Revision as of 05:12, 6 May 2010

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.

CTFTs, DTFTs and DFTs
Eqn for CTFT
A CTFT( continuous time fourier transform) is continuous in both the time and frequency domain. Give example here.
Eqn for DTFT
A DTFT (discrete time fourier transform) is performed on signals which are discrete in the time domain but are continuous in the frequency domain.example needed
Egn for DFT
A DFT (discrete fourier transform) is performed on discrete time signals too but they are also discrete in the frequency domain.
A DFT is almost the same as a DTFT because if the DTFT of a signal is truncated to make it finite length then the resulting waveform will be the same as the result of a DFT being performed on the signal ( this is of course easier said than done because we then have to use a filter of the appropriate length and so on, which means another two pages of math).
Either way you still have to know the formulas for the sums of a geometric series, infinite or not.
Formula for infinite and finite g.ps.

Project Rhea
So Professor Mimi and I decided to apply my new found knowledge on this website. As this is a student oriented website and free (i.e. it is still being established), it is still not as well known or well used as it should be. It works but it is a work in progress and the Rhea development team wants it to be go-to website for the School of ECE here at Purdue. Therefore we needed to see trends in visits and reasons behind those trends. Only after we get an idea of why and who visited Rhea could we implement further changes that would benefit the website.

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