HW

• Signals and Systems

An Introduction to the Fourier Transform

Unfortunately, the Fourier Transform isn't a Transformer. If it was, you would have seen it in the movie that came out lately.
One way to explain a Fourier Transform is to say it's a bunch of sinusoids added to create a just about any function you want. Another way to describe it is to say it's a way of representing a function in the frequency domain instead of the time domain.
For example, a square wave could be represented by: $x_{\mathrm {square} }(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\left((2k-1)2\pi ft\right)} \over (2k-1)}$

That's alot of numbers seemingly out of the blue, at first observance. I bet your questions are:

1. Telling me the difference between a transformer and a Fourier Transform hasn't helped me finish my assignment. What else can you tell me?

Wait till Friday. That's when the deadline for this assignment is, so hopefully I'll panic before then and get something done on this page in addition to the spectacular transformers picture.

2.Can you show me some really easy plug-in formulas, so I can get my homework done faster?

Yes.

3.Why would I bother getting one? I don't know what to do with it.

Fourier Transforms are awesome! They allow continuous functions to be represented by ones and zeros, which means we can implement functions on the computer. Fourier Transforms form the basis of signal processing. They allow us another way to transform a math problem so it's alot easier to solve.

Some Useful Fourier Transform Pairs

${\mathcal {F}}[\alpha (t)]={\frac {1}{\mid \alpha \mid }}f({\frac {\omega }{\alpha }})$

${\mathcal {F}}[c_{1}\alpha (t)+c_{2}\beta (t)]$	$=\int _{-\infty }^{\infty }(c_{1}\alpha (t)+c_{2}\beta (t))e^{-j2\pi ft}\,dt$
	$=\int _{-\infty }^{\infty }c_{1}\alpha (t)e^{-j2\pi ft}\,dt+\int _{-\infty }^{\infty }c_{2}\beta (t)e^{-j2\pi ft}\,dt$
	$=c_{1}\int _{-\infty }^{\infty }\alpha (t)e^{-j2\pi ft}\,dt+c_{2}\int _{-\infty }^{\infty }\beta (t)e^{-j2\pi ft}\,dt=c_{1}\mathrm {A} (f)+c_{2}\mathrm {B} (f)$

${\mathcal {F}}[\alpha (t-\gamma )]=e^{-j2\pi f\gamma }\mathrm {A} (f)$
${\mathcal {F}}[\alpha (t)*\beta (t)]=\mathrm {A} (f)\mathrm {B} (f)$
${\mathcal {F}}[\alpha (t)\beta (t)]=\mathrm {A} (f)*\mathrm {B} (f)$
Some other usefull pairs can be found here: Fourier Transforms

A Second Approach to Fourier Transforms

• Fourier Transforms