Preface Bibliography 1. Interpolation. Introduction The Taylor expansion The finite Taylor series with the remainder term Interpolation by polynomials The remainder of Lagrangian interpolation formula Equidistant interpolation Local and global interpolation Interpolation by central differences Interpolation around the midpoint of the range The Laguerre polynomials Binomial expansions The decisive integral transform Binomial expansions of the hypergeometric type Recurrence relations The Laplace transform The Stirling expansion Operations with the Stirling functions An integral transform of the Fourier type Recurrence relations associated with the Stirling series Interpolation of the Fourier transform The general integral transform associated with the Stirling series Interpolation of the Bessel functions 2. Harmonic Analysis. Introduction The Fourier series for differentiable functions The remainder of the finite Fourier expansion Functions of higher differentiability An alternative method of estimation The Gibbs oscillations of the finite Fourier series The method of the Green's function Non-differentiable functions Dirac's delta function Smoothing of the Gibbs oscillations by Fejer's method The remainder of the arithmetic mean method Differentiation of the Fourier series The method of the sigma factors Local smoothing by integration Smoothing of the Gibbs oscillations by the sigma method Expansion of the delta function The triangular pulse Extension of the class of expandable functions Asymptotic relations for the sigma factors The method of trigonometric interpolation Error bounds for the trigonometric interpolation method Relation between equidistant trigonometric and polynomial interpolations The Fourier series in the curve fitting 3. Matrix Calculus. Introduction Rectangular matrices The basic rules of matrix calculus Principal axis transformation of a symmetric matrix Decomposition of a symmetric matrix Self-adjoint systems Arbitrary n x m systems Solvability of the general n x m system The fundamental decomposition theorem The natural inverse of a matrix General analysis of linear systems Error analysis of linear systems Classification of linear systems Solution of incomplete systems Over-determined systems The method of orthogonalisation The use of over-determined systems The method of successive orthogonalisation The bilinear identity Minimum property of the smallest eigenvalue 4. The Function Space. Introduction The viewpoint of pure and applied mathematics The language of geometry Metrical spaces of infinitely many dimensions The function as a vector The differential operator as a matrix The length of a vector The scalar product of two vectors The closeness of the algebraic approximation The adjoint operator The bilinear identity The extended Green's identity The adjoint boundary conditions Incomplete systems Over-determined systems Compatibility under inhomogeneous boundary conditions Green's identity in the realm of partial differential operators The fundamental field operations of vector analysis Solution of incomplete systems 5. The Green's Function. Introduction The role of the adjoint equation The role of Green's identity The delta function -- The existence of the Green's function Inhomogeneous boundary conditions The Green's vector Self-adjoint systems The calculus of variations The canonical equations of Hamilton The Hamiltonisation of partial operators The reciprocity theorem Self-adjoint problems Symmetry of the Green's function Reciprocity of the Green's vector The superposition principle of linear operators The Green's function in the realm of ordinary differential operators The change of boundary conditions The remainder of the Taylor series The remainder of the Lagrangian interpolation formula

Linear Differential Operators. By C. Lanczos. Pp. 580. 80s. 1961. (Van Nostrand, London)

SPEECH RECOGNITION SYSTEM ABSTRACT Apparatus and method for evaluating the likelihood of a word in a vocabulary of words wherein a total score is evaluated for each word, each total score being the re-sult of combining at least two word scores generated by differing algorithms. In one embodiment, a detailed acoustic match word score is combined with an approxi-mate acoustic match word score to provide a total word score for a subject word. In another embodiment, a polling word score is combined with an acoustic match word score to provide a total word score for a subject word. The acoustic models employed in the acoustic matching may correspond, alternatively, to phonetic el-ements or to fenemes. Fenemes represent labels generated by an acoustic processor in response to a spoken input. Apparatus and method for determining word scores ac-cording to approximate acoustic matching and for deter-mining word scores according to a polling methodology are disclosed.

Speech recognition system

We consider the design of finite impulse response (FIR) filters subject to upper and lower bounds on the frequency response magnitude The associated optimization problems with the filter coefficients as the variables and the frequency response bounds as constraints are in general nonconvex Using a change of variables and spectral factorization we can pose such problems as linear or nonlinear convex optimization problems As a result we can solve them efficiently (and globally) by recently developed interior-point methods We describe applications to filter and equalizer design and the related problem of antenna array weight design

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FIR Filter Design via Spectral Factorization and Convex Optimization

An acoustic signal processing circuit extracts input speech pattern data and subsidiary feature data from an input speech signal. The input speech pattern data comprise frequency spectra, whereas the subsidiary feature data comprise phoneme and acoustic features. These data are then stored in a data buffer memory. The similarity measures between the input speech pattern data stored in the data buffer memory and reference speech pattern data stored in a dictionary memory are computed by a similarity computation circuit. When the largest similarity measure exceeds a first threshold value and when the difference between the largest similarity measure and the second largest measure exceeds a second threshold value, category data of the reference pattern which gives the largest similarity measure is produced by a control circuit to correspond to an input speech. When recognition cannot be performed, the categories of the reference speech patterns which respectively give the largest to mth similarity measures are respectively compared with the subsidiary feature data. In this manner, subsidiary feature recognition of the input voice is performed by a subsidiary feature recognition section.

An improved linear programming algorithm is required for the design of finite impulse response (FIR) digital filters. This algorithm avoids the numerical ill-conditioning problems which commonly occur due to the necessity to sample the frequency response on a very dense grid of points for high-order filters. This technique is applied to the design of equiripple FIR Nyquist filters and equiripple FIR transmit and receive matched-filters for data transmission applications. The use of linear programming insures that the designs are optimal in the sense that they achieve the maximum possible stopband attenuation for a given filter order and stopband edge frequency. The design algorithm for the matched filters consists of a two-stage process of linear programming to design a Nyquist filter with a nonnegative frequency response followed by standard spectral factorization techniques to extract the nonlinear-phase transmit and receive filters. The design software consists primarily of commonly available FORTRAN subroutine packages for linear programming and polynomial factorization, and is numerically well behaved and very accurate. >

On the design of optimal equiripple FIR digital filters for data transmission applications

The paper presents a new method for the optimal design of minimum phase FIR filters. First, a constrained approximation procedure is used to obtain the magnitude function and the transmission zeros in the stopband(s). Next, the zeros of the transfer function inside the unit circle are calculated via a low-degree polynomial factorization. For low-pass filters, a straightforward exchange algorithm is presented which achieves the constrained approximation step; a convergence proof is given and it is shown that the algorithm can be implemented via a simple modification of the Parks-McClellan program. The efficacy of the method is illustrated by a numerical example. Attention is drawn to the fact that bandpass filters may in principle require more sophisticated means.

Optimal design of minimum-phase FIR filters

This paper examines the problem of reducing the number of states in the Markov chain models used in speech recognition algorithms by statistical methods. It is shown that the equivalence requirements between the original and reduced model lead to entirely different reduction equations depending on whether the reduction occurs inside a word model or between the models of different words. In general, the exact solution of these reduction equations is not possible and one shows that a least-squares solution can be reformulated as a matrix approximation problem in Euclidean norm.

State reduction in hidden Markov chains used for speech recognition

The paper presents a new method for the optimal design of minimum phase FIR filters. First, a constrained approximation procedure is used to obtain the magnitude function and the transmission zeros in the stopband(s). Next, the zeros of the transfer function inside the unit circle are calculated via a low-degree polynomial factorization. For lowpass filters, a straightforward exchange algorithm is presented which achieves the constrained approximation step and which can be implemented via a simple modification of the Parks-McClellan program. The efficacy of the method is illustrated by a numerical example. Attention is drawn on the fact that bandpass filters may in principle require more sophisticated means.

Optimal design of minimum phase FIR filters

This paper considers, for a multivariate Gaussian random process, the maximum likelihood estimation (MLE) of a covariance matrix whose structure satisfies some particular constraints. First, one examines the case where the random process is required to satisfy a time varying auto-regressive (AR) model of fixed order p. In particular, one shows that the resulting optimal covariance matrix is a partial reconstruction of the given sample covariance matrix. Next, a linear feature extraction is considered with a slightly unusual criterion which requires that the likelihood of the extracted features should be as large as possible.

Y. Kamp

Papers

Optimal design of minimum-phase FIR filters

State reduction in hidden Markov chains used for speech recognition

Optimal design of minimum phase FIR filters

Some results on constrained maximum likelihood estimation