Una teoría generalizada de señales y sistemas

Abstract

The Signals and Systems Theory (SST) plays a fundamental role in the academic and professional background in different areas of electrical engineering (signal processing, electromagnetics, acoustics, quantum mechanics, etc.) as well as in many other scientific areas. Many authors present this theory following a scheme which is valid for the practical analysis of many systems following the usual scheme of the SST, for instance. This way of presenting this theory usually avoids dealing with more general concepts that are fundamental in the explanations associated with the resolution of a large number of physical problems. These limitations use to be related to the mathematical and physical interpretation of many important concepts underlying the SST, for instance (i) the definition of generalized functions such as the Dirac delta or its derivatives without considering the mathematical rigor of the theory of distributions, (ii) the analysis of linear invariant system in the timefrequency domain by performing spectral analysis under the Fourier transform only, (iii) the analysis of continuous and discrete variable problems separately, (iv) the lack of considering the analysis of linear non invariant systems in a rigorous way, etc. These simplifications leave out many important problems that should be analyzed under the SST. This is particularly important if the analysis focuses on physical problems (usually defined by differential equations plus some boundary conditions) under the SST, for instance: (i) problems in the spatial domain, which are often linear non invariant, (ii) the analysis in the time domain of linear non-invariant systems (amplitude modulator, for instance), (iii) the spectral analysis under other transforms, in connection with the usual representations using different wave functions as base functions (cylindrical waves, spherical waves, Gaussian beams, complex beams, wavelets, etc.), (iv) the analysis of the Green’s functions theory as a particular case of the SST, (v) the consideration of the theory of distributions together with the ordinary functions through the theory of rigged Hilbert spaces (RHS), (vi) the extension of the usual SST to complex variable functions in order to understand the continuation of real coordinates to complex ones, or (vii) the generalization of the analysis of nonlinear operators, as well as many other types of problems. The final aim of the work presented in this paper is to develop a general theory which can include all of these cases in a rigorous way. This leads to a Generalized Signals and Systems Theory (GSST) that has been built keeping in mind that any physical problem may be analyzed particularizing the general concepts of this theory to the concrete parameters of the problem at hand. This scheme is continuously revised and updated with new results. The up-to-date version of this scheme will be presented in this paper and it is used nowadays for presenting the SST to both undergraduates (in a simplified version) and postgraduate students. The GSST scheme is built under finite or infinite dimension signal vector spaces together with the theory of operators and the theory of distributions, considering general variables that may represent any physical magnitude (time, space, etc.). With these initial considerations in mind, several generalized important concepts will be introduced, such as the Generalized Linear Combination (LC), the Generalized Transform (GT), the Generalized Transform Changes (GTC) and the Generalized Spectral Analysis (GSA) of linear (invariant and non-invariant) systems.