GENERALIZATIONS OF FOURIER SERIES: MATHEMATICAL MODELLING OF SPECTRAL REPRESENTATION OF PERIODIC AND NON-PERIODIC SIGNALS ANALYSIS

Abstract

This paper discussed the use of Fourier series frequently in theoretical studies of periodic and non-periodic signals. A periodic signal can be represented by a Fourier series which is formed, by adding together an innite number of harmonics at frequencies which are multiples of the fundamental repetitive frequency of the signal sequence. With the use of Fourier series, we can resolve the signal X(t)=x(t+nT) ∶n=±1,±2,⋯ into an innite sum of sine and cosine terms to obtain spectral representation of a signal. The Fourier Transform is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines. The Fourier Transform shows that any waveform can be re-written as the sum of sinusoidal functions. The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, non-periodic function by a continuous superposition or integral of complex exponentials. This paper shows the spectral representation of non-periodic signals notably, pulsed signals being accomplished by expanding into Fourier integrals.