Fast Recursive Algorithms Research Articles

Easy and effective stabilisation measure for fast recursive least squares algorithms for adaptive transversal filters

Fast recursive least squares (FRLS) algorithms are known for their favourable convergence characteristics. They are of special interest for the fast adaptation of transversal filters of high order N. Unfortunately FRLS algorithms have a tendency towards numeric instability. A new and effective stabilisation measure is presented for a 0(7N) FRLS algorithm which needs only one additional multiplication per iteration step.

Boolean matrix transforms for the parity spectrum and minimisation of modulo-2 canonical expansions

Boolean matrix transforms between the parity spectrum and modulo-2 canonical expansions of a Boolean function are considered. Using a Kronecker matrix product approach, fast recursive algorithms derived for these and other transforms are applied to the minimisation of modulo-2 canonical expansions.

On the least squares tap adjustment algorithm in adaptive digital echo cancellers

Fast recursive algorithms for updating coefficients in digital echo cancellers can be derived from the well-known method of least squares. Unless zero initial conditions are assumed, the exact initialization of these algorithms is yet unknown. For random data of more than twice the order of the filter, the existence of a unique least squares solution is proven. A constructive recursive procedure in time and order for computing the pseudoinverse solution for the initial steps is derived. Since the data matrix is composed of integers, this technique facilitates the implementation of stable tap-update algorithms. >

Modular and Numerically Stable Multichannel Fast Transversal Filters

In this paper, we present scalar implementations of multichannel fast Recursive Least-Squares algorithms in transversal filter form (so-called FTF algorithms). The point is that by processing the different channels sequentially, ie one at a time, the processing of any channel reduces to that of the single-channel algorithm. This sequential processing decomposes the multichannel algorithm into a set of intertwined single-channel algorithms. Geometrically, this corresponds to a modified Gram-Schmidt orthogonalization of multichannel error vectors. Algebraically, this technique corresponds to matrix triangularization of multichannel error covarianre matrices and converts matrix operations into a regular set of scalar operations.Algorithm structures that are amenable to VLSI implementation on arrays of parallel processors naturally follow from our approach. Numerically, the resulting algorithm benefits from the advantages of triangularization techniques in block-processing, which are a well-known part of Kalman filtering expertise. Furthermore, recently introduced stablization techniques for proper control of the propagation of numerical errors in the update recursions are also incorporated.

Recursive two-dimensional median filtering algorithms for fast image root extraction

The authors develop and evaluate two new two-dimensional fast recursive median filtering algorithms, for image root extraction. The first recursive algorithm, which is called the fast binary root algorithm, converges to a root of a binary image in one pass including a number of local rescans. The second is referred to as the fast multilevel root algorithm. It converges to a root of a multilevel image in two passes, the second of which includes a number of local rescans. The number of ordering positions required by the author's algorithms is shown to be significantly less than those required by the standard and regular recursive median filters.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Efficient time-recursive least-squares algorithms for finite-memory adaptive filtering

This paper deals with efficient algorithms in the sense of minimization of the computational complexity for least-squares (LS) adaptive filters with finite memory. These filters obtain the current estimate of the desired response using only a fixed finite number of past data. First, two new fast recursive least-squares algorithms with computational complexities <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">14m</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">15m</tex> multiplications and divisions per recursion (MADPR), respectively, are introduced ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> is the filter order). Then a new estimation-error-oriented recursive modified Gram-Schmidt (RMGS) scheme with a complexity of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m^{2} + 10m</tex> MADPR is given. Finally, the learning characteristics of these algorithms are discussed and some simulation results are included.

Fast and stable recursive algorithms for continuous-time and discrete-time model conversions

Based on the Newton-Raphson method, this paper presents recursive algorithms that are rapidly convergent and more stable for modeling the equivalent continuous-time (discrete-time) model from the available discrete-time (continuous-time) model for a fixed sampling period. The newly developed recursive algorithms relax the constraints imposed upon the existing model conversion algorithms, and, thus, enhance the applications of microprocessors and associated microelectronics to digital control systems. A practical example is presented to demonstrate the effectiveness of the proposed procedures.

Improvement of the fast recursive least-squares algorithms via normalization: A comparative study

This paper deals with the derivation and the properties of fast optimal least-squares algorithms, and particularly with their normalization. It is shown how the well-known fast Kalman algorithm, written in the most general form, can be normalized through a purely algebraic point of view, leading to the normalized least-squares transversal filter derived by Cioffi, Kailath, and Lev-Ari from the geometric approach. An improved form of the algorithm is presented. The different algorithms have been compared from a practical point of view as regards their convergence, initialization procedures, complexity, and numerical properties. Normalized transversal algorithms are shown to be interesting because of their nice structured form, simplicity of conception, and improved good numerical behavior.

Difference Methods for Solving One-Dimensional Inverse Problems

We consider the one-dimensional inverse problem of reflection seismology. An impulsive plane wave in pressure is applied to a horizontally homogeneous elastic half space and the ensuing particle velocity is measured at the surface. The characteristic impedance is recovered as a function of travel time. Several methods of solution of this inverse problem require the construction of an auxiliary solution of the differential equations with a wave front, and the impedance is recovered using the theory of propagation of singularities. The use of difference schemes to construct this auxiliary solution leads to fast algorithms to solve this inverse problem. Fast recursive algorithms for solving discrete integral equations, which are Toeplitz linear systems, can be identified with difference schemes. A downward continuation algorithm developed by the authors which uses difference schemes is also presented.

Fast recursive algorithms for a class of linear equations

In many signal processing applications, one often seeks the solution of a linear system of equations by means of fast algorithms. The special form of the matrix associated with the linear system may permit the development of algorithms requiring 0 (p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) or fewer operations. Hankel and Toeplitz matrices provide well known examples and various fast schemes have been developed in the literature to cover these cases. These techniques have common characteristics so that they may be generalized to cover a wider class of linear systems. The purpose of this paper is to develop fast algorithms that cover this wider set of systems. An important feature of the general scheme introduced here is that it leads to the definition of two broad classes of matrices, called diagonal innovation matrices (DIM) and peripheral innovation matrices (PIM), for which fast schemes can be developed. The class of PIM matrices includes many structures appearing in signal processing applications. Most of them are extensively studied in this paper and Fortran coding is provided. Finally, ARMA modeling is considered and within the general framework already introduced, fast methods for the determination of the autoregressive (AR) portion of the ARMA model are presented.