Chapter 4 - Frequency Analysis: The Fourier Series

Abstract

This chapter discusses the frequency analysis of the Fourier series for periodic signals. The frequency representation of periodic and aperiodic signals indicates how their power or energy is allocated to different frequencies. The Fourier representation of periodic signals is fundamental in finding a representation for non-periodic signals. Complex exponentials or sinusoids are used in the Fourier representation of periodic as well as aperiodic signals by taking advantage of the Eigen-function property of linear time invariant (LTI) systems. Fourier analysis considers the steady state, while Laplace analysis considers both transient and steady state. The Fourier representation is also useful in finding the frequency response of linear time-invariant systems that is related to the transfer function obtained with the Laplace transform. The frequency response of a system indicates how an LTI system responds to sinusoids of different frequencies. Such a response characterizes the system and permits easy computation of its steady-state response, and will be equally important in the synthesis of systems. It is important to understand the driving force behind the representation of signals in terms of basic signals when applied to LTI systems. The Laplace transform is seen as the representation of signals in terms of general Eigen-functions. The frequency representation of signals and systems is extremely important in signal processing and in communications.