OVERVIEW

We discussed in Chapter 1, "Fourier Analysis," the basics of Fourier analysis and established a mathematical way of describing an arbitrary function by using its frequency components. The message was that a periodic function could be described as a composition of simple cyclic frequencies in terms of sine/cosine functions, the fundamental, and its integral multiples called harmonics. We have found complex numbers as the best way to represent points on a Cartesian coordinate system and developed the rules to perform mathematical operations using the Gaussian operator ? ?1. We found that a sinusoidal function can be described as a complex exponent function by using Euler's identity.

In this chapter, we will discuss the transformation of a function from time domain to frequency domain: the culmination of a Fourier series into the Fourier Transform. The frequency domain gives us another dimension of analyzing events that are extremely difficult in time domain or sometimes not even possible. We presented an example in Chapter 1 in which a mysterious output was observed when the input was a periodic function, but with the help of Fourier analysis, the problem was quickly identified. We did not include time in our analysis, and you will see that time is irrelevant when we know that the basic periodic function does not change its shape or form as time goes by.

The aim is to develop software algorithms to achieve the result of the theoretical formulation. But for that, we need to modify the Fourier formula...