Hey guys, I know this might be late, but I'm a new user and this'll be my second post.
Hope it helps.
About the Physical meaning of Fourier Transform and how did J.B. Fourier got the idea:
A very comprehensive approach dealing with this matter is given in the third chapter of "Signals and Systems" by Alan V. Oppenheim and Alan Wilsky with S Hamid Nawab . Prentice-Hall . 2001.
Anyway, here is a summary of what I understood of it from that book and other sources.
First of all, lets take a look at the waves of the sea, they all look a bit like sinusoids moving through the sea or the ocean. When we usually cause a disturbance causing that wave as by throwing a pebble into the water or the periodic attraction force caused by the moon orbiting earth, we find sinusoidal looking waveforms.
So what does this have to do with anything?????
Well, Fourier was a wise guy (8) <--Fourier) !!!! If all the waves were sinusoids, they why do seamen talk about HUGE waves that have a large amplitude and look more like a wall than a slopy sinusoid???? How do they even build up to this huge no sinusoids?? [If you have seen a movie about the Tsunami or The Perfect storm, you can imagine what I'm talking about easly]..
The above paragraph isn't how Forier thought of it, its just an example to make you think about how can WE think of Fourier's ideas. In the real life, Fourier made his theory from the study of the conductive heat transfer through metals. But here is a method to think of it using Fourier's own study but with a clear example. When you solve the exact PDE of the heat transfer by conduction through a body, for example, a metallic bar, this well lead you to the wave equation solution describing the temperature distribution across the bar. Now, if you expose this bar to static thermal conditions, example 100°C at one side and 0°C at the other side, do you get anything that looks like a the sinusoidal wave given by the solution??? Well, then it must be a SUPERPOSITION of many waves, that gave this result. [Remember the Complementary function thing in the solution of PDE's]
That was about the physics, now how did he formulate that in mathematics? Lets see, but this discussion will be made for continuous time signals because they are more easier to imagine:
The last three lines means that generally:
f(x)=∑a1n.sin(bn.x+c1n) [n is the index of the summation]
so to get rid of the phase shift we can say more generally
f(x)=∑(a2n.sin(bn.x)+b2n.cos(bn.x)) [sincesin(a+b)=sin(a)cos(b)+cos(a)sin(b)]
Now, if f(x) was periodic with a period from -\pi to +\pi we could use the orthognality of the series. [Thats a very big subject which I'll assume you know, if not, tell me. It is very important and explains many of the equations] To overcome this condition, a guy called Dirchlet made some modifications to make it applicable for any periodic signal with any general period, but thats not out subject. SO the above form is complete and is known as the Fourier Trigonometric Series. This was the one that was used by Fourier.
A more general form of f(x)=∑an.exp(jwn.x) is prefered, because it reduces the calculations.
Now to get more insight into the transform:
We usually need the transform when we need to deal with non-periodic signals. Now, how to get on with that??
Just use the above form of Fourier Serier as the period tends to infinity (use a limit and make use of Reimann's Sum method to convert it to an integral) and you'll end up with the Forier Transform form.
How to apply the Fourier transform for periodic signals:
The Fourier Transform as well as the Laplace transform are integral transforms, ie they are linear operations. So we can easly just convert any periodic signals to their Fourier series and find their transform so that we can easly find the transform of any periodic signal and they we can concentrate on finding the transform of sin(tx) or cos(tx) or exp(jtx).
To solve that, we make use of complex analysis, their transform can be easly found using the Euler formula and making use of what is known as Cauchy's theory. These integrals are so much encountered that they are called the Fourier Integrals. 8O
:x
Hope I helped. If you don't understand anything of the above or you're confused in anything in it. Just tell me. All your comments are welcome. Hope my tone of speech wasn't bad. I just wanted to be informal so that this post doesn't look like that FORMAL thing which we usually find in books. I didn't write the equations because I'm sure you know them well.