The well-known Liouville, Riemann?Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Grünwald?Letnikov derivative. Particularly, starting from a complex formulation of the Grünwald?Letnikov derivative we establishes a bridge with existing integral formulations and obtained regularised integrals for Liouville, Riemann?Liouville, and Caputo derivatives. Moreover, it is shown that we can combine the procedures followed in the computation of Riemann?Liouville and Caputo derivatives with the Grünwald?Letnikov to obtain a new way of computing them. The theory we present here will surely open a new way into the fractional derivatives computation.

Sleep staging is a crucial step before the scoring the sleep apnoea, in subjects that are tested for this condition. These patients undergo a whole night polysomnography recording that includes EEG, EOG, ECG, EMG and respiratory signals. Sleep staging refers to the quantification of its depth. Despite the commercial sleep software being able to stage the sleep, there is a general lack of confidence amongst health practitioners of these machine results. Generally the sleep scoring is done over the visual inspection of the overnight patient EEG recording, which takes the attention of an expert medical practitioner over a couple of hours. This contributes to a waiting list of two years for patients of the Portuguese Health Service. In this work we have used a spectral comparison method called Itakura distance to be able to make a distinction between sleepy and awake epochs in a night EEG recording, therefore automatically doing the staging. We have used the data from 20 patients of Hospital Pulido Valente, which had been previously visually expert scored. Our technique results were promising, in a way that Itakura distance can, by itself, distinguish with a good degree of certainty the N2, N3 and awake states. Pre-processing stages for artefact reduction and baseline removal using Wavelets were applied.

A software tool for the analysis of the High-Resolution Electrocardiogram (HR-ECG) for Arrhythmia detection is introduced. New algorithms based on Wavelet analysis are presented and compared with the classic Simson protocol over the P and QRS segments of the Electrocardiogram (EEG). A novel procedure based on a two step wavelet analysis and synthesis is performed in order to obtain a frequency description of the P, T or QRS segments. This frequency ?signature? is useful for the detection of otherwise asymptomatic Arrhythmia patients. The tool has been developed in Matlab, and deployed for a standalone C application.

Rail corrugation is a phenomenon that leads to a waving in the rails with wavelengths typically between 3 cm and 100 cm and amplitude levels of several microns. The genesis of this waving is complex. Rail corrugation is a recognized problem that leads to excess vibration on the rails and vehicles to a point of reducing their life span and compromising safety. In urban areas excess vibration noise is also a problem. A software tool was developed to analyze accelerometer signals acquired in the boggies of rail vehicles in order to quantify the rail corrugation according to their frequency and amplitude. A wavelet packet methodology was used in this work and compared with the One Third Octave Filter (OTOF) power representations, which is currently used in the industry. It is shown that the former produces better results.

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.

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.

New BCH-derived PN-EB (pseudo-noise even balanced) sequences suitable to be used in CDMA systems are presented in this paper. It is assumed a new definition for processing gain, which better accounts for the system performance regarding the narrow band noise rejection, and it is shown how to obtain high processing gain values, namely, by using zero mean spreading signals in channels with selective noise. The new sequences have low autocorrelation levels, exist in large numbers and can provide higher processing gain with selective noise.

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