An Interactive Guide To The Fourier Transform

I wonder why most books about periodic phenomena most of the time instead of using circles, use a trigonometric description or even more , a complex exponential description?
How we multiply a circle (representing AC electrical current) by another circle (representing a AC alternating “voltage”) for finding for example the instantaneus AC power. So in other words what is the geometric picture of this two circles multiplied together?
I wonder what the circle based description would be for a two dimensional Fourier transform?

If we have sinusoidal AC voltages and currents, how do we multiply the corresponding two circles for finding the instantaneus AC power? What is the geometric picture?,
I wonder why most books use a trigonometric description or even more a complex exponential for playing with the associated math for both theoretical and practical use?
What would be the “circle” for a two or more dimensionsonal Fourier Transform?

The recipe analogy is really horrible and confusing.

Great explanation! Thanks.

Is there somewhere that explains convolution this well?

I think you should make a similar post for explaining fourier transforms as a change of basis from time to frequency.

@gord: Thanks for the note – the pleasure was all mine, your original sine animation was incredible! So concise and effective.

@sean: Hah, I feel the same!

@Mike, @matt, @Bo: Thanks, I appreciate it :slight_smile:

@josh: Great point. I’d like to do a follow-up going into some of the more advanced math & interpretations. There’s some good overlaps with linear algebra here too.

sthyusr

Better explained is great service to all “scratch heads” who badly wanno feel the concepts behind these great scientific discoveries but unfortunately are somewhere caught tangled… … It takes gr8 effort and flair to be consistent and maintain this network of better explained!! Thank you… I have a question to this article…the first cosine wave we simulate from a circular motion and the sine wave in the referral link provided, vary in explaining the amplitude…the link has an extra perpendicular line dropping from the revolving “radius head” taken to be the amplitude…i clearly understand the amplitude concept there… but the cos signal in this article gives a different explanation for the amplitude…( u had mentioned to turn the circle by 90 degree mentally which i fail to comprehend…) i am probably not in the right perspective…

@Hamppi: I see orthogonal signals as ones whose net “overlap” or contribution to the other is entirely wiped out when you multiply them piece-by-piece.

Imagine two signals driving a car. One controls the speed, the other controls the direction. When the speed is at max, the direction might be null (no direction, the brakes). And when the direction is north, the speed might be null (no speed, the brakes). Sometimes they are both on (North at 10mph, or South at 10mph). Over all time though, the car does not move, because the sum of all contributions cancels.

@Akshar: Thank you!

@Ben: Great feedback, I really like knowing which parts can be clarified. I’ll update the article to make your feedback more clear. Thanks!

@Vic: Glad you enjoyed it

[…] Explained has a page on the Fourier […]

Hi,
Thanks a lot.Best wishes to reach heights.

This could not get better. I have been trying to understand this concept on my own and it has been a long difficult task. But with this post of yours, my life is easy now. Keep up the good work man!

obvious to you, but it gave me some trouble. Now i have a broader understanding.

Sorry for the 3 separate comments

Also: The amplitude of the waveform corresponds to the X value on the circle, as opposed to the Y value like every other example i’ve seen.

I realized it doesn’t matter which you choose, it is just a 90 degree phase shift. Because its a damn circle and X and Y are perpendicular.

This is probably really

The part I missed on the first 2 animations-- the values listed beside “time” are amplitude values, while the timing is implied as divisions of the 1hz cycle.

Further down I was able to use context clues. overall, EXTREMELY HELPFUL

@Stephen: I’d like to do a video to help walk through the animations. Basically, we have two ways to describe a signal: as a series of points (here’s where the signal was at time 0, 1, 2, 3, 4…) or as a series of ingredients (the signal is made from a 0Hz cycle, a 1Hz cycle, a 2 Hz cycle, etc.).

The cycles have various strengths (how much of each ingredient to use). When you change either side, the widget converts the new values to the other. So, if you add a different set of time points (from 1 1 1 1 to 2 2 2 2, for example) then the corresponding cycle ingredients are adjusted. If you change the cycle ingredients, the time points they lead to are similarly adjusted.

It’s a little tricky, but the upcoming video will help clarify.

Thanks for the explanation.
may be this be silly but can you just brief out the terms in the formula like n,N,k sorry to ask this.

@Yves: It would be great if you could post a link which explains better (no sarcastic tone in this line). It would genuinely benefit the many other people who also feel this explanation is not so good. Kalid may include few points from that link to make this article better.

Fantastic article. I have a basic maths understanding but am not a mathematician, and have found most descriptions of Fourier transform to be utterly impenetrable. However this article presented exactly what I needed, for my purposes, and the interactive animations helped greatly too. A huge thanks!

I’ve just been finding some other great articles on this site too. Great work!

[…] Kalid Azad: The Fourier Transform changes our perspective from consumer to producer, turning “What did I see” into “How was it made?”. […]