Fourier Transforms

Introduction

This is a new page (October of 2021) to collect helpful information about Fourier Transforms

Syntax

• This page uses the following definition of the sinc function: \(\begin{align*}\mbox{sinc}(x)&=\frac{\sin(\pi x)}{\pi x}\end{align*}\) Note that other references (such as the zyBook) may omit the $$\pi$$! Given the version above, sinc(0)=1 and sinc(n$$\pi$$)=0 for all integers $$n\neq 0$$. Also note that:\(\begin{align*}\sin(\theta)&=\theta\,\mbox{sinc}\left(\frac{\theta}{\pi}\right)\end{align*}\)

Useful Fourier Transforms

On the Sakai page in Resources, there is a folder called "Ref: Tables" with two files in it. The AllTablesHVV.pdf version is the most relevant to this semester (HVV stands for Haykin and van Veen, who wrote a textbook we previously use and whose notation is most similar to the zyBook). Pages 8 and 9 are related to Continuous Fourier Transforms.

Centered Rectangular Pulse

The pulse of height 1 in the table has width of $$W=2T_1$$; re-writing the Fourier Transform in terms of the width $$W$$ gives:

\(\begin{align*}X(j\omega)&=\frac{2\,\sin(\omega T_1)}{\omega}=\frac{2\,\color{blue}{\sin\left(\frac{\omega W}{2}\right)}}{\omega}\end{align*}= \frac{2}{\omega}\color{blue}{\frac{\omega\,W}{2}\mbox{sinc}\left(\frac{\omega\,W}{2\pi}\right)}= W\mbox{sinc}\left(\frac{\omega W}{2\pi}\right)\)

using the definition of sinc from above.