A new symmetric fractional B-spline

Signal Processing, Vol. 83, nº 11A new de(nition of a symmetric fractional B-spline is presented. This generalises the usual integer order B-spline, that becomes a special case of the new one

The comb signal and its Fourier transform

SignalProcessing, Vol. 81, nº 3In this paper, we study the aperiodic comb signalfrom the point of view of the Fourier transform. The comb is very important in the theory of ideal sampling. The knowledge of its properties is crucial for the establishment of suitable interpolation schemes. Here, we present sufficient conditions so that the Fourier transform of an aperiodic comb is an aperiodic comb. We use this result to propose: (1) an alternative approach to the de"nition of an almost periodic signal and its anharmonic Fourier series; (2) a generalisation of the Shannon}Whittakker sampling/reconstruction for the irregular sampling case. Application of this theory to pulse duration modulation and pulse position modulation is also presented

A new approach to the initial value problem

5th Portuguese Conference on Automatic Control, September, 5-7, 2002, Aveiro, PortugalThe initial condition problem for fractional linear system initialisation is studied in this paper. A new approach that involves functions belonging to the space of Laplace transformable distributions is presented. It is based on the generalised initial value theorem and on reinterpretations of the most common differintegration definitions

On the initial conditions in continuous-time fractional linear systems

Signal Processing, Vol. 83, nº 11The initial condition problem for fractional linear system initialisation is studied in this paper. It is based on the generalised initial value theorem. The new approach involves functions belonging to the space of Laplace transformable distributions verifying the Watson–Doetsch lemma. The fractional derivatives of these functions are independent of the derivative de,nition. This class includes the most important functions appearing in computing the Impulse Response of continuous-time fractional linear systems

An introduction to the fractional continuous-time linear systems

A brief introduction to the fractional continuous-time linear systems is presented. It will be done without needing a deep study of the fractional derivatives. We will show that the computation of the impulse and step responses is very similar to the classic. The main difference lies in the substitution of the exponential by the Mittag-Leffler function. We will present also the main formulae defining the fractional derivatives

Fractional central differences and derivatives

Journal of Vibration and Control, Vol. 14, Nº 9-10Fractional central differences and derivatives are studied in this article. These are generalisations to real orders of the ordinary positive (even and odd) integer order differences and derivatives, and also coincide with the well known Riesz potentials. The coherence of these definitions is studied by applying the definitions to functions with Fourier transformable functions. Some properties of these derivatives are presented and particular cases studied

Comments on ‘‘Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions

Applied Mathematical Modelling, Vol.33Some results presented in the paper ‘‘Modeling fractional stochastic systems as nonrandom fractional dynamics driven Brownian motions” [I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999] are discussed in this paper. The slightly modified Grünwald-Letnikov derivative proposed there is used to deduce some interesting results that are in contradiction with those proposed in the referred paper

Fractional central differences and derivatives

Journal of Vibration and Control, 14(9–10): 1255–1266, 2008Fractional central differences and derivatives are studied in this article. These are generalisations to real orders of the ordinary positive (even and odd) integer order differences and derivatives, and also coincide with the well known Riesz potentials. The coherence of these definitions is studied by applying the definitions to functions with Fourier transformable functions. Some properties of these derivatives are presented and particular cases studied

Introduction to fractional linear systems. Part 2: discrete-time case

IEE Proceedings - Vision, Image, and Signal Processing, Vol. 147, nº 1In the paper, the class of discrete linear systems is enlarged with the inclusion of discrete-time fractional linear systems. These are systems described by fractional difference equations and fractional frequency responses. It is shown how io compute the impulse response and transfer function. Fractal signals are introduced as output of special linear systems: fractional differaccumulators, systems that can be considered as having fractional poles or zeros. The concept of fractional differaccumulation is discussed, generalising the notions of fractal and lif noise, and introducing two kinds of fractional differaccumulated stochastic proccss: hyperbolic, resulting from fractional accumulation (similar to the continuous-time casc), and parabolic noise, resulting from fractional differencing

Pseudo-fractional ARMA modelling using a double Levinson recursion

IET Control Theory & Applications, Vol. 1, Nº 1The modelling of fractional linear systems through ARMA models is addressed. To perform this study, a new recursive algorithm for impulse response ARMA modelling is presented. This is a general algorithm that allows the recursive construction of ARMA models from the impulse response sequence. This algorithm does not need an exact order specification, as it gives some insights into the correct orders. It is applied to modelling fractional linear systems described by fractional powers of the backward difference and the bilinear transformations. The analysis of the results leads to propose suitable models for those systems