Recursive Algorithms for Nonnormal Padé Tables

: Calculating the amount of information about a random function contained in another random function has important uses in communication theory. An expression for the mutual information for continuous time random processes has been given by Gelfand and Yaglom, Chiang, and Perez by generalizing Shannon's result in a natural way. Under a condition of absolute continuity of measures the continuous time expression has the same form as Shannon's result. For two Gaussian processes Gelfand and Yaglom express the mutual information in terms of a mean square estimation error. We generalize this result to diffusion processes and express the solution in a different form which is more naturally related to a corresponding filtering problem. We also use these results to calculate some information rates.

where |y1P #1. These were first discussed by Schur [6]1 with |7pl <1 in connection with functions bounded in the unit circle. In ?2 it is shown that to an arbitrary power series there correspond continued fractions (1.1), and, conversely, to every such continued fraction there corresponds a power series which agrees with the power series for the 2pth and (2p+1)th approximants of the continued fraction up to and including the terms involving zP and zP-1, respectively. An important new feature which is given in ?3 is the simple algorithm for the expansion of a power series into a continued fraction (1.1), and, conversely, for the computation of the corresponding power series from the continued fraction. Heretofore this computation has been exceedingly cumbersome. In ?4 conditions under which these continued fractions converge to the corresponding power series are considered, in particular, if the generating function is meromorphic. In ?5 convergence regions for the expansions (1.1) are found, and in ?6 convergence regions for the continued fraction whose approximants are the even and odd approximants of (1.1). In ?7 certain transformations of (1.1) are discussed.

In 1969, J. Massey published a now-famous paper showing, among other things, that an iterative algorithm introduced by Berlekamp for decoding BCH codes also solved the problem of finding a shortest-length feedback shift register circuit for generating a given finite sequence of digits. This nice physical interpretation opened the door to connections with many other problems, including the minimal partial realization problems of linear system theory, Pade approximations and continued fractions, the fast algorithms of Levinson and Schur for Toeplitz matrices, inverse scattering, VLSI implementations, etc. This paper is an informal account of some of the different contexts in which the Berlekamp-Massey algorithm have been encountered in the work of the author and his students.

The Berlekamp—Massey algorithm (further, the BMA) is interpreted as an algorithm for constructing Pade approximations to the Laurent series over an arbitrary field with singularity at infinity. It is shown that the BMA is an iterative procedure for constructing the sequence of polynomials orthogonal to the corresponding space of polynomials with respect to the inner product determined by the given series. The BMA is used to expand the exponential in continued fractions and calculate its Pade approximations.

The objective of the present paper is the study of models which view the given 1-D data as the impulse response of a minimal order discrete linear filter with data and coefficients confined to a finite field. This representation is known as “minimal partial realization” problem in linear system theory and has been tackled successfully by Ho and Kalman. Interestingly, the same problem is addressed and solved in a coding theory context by the well-known Berlekamp-Massey (bm) algorithm. We present a modified recursive version of thebm algorithm specially suited to our task of signal modelling. By introducing an absolute error bound, one can further solve the approximation problem computationally. The paper includes a number of examples and a mention of the software for this purpose.

Fractional Brownian Motions, Fractional Noises and Applications

An algorithm for shifted continued fraction expansions in parallel linear time

A note on extending Euler's connection between continued fractions and power series

Hankel continued fraction and its applications

BCH code is adopted as a part of the Forward Error Correction subsystem of DVB-S2 system, which is the next generation digital video broadcast system based on satellite wireless communication system. As DVB-S2 system uses very long code length and multiple code modes, full compatible BCH decoder is high area cost. In order to reduce the area cost of the full compatible DVB-S2 BCH decoder, we have modified the VLSI implementation of Berlekamp Massey Algorithm for key equation solver which is the main part of BCH decoder, rebuilt the Galois Filed Multiplications including the constant Galois Field Multiplications and the general Galois Field Multiplications. In order to support all the code modes and Adaptive Coded Modulation, a novel duo-pipeline reconfigurable architecture have been proposed, which can support code mode switching without stalling the symbol stream. We have implemented the decoder with verilog HDL, and have evaluated it by FPGA platform and ASIC library. The results show that the logic area of the decoder is at least 13% fewer than other existing decoders.

: This report deals with research results obtained in the field of mathematical system theory. Special emphasis was given to the following areas: (1) Linear system theory over a field: parametrization of multi-input, multi-output systems and the geometric structure of classes of systems of constant dimension. (2) Linear systems over a ring: development of the theory for very general classes of rings. (3) Nonlinear system theory: basic properties of polynomial (discrete-time) systems and their realization theory. The present report is only a brief summary of findings, which are far too technical to be explained in a few pages. The results are being gradually published in the open literature. For details, refer to previous proposals and/or to the publications listed in the references.

Flat Power Series over a Finite Field

In this part, we review some basic results on frames and the windowed Fourier transform. More detailed information can be found in Daubechies (IEEE Trans. Inform. Theory, 36(5):961–1005, 1990; Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1992) and Daubechies et al. (J. Math. Phys. 27(5):1271–1283, 1986), for example.

Introduction Numerous elementary and higher transcendental functions of complex analysis possess representations in terms of continued fractions of the Stieltjes type. (An abundance of examples are given in the textbooks by WALL [1948] and PERRON [t957]; for some recent examples and applications, see MAEHLY [1960], CONTE and FRIED [t96t], MILLER, MOLMUD and MEECHAM [t962], BLANCH [1964], PATRY and KELLER [1964], W¥1~N [1964].) Similar representations also exist for the remainder terms of the asymptotic expansions of functions defined by certain Laplacian integrals (HESRICI [t963, 1965]). Such representations are of considerable computational value, both because of the ease with which a continued fraction can be evaluated recursively, and because of the fact that continued fraction representations frequently are convergent (and computationally efficient) in wider regions of the complex plane than, say, a Taylor series. In actual computation it is necessary to approximate the value of a continued fraction by some approximant of finite order. This paper makes an a t tempt to s tudy the error committed in this approximation. Both a posteriori bounds depending on the knowledge of a finite number of approximants -and a priori bounds depending only on the elements of the continued fraction or on the coefficients of its corresponding power series are presented. A solution of the problem of finding a posteriori bounds which in a sense is best possible is given in section 2. Several different a priori bounds are given in section 3. Although these bounds are not shown to be best possible, they are shown to have the correct order of magnitude in some important special cases. The tack of a generally accepted terminology and even of clear definitions in the analytic theory of continued fractions made it necessary to recollect, in a unified notation, some basic facts of the theory in section t.

converge to P(z) uniformly over an arbitrary closed region, K, exterior to the real axis. (ii) If P(z) is a Stieltjes power series with radius of convergence R>0, then an arbitrary infinite sequence of distinct approximants converges to P(z) uniformly over the circle Iz -=R-6, a > 0. (iii) If P(z) is a positive definite power series which is summable (Borel) to F(z), then Sk(k > 1) converges to F(z) uniformly over K. If the reciprocal of P(z) is summable (Borel) the same holds for k < 1. (iv) If P(z) is a Stieltjes series possessing minimal extensionsi D=-n Cif(-z)i for every n, and if the series EZiXo(1)ic-ilz-i-' is summable (Borel) to F(z), then in the Pade table for P(z), S-1 converges to F(z) uniformly over an arbitrary closed region exterior to the negative half of the real axis. As a corollary to (iii) we find that when P(z) is summable (Borel) and has a corresponding continued fraction, the latter is necessarily convergent to the Borel sum. The converse is not true. That is, the corresponding continued fraction may converge when P(z) is not summable (Borel). 2. The convergence of the diagonal files in the Pade table for a convergent positive definite power series. We recall that P(z) is a positive definite series if all the quadratic forms j= n= , 1, 2, * , are positive de-