Some results presented in the paper ?Modeling fractional stochastic systems as non-random 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. Keywords: Fractional calculus; Grünwald-Letnikov derivative; Fractional Brownian motion

The time-varying microstructure of sleep EEG spindles may have clinical significance in dementia studies and can be quantified with a number of techniques. In this paper, real and simulated sleep spindles were regarded as AM?FM signals modeled by six parameters that define the instantaneous envelope (IE) and instantaneous frequency (IF) waveforms for a sleep spindle. These parameters were estimated using four different methods, namely the Hilbert transform (HT), complex demodulation (CD), matching pursuit (MP) and wavelet transform (WT). The average error in estimating these parameters was lowest for HT, higher but still less than 10% for CD and MP, and highest (greater than 10%) for WT. The signal distortion induced by the use of a given method was greatest in the case of HT and MP. These two techniques would necessitate the removal of about 0.4 s from the spindle data, which is an important limitation for the case of spindles with duration less than 1 s. Although the CD method may lead to a higher error than HT and MP, it requires a removal of only about 0.23 s of data. An application of this sleep spindle parameterization via the CD method is proposed, in search of efficient EEG-based biomarkers in dementia. Preliminary results indicate that the proposed parameterization may be promising, since it can quantify specific differences in IE and IF characteristics between sleep spindles from dementia subjects and those from aged controls.

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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.

In this paper, the classic Levy identification method is reviewed and reformulated using a complex representation. This new formulation addresses the well known bias of the classic method at low frequencies. The formulation is generic, coping with both integer order and fractional order transfer functions. A new algorithm based on a stacked matrix and its pseudoinverse is proposed to accommodate the data over a wide range of frequencies. Several simulation results are presented, together with a real system identification. This system is the Archimedes Wave Swing, a prototype of a device to convert the energy of sea waves into electricity.

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Fractional 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.

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