Abstract: This paper describes an adaptive filter structure which may be used in multi-channel noise-cancelling applications. The proposed structure differs from those presented previously in that it incorporates a lattice filter framework, rather than tapped-delay-lines. The successive orthogonalization provided by the lattice offers advantages in adaptive convergence rate which cannot be achieved with tapped-delay-lines. In the sections below, we present an explicit description of the general noise-cancelling lattice structure, together with the appropriate adaptive algorithms.

Abstract: The decomposition of vector time series data into orthogonal components can be applied in both temporal and spatial discrete frequency analysis. If the observed multidimensional data is non-stationary, then adaptive procedures can be used for estimation of the eigendata. This paper presents the relationship between multicomponent, spectral signals in noise and the corresponding eigendata. Two adaptive realizations of the eigendata estimation process are considered. Examples are given which allow a comparison between signal detection by data orthogonalization, power spectrum estimation and two channel magnitude squared coherence computation.

Abstract: A general method for adaptive updating of lattice coefficients in the linear predictive analysis of nonstationary signals is presented. The method is given as one of two sequential estimation methods, the other being a block sequential estimation method. The fast convergence of adaptive lattice algorithms is seen to be due to the orthogonalization and decoupling properties of the lattice. These properties are useful in adaptive Wiener filtering. As an application, a new fast start-up equalizer structure is presented. In addition, a one-multiplier form of the lattice is presented, which results in a reduction of computations.

Abstract: As a tool for the interpretation ofab initio SCF calculations, an energy partitioning scheme is presented. When performed within an orthogonalized basis, the scheme allows the deduction of well transferable, almost basis independent two-center terms which characterize bond strengths and non-bonded interactions. The results for a large number of molecules are given. The construction of an orthogonal minimal basis (OMBA) from arbitrary basis sets as a generalization of the symmetrical orthogonalization is described. The transferability of Fock matrix elements is discussed. The energy partitioning quantities are related to the corresponding terms obtained with the semiempirical schemes CNDO and MINDO/3.

Abstract: This chapter discuses two-sample t -tests. In a two-sample t -test, the samples are taken from two distinct populations. The chapter presents examples to explain the two-sample t -distribution. To find the mean vector nearest the observation vector, the concept of the orthogonal projection of a vector onto a given subspace and a method for finding this vector are needed. The orthogonal projection of a vector onto a subspace is given geometrically by dropping a perpendicular from the vector onto the subspace. The vector minus the orthogonal projection of the vector is perpendicular to the subspace. This property is used in defining the orthogonal projection operator and the orthogonal projection of a vector onto a subspace. Subspaces are orthogonal if any two vectors chosen from two distinct subspaces are orthogonal. This is another key idea for multinormal observations with a covariance matrix. The chapter also discusses the noncentral chi-squared distribution.

Abstract: A new stable and efficient implementation of the Lanczos algorithm is presented. The algorithm is a powerful method for finding a few eigenvalues and eigenvectors at one or both ends of the spectrum of a symmetric matrix A. The algorithm is particularly effective if A is large and sparse in that the only way in which A enters the calculation is through a subroutine which computes Av for any vector v. Thus the user is free to take advantage of any sparsity structure in A and A need not even be represented as a matrix et al.

Abstract: The first 21 normal modes and corresponding cutoff frequencies are obtained for the E modes of waveguides with regular hexagonal cross-sections. A previously developed numerical method, using inverse iterations, finite differences, and Richardson’s mesh extrapolation procedure, is employed. The inverse iteration method is modified by an orthogonalization procedure to obtain the higher modes. A block factoring method is used to solve the finite difference equations. This combination of numerical techniques yields excellent accuracy and speed of computation.

Abstract: In this paper, the subspace method of pattern recognition, in which method classification is decided by the largest projection of an unknown pattern vector onto subspaces corresponding to different classes, is applied to the recognition of continuous Finnish speech. Classification is based on phonemic power spectra produced by an analog filter bank. When compared, e.g., with the nearest-neighbor method and the method of direction cosines, the advantages of the subspace method are an improved stability of classification and a more balanced total classification accuracy of the different phonemic classes with respect to their relative frequencies of occurrence. The efficient spanning of the subspaces as well as their mutual orthogonalization are discussed. Furthermore, the close relationship between the phonemic labeling and segmentation when using the subspace method is pointed out.

Abstract: : This report presents a study of adaptive lattice algorithms as applied to channel equalization. The orthogonalization properties of the lattice algorithms make them promising for equalizing channels with heavy amplitude and/or phase distortion. Furthermore, unlike the majority of other orthogonalization algorithms, the number of operations per update for the adaptive lattice equalizers is linear with respect to the number of equalizer taps. (Author)

Abstract: A new computer-oriented algorithm GSO is presented for solving overdetermined systems of linear observation equations according to the principle of the least-squares method. The matrix of the system of observation equations may be of deficient rank. In this case the algorithm leads to the vector of unknowns with a minimum Euclidean norm. Alternatively, it is possible to minimize the norm of a subvector formed by a selected group of unknowns. The weight coefficient matrix, corresponding to the vector (subvector) of unknows, has the least possible trace. The algorithm GSO is based on the Gram-Schmidt Orthogonalization of suitably defined augmented matrices. The establishing and solving of normal equations is not necessary. Apart from the unknowns and residuals, GSO also determines the factorized weight coefficient matrices of the adjusted values.

Abstract: The mathematical framework of the canonical expansion of a random time function has been applied to the analysis of the electroencephalogram. The proposed method has been adapted for a computersuited algorithm by using an orthogonalization concept of the vector space theory. The offered solution avoids the mathematical difficulties of the rigorous theory of Karhunen-Loeve expansion. Moreover, the results of the analysis highlight new aspects of the physical properties of the electroencephalogram. Suggestions with regard to the non-linear character of the oscillators generating electrical activity seem to be valid.

Abstract: The two-dimensional transonic small-perturbation equation is solved by a higher-order finite element method using a combination of a Galerkin orthogonalization procedure and a least-squares method. It is shown that this higher-order method requires a non-trivial formulation of the discrete boundary conditions. The Galerkin orthogonalization leads to an overdetermined non-linear algebraic equation system, that is solved in a least-squares sense by a Newton-Raphson method. The choice of the weights is determined from considerations of the convergence of approximate solutions if the mesh size tends to zero. Results for subsonic flow are given for two cusped aerofoils placed on the side wall of a closed channel. The Newton-Raphson method was found to be efficient. A grid of about 85 grid points (680 unknowns) gives results of comparable accuracy with those of a second-order finite-difference method with about 1400 grid points (1400 unknowns). A considerable effort will be necessary to design a computer code permitting the accurate calculation of shock waves in transonic flows by a higher-order finite-element method.