Krylov Subspace Methods in the Electronic Industry

In most practical cases, the convergence of the GMRES method applied to a linear algebraic system Ax -- b is determined by the distribution of eigenvalues of A. In theory, however, the information about the eigenvalues alone is not sufficient for determining the convergence. In this paper the previous work of Greenbaum et al. is extended in the following direction. It is given a complete parametrization of the set of all pairs {A, b} for which GMRES(A, b) generates the prescribed convergence curve while the matrix A has the prescribed eigenvalues. Moreover, a characterization of the right hand sides b for which the GMRES(A, b) converges exactly in m steps, where m is the degree of the minimal polynomial of A, is given

Performance analysis and improvement of machine learning algorithms for automatic modulation recognition over Rayleigh fading channels

Automatic modulation recognition (AMR) is becoming more important because it is usable in advanced general-purpose communication such as, cognitive radio, as well as, specific applications. Therefore, developments should be made for widely used modulation types; machine learning techniques should be employed for this problem. In this study, we have evaluated performances of different machine learning algorithms for AMR. Specifically, we have evaluated performances of artificial neural networks, support vector machines, random forest tree, k-nearest neighbor, Hoeffding tree, logistic regression, Naive Bayes and Gradient Boosted Regression Tree methods to obtain comparative results. The most preferred feature extraction methods in the literature have been used for a set of modulation types for general-purpose communication. We have considered AWGN and Rayleigh channel models evaluating their recognition performance as well as having made recognition performance improvement over Rayleigh for low SNR values using the reception diversity technique. We have compared their recognition performance in the accuracy metric, and plotted them as well. Furthermore, we have served confusion matrices for some particular experiments

Index-aware Model Order Reduction for higher index DAEs

There exists many Model Order Reduction (MOR) methods for ODEs but little had been done to reduce DAEs especially higher index DAEs. In principle, if the matrix pencil of a DAE is regular, it is possible to use conventional MOR techniques to obtain reduced order models, which are generally ODEs. However, as far as their numerical treatment is concerned, the reduced models may be close to higher index models, that is, to DAEs. Thus the numerical solution of the reduced models might be computationally expensive, or even not feasible. In the worst cases, the reduced models may be unsolvable, i.e. their matrix pencil is singular. This problem is very pronounced for systems with index higher than 1, but it may occur even if the index of the problem does not exceed 1. Thus MOR methods for ODEs cannot generally be used for DAEs. This motivated us to introduce a new MOR method for DAEs which we call the index-aware MOR (IMOR) which can reduce DAEs while preserving the index of the system. This method involves first splitting the DAEs into differential and algebraic parts. Then, we use the existing MOR methods to reduce the differential part. We observed that the reduction of the differential part induces a reduction in the algebraic part. This enabled us to construct a method which reduces both the differential and the algebraic part. As a result a DAE is reduced. This method can also be used as a new method to solve DAEs. In this paper, we generalize the IMOR method to higher index DAEs and we shall call this method the GIMOR method. We use index-3 systems for testing and validating the accuracy of the GIMOR method

Model order reduction for nonlinear IC models

Model order reduction is a mathematical technique to transform nonlinear dynamical models into smaller ones, that are easier to analyze. In this paper we demonstrate how model order reduction can be applied to nonlinear electronic circuits. First we give an introduction to this important topic. For linear time-invariant systems there exist already some well-known techniques, like Truncated Balanced Realization. Afterwards we deal with some typical problems for model order reduction of electronic circuits. Because electronic circuits are highly nonlinear, it is impossible to use the methods for linear systems directly. Three reduction methods, which are suitable for nonlinear differential algebraic equation systems are summarized, the Trajectory piecewise Linear approach, Empirical Balanced Truncation, and the Proper Orthogonal Decomposition. The last two methods have the Galerkin projection in common. Because Galerkin projection does not decrease the evaluation costs of a reduced model, some interpolation techniques are discussed (Missing Point Estimation, and Adapted POD). Finally we show an application of model order reduction to a nonlinear academic model of a diode chain