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In this paper a new least-squares (LS) approach is used to model the discrete-time fractional differintegrator. This approach is based on a mismatch error between the required response and the one obtained by the difference equation defining the auto-regressive, moving-average (ARMA) model. In minimizing the error power we obtain a set of suitable normal equations that allow us to obtain the ARMA parameters. This new LS is then applied to the same examples as in ?R.S. Barbosa, J.A. Tenreiro Machado, I.M. Ferreira, Least-squares design of digital fractional-order operators, FDA'2004 First IFAC Workshop on Fractional Differentiation and Its Applications, Bordeaux, France, July 19-21, 2004, P. Ostalczyk, Fundamental properties of the fractional-order discrete-time integrator, Signal Processing 83 (2003) 2367-2376? so performance comparisons can be drawn. Simulation results show that both magnitude frequency responses are essentially identical. Concerning the modeling stability,both algorithms present similar limitations, although for different ARMA model orders.

Based on two methods recently proposed - the Ranging Criterion (RC) and the Generators Ranging Criterion (GRC) - new (quasi-orthogonal) even BCH-derived sequences are generated which are very attractive for synchronous or quasi-synchronous Code Division Multiple-Access (CDMA) systems. Numerical results show that the new family of BCH-derived sequences can contain a higher number of quasi-orthogonal sequences with lower correlation values and higher processing gains (PGs) than the spreading sequences typically used in the third generation of mobile communications system, UMTS or in the recent large area synchronised CDMA (LAS-CDMA) technology. It is shown that the even BCH-derived sequences are easily generated by a linear shift register generator, allowing the construction of systems with receiver structures of low complexity as compared with those of quasi-synchronous systems using low correlation zone sequences, as for instance the LAS-CDMA system.

The linear scale invariant systems are introduced for both integer and fractional orders. They are defined by the generalized Euler-Cauchy differential equation. The quantum fractional derivatives are suitable for dealing with this kind of systems, allowing us to define impulse response and transfer function with the help of the Mellin transform. It is shown how to compute the impulse responses corresponding to the two half plane regions of convergence of the transfer function.

A look into fractional calculus and its applications from the signal processing point of view is done in this paper. A coherent approach to the fractional derivative is presented, leading to notions that are not only compatible with the classic but also constitute a true generalization. This means that the classic are recovered when the fractional domain is left. This happens in particular with the impulse response and transfer function. An interesting feature of the systems is the causality that the fractional derivative imposes. The main properties of the derivatives and their representations are presented. A brief and general study of the fractional linear systems is done, by showing how to compute the impulse, step and frequency responses, how to test the stability and how to insert the initial conditions. The practical realization problem is focussed and it is shown how to perform the input?ouput computations. Some biomedical applications are described.

The empirical mode decomposition (EMD) is reviewed and some questions related to its effective performance are discussed. Its interpretation in terms of AM?FM modulation is done. Solutions for its drawbacks are proposed. Numerical simulations are carried out to empirically evaluate the proposed modified EMD.

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The 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.

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Fractional centred differences and derivatives definitions are proposed, generalizing to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are obtained. The computations of the involved integrals lead to new generalizations of the Cauchy integral derivative. To compute this integral, a special two-straight-line path was used. With this the referred integrals lead to the well-known Riesz potential operators and their inverses that emerge as true fractional centred derivatives, but that can be computed through summations similar to the well-known Grünwald-Letnikov derivatives.

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