrate the proposed method.

Random unitary maps for quantum state reconstruction

We study the possibility of performing quantum state reconstruction from a measurement record that is obtained as a sequence of expectation values of a Hermitian operator evolving under repeated application of a single random unitary map, U_0. We show that while this single-parameter orbit in operator space is not informationally complete, it can be used to yield surprisingly high-fidelity reconstruction. For a d-dimensional Hilbert space with the initial observable in su(d), the measurement record lacks information about a matrix subspace of dimension > d-2 out of the total dimension d^2-1. We determine the conditions on U_0 such that the bound is saturated, and show they are achieved by almost all pseudorandom unitary matrices. When we further impose the constraint that the physical density matrix must be positive, we obtain even higher fidelity than that predicted from the missing subspace. With prior knowledge that the state is pure, the reconstruction will be perfect (in the limit of vanishing noise) and for arbitrary mixed states, the fidelity is over 0.96, even for small d, and reaching F > 0.99 for d > 9. We also study the implementation of this protocol based on the relationship between random matrices and quantum chaos. We show that the Floquet operator of the quantum kicked top provides a means of generating the required type of measurement record, with implications on the relationship between quantum chaos and information gain.

Hierarchical Harmonic-Balance Methods for Frequency-Domain Analog-Circuit Analysis

As a widely adopted frequency-domain method, harmonic balance (HB) provides efficient steady-state circuit analysis for analog and RF circuits. The conventional matrix-implicit Krylov subspace technique with the block-diagonal (BD) preconditioner has made it possible to compute the steady-state responses of large-scale circuits. However, not all HB problems, particularly strongly nonlinear circuit problems, can be solved reliably or efficiently using the standard BD-preconditioning technique. In this paper, hierarchical HB methods are proposed wherein robust preconditioning is provided via solution of a set of approximate linearized HB problems of progressively smaller size across multiple levels of the problem hierarchy. These subproblems are constructed using the same matrix-implicit formulation to retain the memory efficiency of Krylov subspace methods. Moreover, the number of allocated Krylov subspace matrix solvers, hence the memory usage, is significantly reduced via a recently introduced solver-sharing technique. The efficiency of our hierarchical preconditioning technique is further improved by adopting a one-step correction to the standard BD preconditioner and a multigrid-motivated iterative scheme. It has been shown that the proposed approaches can achieve up to 10 runtime speedup over the popular BD preconditioner and robust convergence even for strongly nonlinear circuits for which the BD preconditioner fails to converge.

Factoring matrices into the product of two matrices

Linear algebra of factoring a matrix into the product of two matrices with special properties is developed. This is accomplished in terms of the so-called inverse of a matrix subspace which yields an extended notion for the invertibility of a matrix. The product of two matrix subspaces gives rise to a natural generalization of the concept of matrix subspace. Extensions of these ideas are outlined. Several examples on factoring are presented.

The Gautschi time stepping scheme for edge finite element discretizations of the Maxwell equations

For the time integration of edge finite element discretizations of the three-dimensional Maxwell equations, we consider the Gautschi cosine scheme where the action of the matrix function is approximated by a Krylov subspace method. First, for the space-discretized edge finite element Maxwell equations, the dispersion error of this scheme is analyzed in detail and compared to that of two conventional schemes. Second, we show that the scheme can be implemented in such a way that a higher accuracy can be achieved within less computational time (as compared to other implicit schemes). We also analyzed the error made in the Krylov subspace matrix function evaluations. Although the new scheme is unconditionally stable, it is explicit in structure: as an explicit scheme, it requires only the solution of linear systems with the mass matrix.

Oversampled filter banks as error correcting codes: theory and impulse noise correction

Oversampled filter banks (OFBs) provide an overcomplete representation of their input signal. This paper describes how OFBs can be considered as error-correcting codes acting on real or complex sequences, very much like classical binary convolutional codes act on binary sequences. The structured redundancy introduced by OFBs in subband signals can be used to increase robustness to noise. In this paper, we define the notions of code subspace, syndrome, and parity-check polynomial matrix for OFBs. Furthermore, we derive generic expressions for projection-based decoding, suitable for the case when a simple second-order model completely characterizes the noise incurred by subband signals. We also develop a nonlinear hypotheses-test based decoding algorithm for the case when the noise in subbands is constituted by a Gaussian background noise and impulsive errors (a model that adequately describes the action of both quantization noise and transmission errors). Simulation results show that the algorithm effectively removes the effect of impulsive errors occurring with a probability of 10/sup -3/.

Software for computing eigenvalue bounds for iterative subspace matrix methods

This paper describes software for computing eigenvalue bounds to the standard and generalized hermitian eigenvalue problem as described in [Y. Zhou, R. Shepard, M. Minkoff, Computing eigenvalue bounds for iterative subspace matrix methods, Comput. Phys. Comm. 167 (2005) 90–102]. The software discussed in this manuscript applies to any subspace method, including Lanczos, Davidson, SPAM, Generalized Davidson Inverse Iteration, Jacobi–Davidson, and the Generalized Jacobi–Davidson methods, and it is applicable to either outer or inner eigenvalues. This software can be applied during the subspace iterations in order to truncate the iterative process and to avoid unnecessary effort when converging specific eigenvalues to a required target accuracy, and it can be applied to the final set of Ritz values to assess the accuracy of the converged results. Program summary Title of program: SUBROUTINE BOUNDS_OPT Catalogue identifier: ADVE Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADVE Computers: any computer that supports a Fortran 90 compiler Operating systems: any computer that supports a Fortran 90 compiler Programming language: Standard Fortran 90 High speed storage required: 5 m + 5 working-precision and 2 m + 7 integer for m Ritz values No. of bits in a word: The floating point working precision is parameterized with the symbolic constant WP No. of lines in distributed program, including test data, etc.: 2452 No. of bytes in distributed program, including test data, etc.: 281 543 Distribution format: tar.gz Nature of physical problem: The computational solution of eigenvalue problems using iterative subspace methods has widespread applications in the physical sciences and engineering as well as other areas of mathematical modeling (economics, social sciences, etc.). The accuracy of the solution of such problems and the utility of those errors is a fundamental problem that is of importance in order to provide the modeler with information of the reliability of the computational results. Such applications include using these bounds to terminate the iterative procedure at specified accuracy limits. Method of solution: The Ritz values and their residual norms are computed and used as input for the procedure. While knowledge of the exact eigenvalues is not required, we require that the Ritz values are isolated from the exact eigenvalues outside of the Ritz spectrum and that there are no skipped eigenvalues within the Ritz spectrum. Using a multipass refinement approach, upper and lower bounds are computed for each Ritz value. Typical running time: While typical applications would deal with m < 20 , for m = 100 000 , the running time is 0.12 s on an Apple PowerBook.

Robust neurofuzzy rule base knowledge extraction and estimation using subspace decomposition combined with regularization and D-optimality.

A new robust neurofuzzy model construction algorithm has been introduced for the modeling of a priori unknown dynamical systems from observed finite data sets in the form of a set of fuzzy rules. Based on a Takagi-Sugeno (T-S) inference mechanism a one to one mapping between a fuzzy rule base and a model matrix feature subspace is established. This link enables rule based knowledge to be extracted from matrix subspace to enhance model transparency. In order to achieve maximized model robustness and sparsity, a new robust extended Gram-Schmidt (G-S) method has been introduced via two effective and complementary approaches of regularization and D-optimality experimental design. Model rule bases are decomposed into orthogonal subspaces, so as to enhance model transparency with the capability of interpreting the derived rule base energy level. A locally regularized orthogonal least squares algorithm, combined with a D-optimality used for subspace based rule selection, has been extended for fuzzy rule regularization and subspace based information extraction. By using a weighting for the D-optimality cost function, the entire model construction procedure becomes automatic. Numerical examples are included to demonstrate the effectiveness of the proposed new algorithm.

A cumulant matrix subspace algorithm for blind single FIR channel identification

Blind identification of discrete-time single-user FIR channels with nonminimum phase is studied here. Exploiting higher order cumulants of output signals of unknown channels, a new closed-form solution to the FIR channel impulse response is derived. The algorithm is simple and fast. It relies only on nullspace decomposition of some cumulant matrices. This method neither involves the difficult task of iterative global minimization of nonunimodal cost functions, nor does it require overparametrization, which poses consistency difficulties. It can be used either as the final channel estimate or as a good initial point in nonlinear cumulant matching techniques. The application of this identification method is broad and not limited to the use of any fixed-order cumulants. Its application in identifying data communication systems shows great potential and promise.

Rational invariant subspace approximations with applications

Subspace methods such as MUSIC, minimum norm, and ESPRIT have gained considerable attention due to their superior performance in sinusoidal and direction-of-arrival (DOA) estimation, but they are also known to be of high computational cost. In this paper, new fast algorithms for approximating signal and noise subspaces and that do not require exact eigendecomposition are presented. These algorithms approximate the required subspace using rational and power-like methods applied to the direct data or the sample covariance matrix. Several ESPRIT- as well as MUSIC-type methods are developed based on these approximations. A substantial computational saving can be gained comparing with those associated with the eigendecomposition-based methods. These methods are demonstrated to have performance comparable to that of MUSIC yet will require fewer computations to obtain the signal subspace matrix.

Identification of the radiating sources in underwater acoustics

In the context of the narrow-band or wideband array processing problem, in this paper a robust algorithm is developed to improve the accuracy of the estimation of the direction of arrival of the sources. It is well known that when the noise cross-spectral matrix is unknown, these estimates may be grossly inaccurate. The band noise spectral matrix, the classical noise model, is used. By means of a linear operator and an iterative algorithm for estimating the spatial correlation length, a new estimator for the direction of arrival of the sources is developed. The proposed algorithm is based upon a particular partition of the received signal vector which leads one to obtain an approximation of the noise subspace matrix without eigendecomposition. Then using both a propagation operator and noneigenvector algorithm to estimate the projection matrix, a new robust algorithm is developed for the source characterization problem in the presence of noise with an unknown cross-spectral matrix. It will be shown that the performance of bearing estimation algorithms improves substantially when this robust algorithm is used. Experimental data results are presented for the band noise spectral matrix.

Weighted parallel problem solving by optimization networks

Based on the analysis of the connections of optimization networks, we present the weighted intrazonal connection (WIZC) method to solve the quadratic assignment problem (QAP). This method improves the dynamics of the network by weighting and controlling the convergence speed of parallel problem solving, and works most effectively with up-to-date techniques such as subspace projection and matrix graduated non-convexity. Then more detailed formulation of the WICZ is given for the traveling salesman problem (TSP) which is an example of the QAP. The results obtained for TSPs show that the proposed method is effective on average, even though its performance depends on the problem itself.

Statistical analysis of the SDMP method for parameter estimation of multiple transients

We present a statistical analysis of the subspace decomposition and matrix pencil (SDMP) method for estimating parameters of multiple transient signals arriving at a uniform linear array. This analysis supports several observations obtained via simulation.

Second-order statistical analysis of totally weighted subspace fitting methods

We analyze the effects of a limited number of snapshots on a general class of multidimensional signal subspace estimation methods, termed totally weighted subspace fitting (TWSF) methods. This class is an extension of the weighted subspace fitting technique that includes row weighting and column weighting of the signal subspace matrix. We quantify the asymptotic statistical properties of the TWSF method by means of a second-order statistical analysis. The main contribution of this article is that it provides, for the first time, the estimator bias. Some simulation results are included to validate our analytical results. >