Manuel Ortigueira

This paper presents a study on the DFT and its implementation, the FFT. It is shown that due to the lack of hermitianity of the trigonometric base, some results are different from those to be expected. With a slight modification the results are improved. An easy implementation based on the Horner algorithm is described.

ABSTRACT A reinterpretation of the classic definition of fractional Brownian motion leads to a new definition involving a fractional noise obtained as a fractional derivative of white noise. To obtain this fractional noise, two sets of fractional derivatives are considered: a) the forward and backward and b) the central derivatives. For these derivatives the autocorrelation functions of the corresponding fractional noises have the same representations. The obtained results are used to define and propose a new simulation procedure.

... Manuel Duarte Ortigueira Faculdade de Ciências/Tecnologia da UNL UNINOVA and DEE Campus da FCT Quinta da Torre 2829-516 Caparica Portugal e-mail: mdo@fct.unl.pt ISSN 1876-1100 e-ISSN 1876-1119 ISBN 978-94-007-0746-7 e-ISBN 978-94-007-0747-4 DOI ...

ABSTRACT Starting from the discrete-time nabla (forward) and delta (backward) derivatives we introduce a two-sided derivative valid for any order. Its eigenfunction is the normal discrete exponential. This derivative leads to discrete non causal linear systems.

ABSTRACT A derivative based discrete-time signal processing is presented. Both nabla (forward) and delta (backward) derivatives are studied and generalised including the fractional case. The corresponding exponentials are introduced as eigenfunctions of such derivatives.

ABSTRACT In this paper we deal with a frequency estimation problem using a zoom transform algorithm. In many a signal processing problems, it is of paramount importance the exact determination of the frequency of a signal. Some techniques derived from the FFT, just pad the signal with enough zeros in order to better sample its Discrete-Time Fourier Transform. Our approach, although FFT-based, does not rely on such method, using instead a "zoom" function built around the well-known sinc(.) function, resulting in an exact and deterministic method. Its analytic formulation is presented and illustrated with some simulation results over short-time based signals.

A new ARMA estimation algorithm is proposed. It is based on a fundamental relationship which shows that the AR polynomial of an ARMA(N, M) model belongs to the linear space spanned by the forward and backward linear predictors. This relationship allows us to construct an equivalent linear system with two inputs and the same output of the ARMA system. The inputs of this new system are the forward and backward linear prediction errors. As in this case the inputs and output are known, a least-squares identification algorithm is used to obtain the parameters of the system. These parameters define three polynomials. One of them is the AR polynomial. The other two converge asymptotically to the MA polynomial and to zero. Simple recursions are available to perform such a limiting operation. Zusammenfassung. Ein neuer Algorithmus zur Spektralschatzung mit ARMA-Modellen wird vorgeschlagen. Er beruht auf einer grundlegenden Beziehung, die besagt, dap das Nennerpolynom eines ARMA(N, M)-Modells zum linearen Raum gehort, der durch den Vorwarts-und den Ruckwartspradiktor aufgespannt wird. Dieser Zusammenhang erlaubt es, ein aquivalentes lineares System mit zwei Eingangen und einem unveranderten Ausgang zu kontruieren: Die Eingangssignale des neuen Systems sind dabei die Fehlersignale der beiden Pradiktoren. Da in diesem Fall Ein-und Ausgange bekannt sind, kann ein Minimal-Fehlerquadrat-Schatzalgorithmus benutzt werden, um die Systemparameter zu bestimmen. Diese Kenngropen definieren die Polynome. Eines davon ist das AR-Polynom; die beiden ubrigen konvergieren asymptotisch gegen das MA-Polygon und gegen Null. Einfache Rekursionsbeziehungen fur diesen Grenziibergang werden vorgelegt. RCsurne. Un nouveau algorithme d'estimation ARMA est propost. I1 est bast sur une rtlation fondamentale qui montre que le polynome AR d'un modtle ARMA(N, M) appartient a I'espace lintaire engendrt par les predicteurs lintaires direct et retrograde. Cette rtlation nous permet de construire un systbme lintaire tquivalent,avec deux entrtes et la mbme sortie que le systbme ARMA. Les entrtes du nouveau systtme sont les erreurs de prediction lintaires directe et rttrograde. Comme les entries et sorties sont connues, un algorithme &identification des moindres carrts est utilist pour obtenir les paramttres du systbme. Ces paramttres definissent trois polyn8mes. L'un est le polyn8me AR. Les deux autres convergent asymptotiquement vers le polyn8me MA et vers ztro. De simples rtcursions sont disponibles pour effectuer le passage a la limite.

RESUMEN Se estudian las matrices elementales de rango 1 (diadas). Para estas matrices se presentan fórmulas para su factorizacion, inversion, descomposition en valores própios y valores singulares. Estos resultados son aplicados en análisis recursivo de qualquiera que sea la matriz, siempre que se descomponga en una suma de matrices de rango 1. ABSTRACT T h e e l e m e n t a l r a n k-o n e m a t r i c e s (d y a d s) a r e s t u d i e d. F o r t h e s e m a t r i c e s w e p r e s e n t formulas for their factorization, inversion, eigendescomposition and singular descomposition. These results are applied in the recursive analysis of every matrix, as soon as it is assumed decomposed as a sum of rank-one matrices.

Sleep spindles are a hallmark of stage 2 sleep and are promising indicators of neurodegenerative disorders such as schizophrenia and dementia. In this paper two sleep spindle detectors are presented. The first is based on the Short Time Fourier Transform (STFT), the second is a novel algorithm and is based in the wave morphology of sleep spindles. Finally, a combination of the previous is proposed in a novel mixed algorithm. Performance results are presented applying the algorithms to a signal scored by two human experts. It is showed in that the combination of two algorithms, which separately provided seasonable results (around 91% sensibility), improves when they are mixed using the approach proposed (93%sensibility).

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. The fractional calculus is a generalization of the traditional calculus that leads to similar concepts and tools, but with wider generality and applicability. By allowing derivative and integral operations of arbitrary real or complex order, it is to traditional calculus what the real or complex numbers are to the integers [1,2]. This means that we must recover the traditional calculus when the order is a positive integer number. In ''Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions " [3] the introduction to fractional calculus presented there leads to several statements and results that deserve some comments, because they are in contradiction with the classic results and also with its own starting point (Eq. (1)). Let us start as in [3] with the following definition of fractional derivative:

T w o i m p o r t a n t q u e s t i o n s i n a r r a y s i g n a l p r o c e s s i n g a r e a d r e s s e d i n t h i s p a p e r : t h e d a t a m a t r i x v s autocorrelation matrix alternative and the recursive implementation of subspace DOA methods. The discussion of the first question is done in face of the proposed class of recursive algorithms. These new algorithms are easily implementable and have a high degree of parallelism, suitable for on-line implementations. Algorithms for recursive implementation of the eigendecomposition (ED) of the autocorrelation matrix and SVD of the data matrix are described. The ED/SVD trade-off is discussed.

—Sleep spindles are the most interesting hallmark of stage 2 sleep EEG. Their accurate identification in a polysomnographic signal is essential for sleep professionals to help them mark Stage 2 sleep. Sleep Spindles are also promising objective indicators for neurodegenerative disorders. Visual spindle scoring however is a tedious workload. In this paper three different approaches are used for the automatic detection of sleep spindles: Short Time Fourier Transform, Wavelet Transform and Wave Morphology for Spindle Detection. In order to improve the results, a combination of the three detectors is presented and comparison with human expert scorers is performed. The best performance is obtained with a combination of the three algorithms which resulted in a sensitivity and specificity of 94% when compared to human expert scorers.

The problem of steady state output of the discrete-time fractional differential systems is studied in this paper. Based on the fact that the exponentials are the eigenfunctions of such systems, a general algorithm for the output computation when the input is the product ''rising factorial. exponential'' is presented. The singular case is studied and solved.

In this paper we formulate a coherent discrete-time signals and systems theory taking derivative concepts as basis. Two derivatives – nabla (forward) and delta (backward) – are defined and generalized to fractional orders, obtaining two formulations that are discrete versions of the well-known Grünwald–Letnikov derivatives. The eigenfunctions of such derivatives are the so-called nabla and delta exponentials. With these exponentials two generalized discrete-time Laplace transforms are deduced and their properties studied. These transforms are back compatible with the current Laplace and Z transforms. They are used to study the discrete-time linear systems defined by differential equations. These equations although discrete mimic the usual continuous-time equations that are uniformly approximated when the sampling interval becomes small. Impulse response and transfer function notions are introduced and obtained. The Fourier transform and the frequency response are also considered. This implies a unified mathematical framework that allows us to approximate the classic continuous-time case when the sampling rate is high or obtain the current discrete-time case based on difference equation.

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In Sharma (SIViP 4:377–379, 2010) a fractional Laplace transform assumed to generalize the fractional Fourier transform was proposed. Here, it is shown that its region of convergence degenerates to the imaginary axis. So it is not a generalization of the fractional Fourier transform.

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.

The main objective of this paper is to propose a new generalisation of the Helmholtz decomposition theorem for both fractional time and space, which leads to four equations generalising the Maxwell equations that emerge as particular case. To get these results the well-known classical vectorial operators, gradient, divergence, curl, and laplacian are generalised to fractional orders using Grünwald–Letnikov approach.

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.