@RR: Thanks
@Satheesh: No problem, just updated the post (the Discovering the Full Transform section)
Thanks! Really good!
Kalid, could you possibly tell us what software you use to create these animations? How do you accomplish these?
Goodness me, what a splendid article!
I just wanted to learn digital signal processing… After a couple of chapters in my dsp book i noticed that i have to study signals and systems first in order to understand fully dsp. It became clear very soon that i need to learn more math especially fourier analysis to make sense of everything in my signals and systems book. I’m on chapter three now in my signals and systems book (fourier series) and it will probably take my whole life to get to a fourth.
@Angel: Cool background – it seems to only take a few components before the shapes get really intricate.
Thanks, Great article
I like your site. This is the way teaching math should be done, in order for anybody to “get” it.
Simple thanks.
Somehow my attribution got bombed from the last post
1+sin(t) can be evaluated very easily at w=0 if you remember that e raised to the zero power is always exactly equal to 1. On the otherhand, if you write the integral in expanded form (i.e. as integral of x(t)Cos(wt)-jx(t)Sin(wt)) and integrate piecewise.Then you are only ingtegrating Cos(wt), iSin(wt), Sin(t)Cos(wt) and iSin(t)Sin(wt) and combining the results. Since you have no phase shift and only a single frequency, the integrals of iSin(wt) and Sin(t)Cos(wt) evaluate to exactly zero. The integral of Cos(wt) also evaluates to exactly zero everywhere except at w=0. This is easily seen since Cos(0)=1, so at w=0, you are taking the integral of 1. The integral of iSin(t)Sin(wt) is non-zero when w=1 and zero everywhere else.
When you actually do the integral, at least for the value of the integral of Cos(0) dt, there is this irritating problem that the integral actually evaluates to “t”, not “1” and this is probably why Maxima blew up calculating the definite integral: t would have been replaced by infinity in the calculations! This is exactly the problem that Lipot Fejer resolved by proving that the series converged only when cast in terms of the means. So whenever you are attempting to recover amplitudes, you need to take the mean, which essentially means dividing by “t”. When you do this, you get the expected result. You may still need to massage the equations in Maxima to prevent the possibility of dividing Infinity by infinity. Intuitively you (a human) may think that this should be equal to 1, however, in reality it is mathematically undefined This is because infinity+x is still infinity so the ratio of 2 infinities must be indeterminate since, by definition, you can never say that 2 infinities have the same value and there is no way to find out if they do.
More complex periodic functions can be analyzed in a similar fashion by first applying trigonometric transforms to the functions and then integrating the components. Not always easy, but it works.
Well explanation is very good but i m stuck at some points. I don’t understand how the Yellow points are placed and what is meant by 0 Hz at strength 0/1? plz reply asap. Thank u!
I just figured out how the transform works on my own. I think its “better explained” by showing how multiplying f(t) by e^iwt is really rotating in the complex plane with frequency w/2pi, and the larger the value you get integrating this over all time, the better f(t) “matched the rythm of w”. Further more, depending on where in e^iwt’s phase f(t) keeps getting bigger at periodically, that will be contributing the most to the fourier transform’s complex value. So the final value, the sum over all time of f(t)'s complex position when rotated by multiplication with e^iwt, tells you about the phase and magnitude of the match-up between w, the rotation speed, and f(t), the function being rotated. Its wierd, its very much a sort of circular integral, where depending on w, you get really far from where the value of the integral starts, or depending on how f(t) matches up with the phase of e^iwt, gives the integral’s angle in the complex plane. Its a little mathematical machine, and it is an extremely intuitive one. For instance, why divide by 2pi? Because that is how much further a rotation in the complex plane moves a value of f(t), it moves it more by a factor of 2pi.
Spectacular article - I’ve always wanted to see it explained in an intuitive way and you’ve finally done it. Congrats on all the hard work.
This is really, really awesome. Thank you for sharing!
Absolutely fantastic article, thanks for this!
awesome! if the world does come to an end today, i can at least die having understood the basics of DFT
Hey, great use of animations to explain and edify…
and I much appreciate your crediting and linking to my HTML5 sine animation… Kudos!
Nice article!
gord.
@A. Scarlat MD: Really glad it clicked, thanks for the note! Yes, often times people jump into extremely technical discussions of math without laying an intuitive foundation. I’d like to do more signal processing posts later on :).
A nice description! You might enjoy the "full"
mathematical story at
www.civilized.com/files/newfourier.pdf