Many methods for signal analysis are based in the framework of time-frequency analysis, and some of them are supposed to be able to deal with non-stationary signals, as the Hilbert-Huang transform, which is based on the Hilbert Spectrum Analysis (HSA). HSA is a classical methodology which allows to assign an amplitude-phase representation, and hence an instantaneous energy and frequency, to a signal, which are supposed to be meaningful when applied to some specific signals. However, the resulting methods not always produce consistent or satisfactory frequential analyses. In this work, we propose a new method based on the HSA for time-frequency analysis, which is easy to implement, computationally cheap, and highly customizable to adapt the results to some specific requirements on the frequential behaviour of the signal. To do that, some previous theoretical analysis are performed, which are interesting by themselves. We investigate the time-frequency localization problem of the Fourier transform, defining the support of measurable functions, and characterizing the spaces of functions with support contained into a specific set. This leads us to consider the Hardy-type spaces, composed by square-summable functions such that the support of its Fourier transform is contained into a specific set. On the other hand, we approximate some time-frequency components of a signal, linked to specific time windows and frequency ranges, with generalized trigonometric polynomials, with frequencies into these specific ranges. This leads us to introduce the Tremolo-Vibrato spaces, intimately linked to the Hardy-type spaces, and to study the Hilbert spectrums of generalized trigonometric polynomials, which can be performed directly by using explicit general expressions. Finally, we propose a simple method for quantifying the time-varying relative importance of each component in the frequential decomposition. This method is applied to the analysis of medical signals, as EEG and ECG.

Frequency is a central concept in Mathematics, Physics, and Signal Processing. It is the main tool for describing the oscillatory behavior of signals, which is usually argued to be the manifestation of some of their key features, depending on their nature. For instance, this is the case of Electroencephalographic signals. Hence, frequency is substantially present in the most common methodologies for analyzing signals, as the Fourier Analysis or the Time-Frequency Analysis. However, in spite of its importance as a keystone in Signal Processing, and its seemingly simple meaning, its mathematical foundation is not as straightforward as it may seem at first glance. A naive interpretation of the different mathematical concepts modelling frequency can be misleading, as their actual meanings essentially differ from the intuitive notion which are supposed to represent. In our opinion, this circumstance should be taken into account to develop and apply appropriate signal analysis and processing tools in some applications. In the current text we discuss this topic, with the main goal to draw the attention of the mathematical and engineering community to this point, often overlooked.