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The normal way of introducing the notion of derivative is by means of the limit of an incremental ratio that can assume three forms, depending the used translations as we saw in Chaps. 1 and 4. On the other hand, in those derivatives the limit operation is done over a set of points uniformly spaced: a linear scale was used. Here we present an alternative derivative, that is valid only for t {\ensuremath{>}} 0 or t {\ensuremath{<}} 0 and uses an exponential scale

Generalised fractional centred differences and derivatives are studied in this Chapter. These generalise to real orders the existing ones valid for even and odd positive integer orders. For each one, suitable integral formulations are presented. The limit computation inside the integrals leads to generalisations of the Cauchy derivative. Their computations using a special path lead to the well known Riesz potentials. A study for coherence is done by applying the definitions to functions with Fourier transform. The existence of inverse Riesz potentials is also studied.

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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.

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