CHAPTER 3: FOURIER TRANSFORM AND APPLICATIONS

Introduction

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies. The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in the original function, and whose complex argument is the phase offset of the basic sinusoidal in that frequency. The Fourier transform is called the frequency domain representation of the original signal. 

The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, the domain of the original function is commonly referred to as the time domain. For many functions of practical interest, one can define an operation that reverses this: the inverse Fourier transformation, of a frequency domain representation combines the contributions of all the different frequencies to recover the original function of time.

Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency, so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, can be expressed relatively simply as an operation on frequencies.

After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.

Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. [source: wikipedia-Fourier Transform]

For Chapter 3, students need to understand how to perform the Fourier Transform whether with manual calculation or using Table. Then, students should understand why Fourier transform is needed and how to apply Fourier Transform for several application such as filter.

In Summary, this topics will be covered in Chapter 3:

• Introduction to Fourier Transform

• Fourier Transform properties

• Fourier Transform application

• Fourier Transform analysis

• Inverse Fourier Transform