A Novel Way for Fourier Expansion Using Similarity Matching in Time Domain

Abstract

This paper discusses the inherent difficulties presented by Fast Fourier Transform in factoring the time domain signal into its frequency domain and presents an alternative method to determine them by analysing the data in time domain itself. The frequencies identifiable in FFT implementation is limited by the nyquist criteria, which enables to identify the higher harmonics albit low amplitudes by increasing the sampling frequency and number of samples passed on to the FFT algorithm. The frequencies whether meaningful or not will be identified in this implementation, thus presenting typical problem of frequency flow. But, if the participating harmonics are known a priori, the problem can be addressed by matching the contour of the participating harmonic sinusoids. As since only the known harmonics are addressed in this approach, the difficulty of frequency flow and appearance of fractional frequencies, which otherwise would have no meaning and interpretation are taken care of. The codes are based on the time domain split of information into a matrix vector form and obtaining the solution by least square method, via regression. Complete mathematical formulation for the proposed method and an example is presented. For ascertaining the applicability, the initial signal has been compared with the reconstructed signal, which is in good agreement.

Keywords: Fast Fourier Transform, Time domain harmonic extraction, Linear Regression, Alternative to FFT