Extraction of harmonics from trigonometric polynomials by amplitude and phase operators

TL;DR

This paper introduces a novel amplitude and phase transformation method for extracting specific harmonics from trigonometric polynomials, offering precise analytical formulas and potential advantages over traditional spectral approaches.

Contribution

The paper presents a new harmonic extraction technique using amplitude and phase operators based on simple transformations, differing from existing approximative spectral methods.

Findings

Provides exact analytical formulas for harmonic extraction.

Uses simple amplitude and phase operations for polynomial similarity.

Offers an alternative to Fourier-based spectral methods.

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.