Spectral representation of signals

In addition to the usual dynamic representation of signals and functions in the form of the dependence of their values ​​on certain arguments (time, linear or spatial coordinates, etc.), in the analysis and processing of data, a mathematical description of signals is widely used by the arguments inverse to the arguments of the dynamic representation. So, for example, for time, the inverse argument is frequency. The possibility of such a description is determined by the fact that any signal, arbitrarily complex in its shape, which does not have discontinuities of the second kind (infinite values ​​in the interval of its assignment), can be represented as a sum of simpler signals, and, in particular, as a sum of the simplest harmonic oscillations , which is performed using the Fourier transform. Respectively,mathematically, the decomposition of a signal into harmonic components is described by functions of the values ​​of the amplitudes and the initial phases of oscillations in a continuous or discrete argument - the frequency of changes in functions at certain intervals of the arguments of their dynamic representation. The set of amplitudes of harmonic oscillations of the decomposition is called the amplitude spectrum of the signal, and the set of initial phases is calledphase spectrum . Both spectra together form the full frequency spectrum of the signal, which is identical in mathematical accuracy to the dynamic form of signal description.

Linear signal conversion systems are described by differential equations, and the principle of superposition is true for them, according to which the response of systems to a complex signal consisting of a sum of simple signals is equal to the sum of responses from each component of the signal separately. This allows, with a known response of the system to a harmonic vibration with a certain frequency, to determine the response of the system to any complex signal, expanding it into a series of harmonics over the frequency spectrum of the signal. The widespread use of harmonic functions in signal analysis is explained by the fact that they are fairly simple orthogonal functions and are determined for all values ​​of continuous variables. Moreover, they are the proper functions of time,retaining their shape when oscillations pass through any linear systems and data processing systems with constant parameters (only the amplitude and phase of oscillations change). Of no small importance is the fact that a powerful mathematical apparatus has been developed for harmonic functions and their complex analysis.

Examples of frequency representation of signals are given below (Fig. 1.1.5 - 1.1.12).

In addition to the harmonic Fourier series, other types of signal decomposition are also used: by the functions of Hartley, Walsh, Bessel, Haar, Chebyshev, Lagger, Legendre polynomials, etc. The main condition for the uniqueness and mathematical identity of signal display is the orthogonality of the decomposition functions. However, qualitative analysis of signals can also use non-orthogonal functions that reveal any characteristic features of signals that are useful for interpreting physical data.