Application of the Continuous Fourier Transformation in Signal Theory

There are a number of methods in systems analysis that are used to solve a wide range of applied problems, including digital signal processing. The method of discrete Fourier transformation is one of the widely used methods. The method of discrete Fourier transformation allows us to analyze, convert and synthesize signals. It is not possible using analog processing. The discrete Fourier transformation is particularly used to work with digital signals processing. However, sometimes it is necessary to work with continuous signals, so it is impossible to use the Fourier discrete transformation. In such cases, we use the so-called continuous Fourier transformation, which is a continuous analogue of a discrete transformation and has a similar content. Due to the specifics of the continuous Fourier transformation as a method for solving classical problems of systems analysis, there is a need to study the number of its properties, including its convergence. The relation between the smoothness of the summing Abel-Poisson function, which models the studied signal, and the convergence rate of its continuous Fourier transformation has been established. Since the summing Abel-Poisson functions are harmonic, as well as the signals that occur in real conditions, the results obtained in this paper take into account not only the smoothness of the function, but also its periodicity. Having taken into account this aspect, we could indicate the method of signal recovery, which gives better approximate properties than those previously studied in foreign and domestic scientific works.