It is very difficult to get an intuitive understanding of Fourier analysis through college classes because those classes undoubtedly get bogged down in the math and you never get the big picture. What I am going to do here is explain the concepts of Fourier analysis so that you can get a "big picture" intuitive understanding of what it is and why it's important.
Basically Fourier analysis relates functions of TIME and FREQUENCY.
The math behind Fourier analysis involves Fourier series. The general concept is that ANY function can be created with a superposition of different types of sine waves. You can take a Pink Floyd recording and recreate it by super imposing many, many different sine waves (with varying frequencies and amplitudes). This is all you need to know for the math.
The Fourier transform is a tool that uses the concept of Fourier series.
WHAT THE FOURIER TRANSFORM DOES is convert a function of time to a function of frequency. There is also an inverse Fourier transform that converts a function of frequency to a function of time. Actually calculating these transforms by hand is mathematically difficult and usually on required if you are an electrical or computer engineering major. Since you are not, here's a big picture example:
Imagine a function of time, x(t) = sin(t). A simple sine wave, right? Visualize it. It is just a tone, an oscillation occurring throughout time that goes up and down in amplitude with a constant period. When you play a note on an electric keyboard, lets say A, it plays at 440 Hz. The speaker of that keyboard moves up and down 440 times each second and your ear registers those air pressure variations as a sharp tone.
Now, our objective is to convert this sine wave (oscillating at 440 times per second and continuing indefinitely throughout time) to a function of FREQUENCY instead of time. Now, what are all the spectral (frequency) components in this note? There is only 1 and it is at 440 Hz. So, to convert this function into frequency, the x-axis changes from time to frequency and instead of a sine wave, we have a single impulse at 440 on the x-axis. Additionally, there will be another impulse at -440 hz, because the Fourier transform cannot distinguish between positive and negative frequency.
You have just performed an x(t) -> X(f) conversion, a Fourier transform!
Here is a list of common transforms:
http://www.astro.umd.edu/~lgm/ASTR410/ft_ref3.pdf
Scroll down to the 7th row and you can see the transform we just did. We converted a sine wave to two delta functions (impulses) located at +/- 440 Hz.
Lets look at another one of these common transforms. Look at the 2nd row, the first transform listed. We have the function x(t) = 1. This is just a horizontal line crossing the y-axis at 1. It is 1 for every value of x. What are the spectral components of this function? Well, there aren't any. The function is not changing, it is constant. The are not oscillations or variations anywhere. So the Fourier transform is simply an impulse at 0 Hz. It has NO frequency.
One more example. Look at the function below the sine example, x(t) = rect(t). This is a rectangle function (
http://en.wikipedia.org/wiki/Rectangular_function). It means that there is a rectangle on the x-axis going from -.5 to +.5 with an amplitude of 1.0. Now, in the previous two examples, we have looked at functions that continue forever (sine wave and constant 1). This is only non-zero for a finite period of time. What is the spectral content of this function? You may think, it's flat - it's just like the previous example, it doesn't have any spectral content! But you are wrong. It is possible to recreate this rectangle pulse by using many, many sine waves of different frequencies and amplitudes (Fourier series). Each of these sine wave's frequnecies contribute to the spectral content of the time signal. So the Fourier transform of this signal, the frequency domain function, is actually a sinc function, which looks like this:
http://en.wikipedia.org/wiki/File:Sinc_function_(both).svg
As you can see, a finite signal in the time domain (like our rectangle function), has an infinite signal in the frequency domain (that sinc function goes on forever - it has spectral content continuing out to +/- infinity Hz). But an infinite signal in the time domain (like our sine wave from the keyboard that plays forever), has a finite signal in the time domain (a simple impulse at 1 frequency). Take a look at the sinc signal in the time domain. It is infinite and goes on for all time. Consequently, it has a finite signal in the frequency domain. It is made up of exactly all the frequencies from -0.5 to 0.5 Hz, nothing else.
Now consider the case where we play our note on the keyboard at 440 Hz, but we only play it for a few seconds. after 2 seconds, the sine wave disappears and the function is 0. We now have a finite signal in the time domain. What do you think happens in the frequency domain now? Well, We still have that impulse at 440 Hz, but now because we cut the signal off), there is spectral smearing all around that 440 Hz, so there will be spectral content extending out to infinity (although it will be extremely small).
The practical applications of the Fourier Transform are most commonly used by a digital method called the Fast Fourier Transform (FFT). The FFT is what is used in your course. You can take samples in time, do an FFT on the data, and visually see how often stuff is occurring (the frequency). When you play an MP3 on winamp and are looking at the spectrum, you're seeing an FFT. You're seeing the spectral components of the song you're listening too. When you hear a bass note, you see a spike on the low end of the spectrum, and when you hear the singer screech his voice, you see a spike on the high end of the spectrum.
So, there you have an explanation of the Fourier Transform without using ANY math. I hope this has been helpful. Let me know if you need any clarification.
