Application of Trigonometric Interpolation Polynomials to Signal Processing

At the moment, signal theory and signals processing occupy very important place in the scientific and technical progress of society. The results of the research in this field are used in many areas of science and technology to solve a wide range of problems. Many of these problems require the use of approximation theory methods for signal processing. The use of approximation methods for signal theory is one of the most necessary nuances related to their digital processing. Classical approximation theory methods used to construct an approximation of a continuous signal, as a rule, are based on the use of a set of certain values of the function representing the studied signal, at predetermined values of the independent variable. When working with signals, we use interpolation to solve such problems. In such cases, classical interpolation methods such as Lagrange interpolation polynomials or other interpolation methods, such as those based on Fourier series are used. However, most of the interpolation methods of the type indicated above do not always give optimal results in terms of approximation quality or require a lot of calculations compared to other approximation methods. In this work, a new, more effective method of interpolation has been proposed. This method makes it possible to restore a continuous signal qualitatively using its discrete representation. We found that the introduced trigonometric interpolation polynomial allows obtaining a higher quality of signal restoration in comparison with the often considered Lagrange interpolation polynomials. Also in the paper, a comparative analysis of the proposed introduced trigonometric interpolation polynomial with the widely used Lagrange interpolation polynomial has been carried out. We have shown that the trigonometric interpolation polynomial introduced in the work gives the better approximation and requires significantly fewer calculations.