Extraction of harmonics from trigonometric polynomials by amplitude and phase operators

Abstract

Extraction of harmonics of a given order from real trigonometric polynomials (signals) is one of the main problems in harmonic analysis. It has many applications in physics, radio and electrical engineering, in particular, in filtration of harmonic signals of different nature. There exist many methods (mainly, approximative) for solution of this problem. The most common ones are spectral methods based on Fourier transform and other resonance principles. In this paper we propose a new method for extracting harmonics by amplitude and phase transformation of trigonometric polynomials. These transformations use the two simplest operations -- multiplication by a real constant and phase shift -- to obtain polynomials similar to the initial ones. A harmonic is extracted by an amplitude and phase operator that simply overlays (sums up) a finite number of such similar polynomials. The overlay method enables us to obtain precise analytical formulas for calculating harmonics of a given order.