Educational software tool for signals and systems theory

Abstract

One of the most important subjects in the educational background of electrical engineering as well as in other important scientific areas of knowledge is the Signals and Systems Theory (SST). Taking into account the connection of the SST with the algebraic representation and the mathematical understanding of physical problems — electromagnetic radiation, scattering and inverse problems in our particular case —, a generalized scheme of the SST (GSST) is currently under development. While the usual point of view of classical SST is valid for practical purposes, it usually avoids many important concepts which become fundamental in the generalization of the analysis of physical problems. Some examples of this generalization are concerned with representations of physical problems different from those in the usual time-frequency domains — using a general variable —, the description of the problem in terms of infinite dimensional vector spaces and the algebra associated, the description of the Dirac delta function in terms of the distribution theory, etc. The most important purpose of obtaining a GSST is to connect the usual SST with important mathematical representations that lead to obtain more general representations that may be rigorously applied to model general physical problems. This generalized theory is focused on the algebraic concepts of signals and operators spaces, identifying each problem into their corresponding vector spaces and representing it in terms of the GSST scheme. One important point within this generalization is the concept of the Generalized Transform. Choosing a set of functions which play the role of a base, each signal of the space may be expressed in terms of a Generalized Linear Combination operator of this base weighted by a set of coefficients which identify the transform — their expression is directly related to the metric defined initially in the associated algebra —. Spectral analysis of systems is also contemplated, representing the signals in terms of their coefficients once a set of basis functions is chosen. The generalized theory let to obtain integral characterizations of linear systems — invariant and non invariant —, leading to the Generalized Spectral Analysis concept, closely connected to the operator theory. While the mathematical analysis of the GSST and its application, for instance to generalize the Green's function theory, has already been presented in several symposium — like 2009 IEEE AP and URSI Meeting —, a software tool based on the GSST is being worked out in parallel to the theoretical works. The aim of this tool is to facilitate both students and practical scientific and engineers the understanding of the main algebraic concepts of this generalized scheme, providing some examples of many signals and many systems in five different signal spaces. The user can select the signal and the system with their appropriate parameters and analyze the properties of the systems as well as the output signal both in real and different spectral domains. The last version of this tool, SSTv4.5, lets to analyze linear non invariant systems and to visualize the set of impulse responses as well as the spectral analysis under some transformations such as Fourier, Hilbert or Bessel Transforms — depending on the signal space —, thus exemplifying some particular cases of the Generalized Spectral Analysis. The description of the current version of this software tool is presented in this paper.