Fourier Analysis

Abstract

In this chapter, the first major challenge is to understand the difference between two descriptions of a signal: one in the time domain and another in the frequency domain. We initially use a gradual increase in complexity to help the reader grasp the difference. We then use phasors in order to introduce positive and negative frequencies, a detail that is encountered later. The formal mathematical Fourier transform and inverse transform are then introduced as well as Fourier series. The remainder of the chapter is devoted to discrete Fourier transform in the form of fast Fourier transform (FFT). All exact details on intervals in time and frequency are stated with great care. Important details like aliasing/folding and sampling theorem are given. We also analyse a time-limited oscillating signal and get our first encounter with the bandwidth theorem, and a theme we will recur to in several later chapters of this book.