Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations.

Area of Science:

• Electrical Engineering
• Applied Mathematics
• Signal Processing

Background:

• Pulse Width Modulation (PWM) aims to represent harmonic waveforms using fixed-height square pulses.
• Approximation accuracy is crucial, requiring elimination of high-order harmonics in the error term.
• Reproducing sine-waves accurately with a limited number of pulses is a key challenge.

Purpose of the Study:

• To provide a complete mathematical solution to the general Pulse Width Modulation problem.
• To characterize discrete pulses for accurate harmonic waveform approximation.
• To leverage advanced mathematical concepts for improved PWM signal generation.

Main Methods:

• Utilizing Pade approximations for function approximation.
• Employing orthogonal polynomials in the analysis.
• Applying soliton theory and solutions to Korteweg-de Vries equations.
• Characterizing discrete pulse solutions within the framework of rational solutions to KdV equations.

Main Results:

• A comprehensive solution to the PWM problem is presented.
• Discrete pulse characteristics are defined for accurate analog signal approximation.
• The solution connects PWM to the theory of rational solutions of Korteweg-de Vries equations.

Conclusions:

• The study offers a novel mathematical framework for solving the PWM problem.
• The findings enable more accurate reproduction of harmonic waveforms using digital pulses.
• This research bridges advanced mathematical concepts with practical signal processing applications.