HW

• Signals and Systems

Fourier Transform Introduction and focus on Applications

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 particular square wave could be represented by the following discrete fourier transform: $x_{\mathrm {square} }(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\left((2k-1)2\pi ft\right)} \over (2k-1)}$

Fourier Transform Applicaitons

The "Fast" Fourier Transform

Cooley-Turkey Algorithm

What is a Fast Fourier Transform? (FFT)

It's an algorithm that can compute the discrete Fourier transform faster than other algorithms. In digital systems, continuous Fourier Transforms are sampled, turning them into discrete Fourier Transforms which then can be computed and manipulated using Digital Signal Processing.

An intuitive brute force way of computing a Fourier Transform means rearranging the the summation so that you don't compute the transform in sequential order - you group similar elements together and simplify before combining them. This cuts down the adding and multiplying, thus cutting computation time down by about 100 times.

One of the most popular FFT algorithms is called the Cooley-Turkey algorithm. Which I will explain on Friday