My use for the DFT
In ECE 301 students were introduced to the concept of Fourier Transforms. No matter whom you had been taught by it seemed intimidating and confusing. And irrelevant. They tried to show you where and how it was used but honestly how long did it take for you to finally realize its use? For me two semesters! There were CTFTs, DTFTs and DFTs (and I am sure there are more that I have not been introduced to yet) and they were all a blur of equations that sometimes gave me nightmares. I am not a very mathematical person: I like engineering concepts and ideas but dislike the math behind it because the equations often stop making sense after the fifth Greek letter has been added. So the goal should be to make such math (these transforms rather) and relate them to real life, not just by telling us where it used but by telling us how it is. So I am going to attempt to do that with my limited knowledge of the math that goes one behind the screen.
What is a Fourier Transform?
A Fourier Transform (FT for short) is just an equation that allows us to from the time domain to the frequency domain and the inverse FT performs the reverse. The time domain is just the real time representation of our signal; how the signal varies over time whereas the frequency domain shows us the number of times the main component of a signal repeats per second.
In the time domain you look at the value of something as it changes over time - a series of snapshots, if you will. In the Fourier domain you look at the entire lifetime of the signal all at once - and analyze it in terms of the underlying frequencies that made it up. This means you can no longer see the value at any one time, or the rate at which the signal is changing at any one time. Instead, for each possible frequency, you see the amplitude of the signal at that frequency (such a distribution is called a frequency spectrum).
It is thus a technique that can be used to describe almost anything in the world be it an electric signal or the stock market. Did you know that our brain picks up different frequencies around us and performs a Fourier analysis (really quickly and effectively) on data: for example on different voices, or recognizes differences in high and low notes or just perceives different colors? Scientists haven’t yet found out how all that is done but they know for sure that something of that sort goes on in our highly complex brains.
