Mathematical description of signals

Signals can be objects of theoretical research and practical analysis only if the way of their mathematical description is specified - a mathematical model of the signal. The mathematical description allows one to abstract from the physical nature of the signal and the material form of its carrier, to classify signals, perform their comparison, establish the degree of identity, and simulate signal processing systems. As a rule, the description of the signal is given by the functional dependence of a certain information parameter of the signal on the independent variable (argument) - s (x), y (t), etc. This form of description and graphical representation of signals is called dynamic (a signal in the real dynamics of its behavior by arguments). The functions of mathematical description of signals can be both real and complex.The choice of the mathematical apparatus of description is determined by the simplicity and convenience of its use in the analysis and processing of signals.

Note the duality of using the description of signals by functions of the type s (t), etc. On the one hand, s (t) is a value equal to the value of the function at time t. On the other hand, we denote by s (t) and the function itself, i.e. that is the rule according to which each value of t is associated with a certain value s. In most analytical expressions, this does not cause misunderstandings, and if the signal values ​​unambiguously correspond to their analytical expressions, it is accepted by default.

Let us also make one remark on the terminology for describing signals. In theoretical works on signal analysis, specific values ​​of the signal magnitude (readings of values ​​by argument) are often referred to as signal coordinates. In the fields of knowledge related to geology and mining, and in geophysical practice as well, this term is used for its direct semantic purpose - the spatial coordinates of measurement results, and is an invariable attribute of all geological and geophysical data. Taking into account the latter factor, let us agree to use the term “coordinate” according to its traditional semantic purpose as a generalizing term for independent variable signal functions. Moreover, under the concept of coordinates values ​​of the signal, we will understand not only any spatial coordinates, as is directly the case for the measurement results during geological and geophysical surveys, but also any other arguments, on the numerical axis of which the values ​​or readings of the signal are plotted and the dynamics of its change is considered (an example in Fig. . 1.1.1).