Mamadou Mboup

Numerical differentiation with annihilators in noisy environment

Numerical differentiation in noisy environment is revised through an algebraic approach. For each given order, an explicit formula yielding a pointwise derivative estimation is derived, using elementary differential algebraic operations. These expressions are composed of iterated integrals of the noisy observation signal. We show in particular that the introduction of delayed estimates affords significant improvement. An implementation in terms of a classical finite impulse response (FIR) digital filter is given. Several simulation results are presented.

A revised look at numerical differentiation with an application to nonlinear feedback control

We are presenting new and efficient methods for numerical differentiation, i.e., for estimating derivatives of a noisy time signal. They are illustrated, via convincing numerical simulations, by the analysis of an academic signal and by the feedback control of a nonlinear system.

Parameter estimation for signals described by differential equations

An estimation method is presented for signals described by linear differential equations whose coefficients are functions of the unknown parameters. Various types of identifiability are defined according to these functions. In the linear case, an estimation algorithm is derived directly from the identifiability conditions, by use of elementary rules of operational calculus. The main steps of the algorithm are first presented through an introductory example of a damped sinusoid estimation. Unlike the modified Pronys method (used as reference), the presented one (1) provides explicit closed-form expressions and (2) remains valid for linear differential equations with non-constant coefficients. These expressions depend only on iterated integrals of the observed signals. Some basic features of the method are analysed and, in particular, we show that a least squares interpretation can be attached to it. Application to the estimation of a chirp signal is also presented with numerical simulations.

LMS coupled adaptive prediction and system identification: a statistical model and transient mean analysis

The LMS algorithm has been successfully used in many system identification problems. However, when the input data covariance matrix is ill-conditioned, the algorithm converges slowly. To overcome the slow convergence, an adaptive structure is studied, which incorporates an LMS adaptive predictor (prewhitener) prior to the LMS algorithm for system identification (canceler). Since the prewhitener is also adaptive, the input to the LMS canceler is nonstationary, even when the input is stationary. Because of the coupling and the nonstationarity of LMS canceler input, analysis of the performance of the two adaptations is extremely difficult. A simple theoretical model of the coupled adaptations is presented and analyzed. First and second moment analysis indicates that the adaptive predictor significantly speeds up the LMS canceler as compared to a system without prewhitening and enlarges the stability domain of the canceler (larger allowable /spl mu/). Monte-Carlo simulations are presented which are in good agreement with the predictions of the mathematical model. >

An error analysis in the algebraic estimation of a noisy sinusoidal signal

The classic example of a noisy sinusoidal signal permits for the first time to derive an error analysis for a new algebraic and non-asymptotic estimation technique. This approach yields a selection of suitable parameters in the estimation procedure, in order to minimize the noise corruption.

A gradient search interpretation of the super-exponential algorithm

This article reviews the super-exponential algorithm proposed by Shalvi and Weinstein (1993) for blind channel equalization. The principle of this algorithm-Hadamard exponentiation, projection over the set of attainable combined channel-equalizer impulse responses followed by a normalization-is shown to coincide with a gradient search of an extremum of a cost function. The cost function belongs to the family of functions given as the ratio of the standard l/sub 2p/ and l/sub 2/ sequence norms, where p>1. This family is very relevant in blind channel equalization, tracing back to Donohos (1981) work on minimum entropy deconvolution and also underlying the Godard (1980) (or constant modulus) and the earlier Shalvi-Weinstein algorithms. Using this gradient search interpretation, which is more tractable for analytical study, we give a simple proof of convergence for the super-exponential algorithm. Finally, we show that the gradient step-size choice giving rise to the super-exponential algorithm is optimal.

Coupled adaptive prediction and system identification: a statistical model and transient analysis

A significant drawback of the least mean square (LMS) algorithm is slow convergence speed when the input covariance matrix is ill-conditioned. Two structures are presented and studied for increasing the convergence speed for this case. The structures incorporate a prewhitening filter prior to the usual LMS adaptation. When the prewhitening filter is also adaptive the input to the LMS algorithm is nonstationary. An analysis of the coupling effect between the two adaptive algorithms show that the adaptive prewhitener has the capability of significantly speeding up to LMS adaptation as compared to a system without prewhitening. When the prewhitening filter is fixed (nonadaptive), the structure is shown to be equivalent to the filtered-X LMS algorithm. Stability conditions and transient means behavior are given in the time domain, in terms of the parameters of the pre-whitening filter.<<ETX>>

On the Structure of Self-similar Systems: A Hilbert Space Approach

This paper investigates the structural properties of linear self-similar systems, using an invariant subspace approach. The self-similar property is interpreted in terms of invariance of the corresponding transfer function space to a given transformation in a Hilbert space, in a same way as the time invariance property for linear systems is related to the shift-invariance of the Hardy spaces. The transformation in question is exactly that defining the de Branges homogeneous spaces. We show that any de Branges homogeneous space of order \(\nu \geqslant - \frac{1} {2}\) belongs to the Paley-Wiener space so that each element of such a space may be viewed as the transfer function of some linear self-similar system of parameter v. The explicit form of the corresponding impulse response, which is shown to be described by a hyperbolic partial differential equation, is given. Finally, we emphasize on the infinite dimension nature of self-similar systems through an abstract state space description.

A new QRD-based block adaptive algorithm

In this paper we present a new robust adaptive algorithm. It is derived from the standard QR decomposition based RLS (QRD-RLS) algorithm by introducing a non-orthogonal transform into the update recursion. Instead of updating an upper triangular matrix, as it is the case for the QRD-RLS, we adapt an upper triangular block diagonal matrix. The complexity of the algorithm, thus obtained, varies from O(N/sup 2/) to O(N) when the size of the diagonal blocks decreases. Simulations of the new algorithm have shown a better robustness than the standard QRD-based algorithm in the context of multichannel adaptive filtering with highly inter-correlated channels.

On-line frequency and damping estimation in a single-link flexible manipulator based on algebraic identification

The problem of estimating the defining parameters in a sinusoidal signal corrupted by noise is very important in the vibration analysis of flexible structures. Such estimations can be used in Health Monitoring or in adaptive control applications. Usually, pure sinusoidal models are used in order to represent the mechanical vibration. The main problem of these models is they do not consider the damping phenomenon of the flexible structures. In this work, an algebraic approach is used for the fast and reliable, on line, identification of the natural frequency and its damping coefficient of a flexible-link manipulator. This method uses an exponential damping sinusoidal model in combination with the algebraic derivative method for parameter identification. The computations are performed in a time interval inferior to the first full cycle of the measured vibration signal. A single-link flexible manipulator is used to verify the estimator performance.