Modified Berlekamp-Massey algorithm and modelling of discrete-time signals

On parametrizations for the minimal partial realization problem

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.

Minimal state-space realization in linear system theory: an overview

The Berlekamp-Massey algorithm [L. S. De Jong, Numerical aspects of realization algorithms in linearsystems theory, Ph.D. thesis, T. H. Eindhoven, the Netherlands, 1975] and [J. L. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Information Theory, IT-15 (1969), pp. 122–127] for minimal realization problems is a special case of the Padé approximation problem. As a matter of fact, it computes among other polynomials the denominators of the elements of the Padé table that are on the descending diagonal $\{ [0/1],[1/2], \cdots ,[k/k + 1], \cdots \} $ as far as they exist and this algorithm works for nonnormal Padé tables too. This algorithm does not seem to be well-known in Padé approximation literature. It is not very difficult to generalize this algorithm so as to compute the other Padé approximants of a nonnormal table. Some variants will lead to a generalization of the algorithm of Brezinski [Computation of Padé approximants continued fractions, J. Comput. Appl. Math., 2 (1976), pp. 113–123], which computes the descending diagonals of a normal Padé table, and of the algorithm of Watson [D. Bussonnais, “Tous” les algorithmes de calcul par recurrence des approximants de Padé d’une serie, Construction de fractions continues correspondantes, Séminaire d’Analyse Numérique, No. 293, Grenoble,1978], [G. Claessens, new look at the Padé table and the different methods for computing its elements, J. Comput. Appl. Math., 1 (1975), pp. 141–152], [P. J. S. Watson, Algorithms for differentiation and integration, Padé Approximants and their Applications, P. R. Groves-Morris, ed., Academic Press London, 1973, pp. 93–98] that computes the descending staircases of normal tables. This work is the dual of Cordellier’s [algorithmes de calcul recursif des elements d’une table de Padé non normale, Conference on Padé approximation, Lille, France, 1978], and of McEliece and Shearer [A property of Euclid’s algorithm and an application to Padé approximation, SIAM J. Appl. Math., 34 (1978), pp. 611–616] deriving similar results for ascending diagonals and staircases. A more continued fraction-like approach would generalize the Thatcher algorithm and may be found in [J. A. Murphy and H. R. O’Donohoe, A class of algorithms for obtaining rational approximants to functions which are defined by power series, Z. Angew. Math. Phys., 28 (1977), pp. 1121–1131] and [A. Bultheel, Division algorithms for continued fractions and the Padé table, Applied Mathematics and Programming Division, Rep. TW41, Katholicke Universitiet Leuven, Heverlee, Belgium, August1978].

The partial realization problem for 2D discrete linear shift-invariant systems is discussed. The authors attempt to establish a reasonable definition of minimality and propose a method for finding the degree and the system function of a minimal realization of a 2D linear system characterized by a given 2D impulse response array. The method used is based on the 2D Berlekamp-Massey algorithm which has a close connection with the 2D Hankel matrix derived from the 2D array. While it compares in efficiency with other methods for identification and approximation of 2D linear systems, it gives a novel approach to the problem in the sense that one does not need any knowledge or assumptions concerning the system degree. >

The realization of linear time-varying systems specified by an analytic weighting pattern is approached in a novel manner using an algebraic framework defined over the ring of analytic functions. Realizations are given by a state representation consisting of a first-order vector differential equation and an output equation, both with analytic coefficients. Various new criteria for realizability are derived, including conditions given in terms of the finiteness of modules over the ring of analytic functions generated by the elementary rows or columns of a (generalized) Hankel matrix. These results are related to local criteria for realizability specified in terms of the rank of matrix functions, as developed in the work of Silverman and Meadows [5], [8], [9] and Kalman [7]. It is shown that the construction of minimal realizations reduces to the problem of computing a basis for a finite free module defined over the ring of analytic functions. A minimal realization algorithm is then derived using a constructive procedure for computing bases for finite free modules over a Bezout domain. The Silverman-Meadows realization algorithm [5] is a special case of the procedure given here. In the last part of the paper, the realization algorithm is applied to the problem of system reduction.

In this paper proposes a receiver for direct sequence-code division multiple access (DS-CDMA) multi-user communication system based on Laguerre filter and the recursive least squares (RLS) adaptive algorithm. Intersymbol interference (ISI) is usually caused by multipath propagation and the inherent non-linear frequency response of a channel. Therefore, the objective of this paper is to minimize the effects of ISI with a long impulse response environment based on the proposed receiver. Because of the stability of Laguerre structure is compromise between the finite impulse response (FIR) filter and infinite impulse response (IIR) filter. It can improve the bit error rate (BER) performance of the proposed receiver compared to the conventional FIR filter receiver. The simulation results show that the performances of the proposed receiver in terms of BER and the stability are better than does the conventional FIR filter receiver.

In this paper it is shown that a natural representation of a state space is given by the predictor space, the linear space spanned by the predictors when the system is driven by a Gaussian white noise input with unit covariance matrix. A minimal realization corresponds to a selection of a basis of this predictor space. Based on this interpretation, a unifying view of hitherto proposed algorithmically defined minimal realizations is developed. A natural minimal partial realization is also obtained with the aid of this interpretation.

The problem of distortion of the autoregressive model of the voice signal under the influence of additive background noise in digital speech processing systems with data compression based on linear prediction is considered. In the frequency domain, these distortions are observed in the weakening of the main formants responsible for the intelligibility of the speaker’s speech. To compensate for formant attenuation, it is proposed to modify the parameters of the autoregressive model (linear prediction coefficients) using the impulse response of a recursive shaping filter. Along with the amplitude amplification of the formants, their frequencies remain unchanged to make the speaker’s voice recognizable. The effectiveness of the method was studied experimentally using specially developed software. Based on the experimental results, conclusions were drawn about a significant increase in the relative level of formants in the power spectrum of the corrected voice signal.

The state‐space realization of linear systems is of utmost importance in linear systems theory. After the realization problem for the time‐invariant case was solved, particular attention was paid to the case of linear periodic systems. Recently, such systems have regained importance, for instance, in the context of coding theory where periodic convolutional encoders play an important role. The majority of the contributions within this area only concern the realization of transfer functions (or impulse responses), thus excluding the case of input/output linear systems without coprime representations. By the end of the eighties of the last century, Jan C. Willems suggested an approach (nowadays known as the behavioral approach) that considers a wider class of systems and allows to overcome this drawback. According to this approach, the central object in a system is its behavior which consists of all the signals that satisfy the system laws (also called system trajectories). Consequently, the behavior of a system with an input/output representation that is not coprime contains more trajectories than the set of input/output signals defined by the system transfer function. Our work takes this fact into account. In previous research conducted by the authors, some results on the realization of periodic behavioral systems were obtained, which, despite their importance, are conservative with respect to the dimension of the obtained realizations. In this paper, we revisit the behavioral periodic realization problem and give further insight into this question. This allows to derive new results and set up an algorithm to compute low‐dimensional state‐space realizations.

For the further development we use the state-variable description of linear time-invariant systems. This is the standard form used in linear system theory. Introducing this notation brings some advantages. The state-variable description is the most general form and is equally applicable for scalar and for multivariable systems. Matrix calculus is employed instead of working with polynomials. The whole theory of linear systems is available.

Given a discrete-time linear system C, a shortest basis for C is a set of linearly independent generators for C with the least possible lengths. A basis B is a shortest basis if and only if it has the predictable span property (i.e., has the predictable delay and degree properties, and is non-catastrophic), or alternatively if and only if it has the subsystem basis property (for any interval J, the generators in B whose span is in J is a basis for the subsystem CJ). The dimensions of the minimal state spaces and minimal transition spaces of C are simply the numbers of generators in a shortest basis B that are active at any given state or symbol time, respectively. A minimal linear realization for C in controller canonical form follows directly from a shortest basis for C, and a minimal linear realization for C in observer canonical form follows directly from a shortest basis for the orthogonal system C⊥. This approach seems conceptually simpler than that of classical minimal realization theory.

Based on Takagi-Sugeno fuzzy model, employing parallel distributed compensation, the multivariable integral control and state observer in the linear system theory are applied to nonlinear systems for the problem of output tracking and rejecting disturbance with reference and/or disturbance signals in the form of step, and the fuzzy integral controller and fuzzy state observer are synthesized. The stability condition of continuous closed-loop fuzzy control system and a similar separation principle to the counterpart on observer design in the linear system theory are obtained.

On the generalized Sylvester mapping and matrix equations

We present a theoretical approach to analysis of toxicokinetics and toxicodynamics using linear systems theory. In our approach, we define two impulse response functions that characterize the kinetic behavior of an environmental agent in the body and the dynamic time-course behavior of its effect on the body. This approach provides a formalism for understanding the relation among exposure, dose, and cumulative biologically effective dose and for understanding the implications of an effect time-course on cross-sectional and longitudinal data analyses. We use lead-associated cognitive decline as a specific example where the approach may be applied.