Chapter 4 - Frequency Analysis: The Fourier Series

Abstract

In this chapter we consider the frequency representation of periodic signals by means of the Fourier series, an infinite expansion in terms of orthonormal complex exponentials or sinusoids. Representing aperiodic signals in terms of periodic signals will permit us to extend the Fourier series representation to the Fourier Transform valid for periodic and aperiodic signals. The Fourier series coefficients are obtained using the orthonormality of complex exponentials or sinusoidal bases and efficiently computed using the Laplace transform of a period. The line spectrum, obtained from the Fourier series coefficients, indicates how the power of the signal is distributed to harmonic frequency components in the series. Properties of the Fourier series allow visualization of the power distribution over frequency, the symmetry of the spectrum, and the nature of the Fourier coefficients depending on the symmetry of the signal. Taking advantage of the eigenfunction property of linear time-invariant (LTI) systems, the steady-state response of these systems to periodic signals is easily obtained. MATLAB is used to represent and process periodic continuous-time signals.