Introduction to signal processing: spectral representation

Abstract

The article introduces powerful common tool for the signals analysis – basics of a spectral analysis. It focuses on decomposition into complex sine waves (harmonic sinusoids) called the Fourier transform. Based on qualitative review of the natural phenomenon of visible light (that is the electromagnetic field oscillation in a working range of the eyes) decomposition into a clear colors in a prism, known as Newton’s experiments, corresponding quantitative procedure of Fourier transform is announced. It translates the arbitrary signals specified in time (usually real, but not necessarily) into a set of complex sine waves in a mathematical form. The reversibility of the transformation, similar to the second prism that collects light rays back into a single white beam, is briefly shown. For this purpose, one of the key mathematical constructions of the theory of signals (the Dirac delta function and its properties) is considered more detailed. The phenomenon of negative frequencies is also discussed (which are not a fiction, but are characterized by standard propagating complex sinusoids twisted in different directions). It is shown that real physical signals have an excess in a spectrum (negative frequencies correspond to slightly modified positive ones), and they can be replaced with corresponding analytical ones. All arguments are presented at a descriptive level and in mathematical form, using visualizations and in the form of MatLab scripts. This gives a more complete picture of the subject matter, allowing looking at that from various points of view