ABSTRACT
IN THIS CHAPTER WE TURN TO THE STUDY OF FOURIER TRANSFORMS, which provide integral representations of functions defined on the entire real line. Such functions can represent analog signals. Recall that analog signals are continuous signals which are sums over a continuous set of frequencies. Our starting point will be to rewrite Fourier trigonometric series as Fourier exponential series. The sums over discrete frequencies will lead to a sum (integral) over continuous frequencies. The resulting integrals will be complex integrals, which can be evaluated using contour methods. We will investigate the properties of these Fourier transforms and get prepared to ask how the analog signal representations are related to the Fourier series expansions over discrete frequencies which we had seen in Chapter 2.
