# An Interactive Guide To The Fourier Transform

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Our cycle ingredients must start aligned at the max value, 4 and then "explode outwards", each cycle with partners that cancel it in the future. Every remaining point is zero, which is a tricky balance with multiple cycles running around we can't just "turn them off". At time 0, the first instant, every cycle ingredient is at its max.

Ignoring the other time points, 4? Imagine a constellation of points moving around the circle. Here's the position of each cycle at every instant:. When our cycle is 4 units long, cycle speeds a half-cycle apart 2 units will either be lined up difference of 0, 4, 8… or on opposite sides difference of 2, 6, 10….

When every cycle has equal power and 0 phase, we start aligned and cancel afterwards. I don't have a nice proof yet -- any takers? Try [1 1] , [1 1 1] , [1 1 1 1] and notice the signals we generate: In my head, I label these signals as "time spikes": Here's where phase comes in.

Imagine a race with 4 runners. Normal races have everyone lined up at the starting line, the 4 0 0 0 time pattern. What if we want everyone to finish at the same time? Just move people forward or backwards by the appropriate distance.

Maybe granny can start 2 feet in front of the finish line, Usain Bolt can start m back, and they can cross the tape holding hands. Phase shifts, the starting angle, are delays in the cycle universe. Here's how we adjust the starting position to delay every cycle 1 second:.

If time points 4 0 0 0 are made from cycles [1 1 1 1] , then time points 0 4 0 0 are made from [1 1: I'm using "1Hz", but I mean "1 cycle over the entire time period". If we merge the recipes for each time spike, we should get the recipe for the full signal. This was my most challenging article yet.

I was constantly bumping into the edge of my knowledge. But there's always simple analogies out there -- I refuse to think otherwise. The analogy is flawed, and that's ok: I realized how feeble my own understanding was when I couldn't work out the transform of 1 0 0 0 in my head. Shouldn't we have an intuition for the simplest of operations? That discomfort led me around the web to build my intuition.

In addition to the references in the article, I'd like to thank:. Today's goal was to experience the Fourier Transform. We'll save the advanced analysis for next time. Stuart Riffle has a great interpretation of the Fourier Transform:. Imagine spinning your signal in a centrifuge and checking for a bias. I have a correction: You already know why: Lucas Vieira , author of excellent Wikipedia animations , was inspired to make this interactive animation:.

Fourier Toy - Click to download, requires flash. Detailed list of control options. The Fourier Transform is about cycles added to cycles added to cycles. Try making a "time spike" by setting a amplitude of 1 for every component press Enter after inputting each number.

Congrats on all the hard work. I think you should make a similar post for explaining fourier transforms as a change of basis from time to frequency. Thanks for the note — the pleasure was all mine, your original sine animation was incredible! So concise and effective. Glad you enjoyed it!

The transform lets you switch between the two. But there were an extra step of loop through every frequency recipe to get full transformation, I thought that all are handled already. I recently wrote a tutorial on the DFT as well, though I came at it from a different point of view.

My understanding of it is based on correlation between the time domain signal and a series of sinusoids of increasing frequency: Hey, man, this is really great. Rarely seen such a complicated and confusing presentation of the Fourier transform. I learned it as a way to approximate the solution to conductive heat transfer integral math. The length of the series that substitutes for the equation has diminishing changes after only a few members of the series are calculated.

It was at that time taught for computerized solutions as finite difference method rather than finite element method. Bessel functions ultimately similar- just more obtuse 3rd order nonlinear partial diff eq…. Glad you liked it! I believe heat transfer was the original use case for the transform. If there are any parts that are confusing after the 2nd reading, I probably need to reword them: This explanation is awesome and it would be great to have it expanded upon!

Best things in life are not free-they just make you happy to think they are. Engineering is the art science? Kalid, your article appeared in my mailbox and I had no intention of reading it at that moment, but read the first bit and I was totally, totally hooked!

I was flat out excellent and we all certainly appreciate the seriously hard work and thought that you obviously put into this. It was riveting, and helped me understand in a much different and wholly more satisfying way, than my college math days, The Fourier Transform. I think the best way to intuit why a spike can be built in the way you describe is by going back to the circle. I am going to speak extremely loosely in the spirit of your blog.

As you said, at time 1, the dots will be all evently spaced out throughout the circle, spaced by 1 unit representing each frequency increment. Then at time 2 they will be spaced out by 2, at time 3, 3, etc. In fact, this will always happen unless the dots are placed with a one unit spacing N slots, N dots, so either you go in steps of one or you have to loop around.

What I want to convince you of now is that whether or not there is looping, the resulting occupied slots form a regular polygon on the unit circle, and further, every occupied slot is occupied equally with dots.

This is an N-sided regular polygon, each slot gets hit the same number of times. We are left with a regular polygon, and it also must have each vertex occupied evenly, since repetition happens after a number of times divisible into N.

So the result is always expressible as the sum of the vertices of a regular polygon times some integer to account for some integer number of layers.

But it is visually obvious that the sum of vertices of a regular polygon sum to the center of the polygon zero. There is one exception to this rule: And I hope this is what I have done. It must have been hard work. And fun in putting in a lot of imagination to get this done. Glad to see it explained clearly here. This is all very interesting and I do rather like the use of the unit circle definition of the sine wave as an illustrative tool.

A few years back, when working on a software algorithm for synchronous signal detection, I once again became intrigued by the FFT and why it actually works.

At that time, I set upon the task of figuring it out. A couple examples of using the Fourier integrals and series on a known signal e. I tried and failed to get what I expected e. I look forward to you tackling the Laplace transform one of these days: On the otherhand, if you write the integral in expanded form i.

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. This is exactly the problem that Lipot Fejer resolved by proving that the series converged only when cast in terms of the means. 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. 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. Stephen, Thanks for replying. The formula in the first bullet of section 2. At least I finally got the answer I was looking for, but needless to say I would have failed the exam which is why I like this blog in the first place: In the case of the zero frequency component, we expect zero anywhere away from zero, but an infinitely thin spike around zero.

Glenn, I should have mentioned the 1 cycle issue. Since Sin and Cos are periodic, the integral over one cycle is exactly the same as the integral over an infinite number of cycles. In the limit of infinity, this clearly becomes exactly equal to 1 — not to mention that cos t can never exceed one anyway, so this integral can never blow up — at least not with real valued t.

You might also consider forgetting about the negative frequency part of the spectrum. For all real valued data that is, all real data!

Daniel, Are you sure about this? If this were not to be the case, the Fourier transform would not really be very useful for AC signal analysis. I suspect that you are thinking about some other interesting property of the transform and I would like to have you clarify what you are thinking about — perhaps the transform of the delta function?

I say the Fourier Transform of has three spikes, one at and one at both and respectively. According to the definition given at the beginning of the post of the Inverse Fourier Transform, we have. Then when we perform the integration, the delta functions yield the integrand evaluated at the value of s that makes the argument of that delta function zero. It is not at all that I disagreed with your assertions, especially since they are formally correct in every way.

They are also really cool and insightful and your clarification is very adequate,. Also, invoking this interpretation involves understanding a lot of really difficult math involving distributions.

My only intent is to help clarify the workings of the Fourier concept using simple, approachable math. I think I can answer that question also. When the frequency domain is discrete, it makes sense to talk about non-infinite amounts of each component, as you describe. After all, in the real world there is not really such thing as a pure sine tone; everything has some width in frequency space. This is the same principle why discussing the mass of an object in detail requires a notion of a local mass density, because if every POINT in the object had its own noninfinitesimal mass, the whole thing would have infinite mass.

The Dirac delta is a formal way to revive the concept of a pure sine frequency even after we have evolved to the notion of densities. The problem with this statement is that the width is finite and therefore there are only a finite number of finite amounts and the sum is therefore finite! If I recall my history,this is precisely the polemic that raged for years regarging whether in the limit, the differential becomes vanishingly small on the one hand or 0 on the other..

The argument is whether an infinite number of infintely small pieces totals up to a finite value. Certainly we know that when we integrate a simple function that the answer almost always gives us the area bounded by the curve, not an infinite value, and in this regard, for all practical considerations it does not seem to matter if we consider that the differential truly reaches 0 in the limit or not.

I suppose that we could argue about this for years, just as mathematicians did in the 19th century, However, it appears that Paul Dirac must have finally resolved this debate with the invention of the Delta Function.

In the interest of all the other readers of this posting, I think we should put the whole matter to rest. I am just explaining why setting the amount at each piece to a constant non infinitesimal would give trouble. 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.

Math is as easy as pie! I had asked Kalid about the Fourier Transform a while back and he had emailed a great brief explanation. This is a fantastic website! Hi Genius, Thanks for the article ,It was awesome like always but i have a doubt. Hah, just a curious learner here. Negative values in time, for the signal you mean? I just figured out how the transform works on my own.

Its a little mathematical machine, and it is an extremely intuitive one. Thanks for the explanation. The old Yamaha DX synthesizers series used frequency modulation to create, from 4 to 6 sine waveforms, quite complex sounds. Cool background — it seems to only take a few components before the shapes get really intricate.

Is there anyway to visualize orthogonal signals? Or understand it intuitively instead of just saying that their dot product is zero. 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.

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! Because its a damn circle and X and Y are perpendicular.

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. Great feedback, I really like knowing which parts can be clarified. I was struggling with Fourier for quite some time. Wikipedia and other web based explanations are way too complicated for my rudimentary knowledge — and thus, useless.

Thanks for explaining a difficult concept so elegantly. Would you please be kind enough and consider doing the same magic and explain the concept of Claude Shannon Entropy? Really glad it clicked, thanks for the note!

Yes, often times people jump into extremely technical discussions of math without laying an intuitive foundation. What is a circle? Just a circle itself? Why not considering a circle the son of a cone? Or maybe a circle is just a straight line for bug living in an infinite radius circle. Thank you very much for this work sharing your insights, there has been very practice for my math career, keep going! 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?

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? Not sure why most books jump to the most technical definition first: Intuitively, I imagine a circular path, and on that circular path, another circle is traveling [a bit like how the Earth moves around the sun, and the moon moves around the Earth].

The combined effect of the two positions is the net power seen. What is the practical use of this circle view approach in solving practical problems? This is the best ever intuitive presentation of Fourier! And the animations…gr8 work…Thanks for the effort…Similar insights on Wavelets might be of gr8 help too…pls consider it…. Thanks again…keep it going!!! This is a work of a an extremely talented and gifted person!!!

Because characteristic function of a probability density is Fourier Transform, so it needs to be time and cycle driven, but I just not sure what does cyclicality have to do with probability.. Thanks for the note! Wavelets would be a good follow-up. I need to learn more about them. Really appreciate the kind words. All I have to say is that you have put together a wonderful article. Your style of writing immensely helps in removing the apprehension in the mind of the reader of having to deal with a complex topic.

I plan to write articles in this domain the help students and professionals maneuver complex topics in these subjects by presenting them in a easy to grasp manner. My article on Fourier Transforms http: Would be very thankful if you can provide your feedback. This explanation of the Fourier Transform is an excellent example of it. Please write some textbooks. Wonderful and revelatory stuff — you make learning it a delightful experience with all the visual metaphors and animations gradually building to the abstract formulas.

The formulas themselves just confused me. Now i really feel I have a handle on it. Thank you for the effort in making this. Hope everybody who gets swamped in this domain comes here. I like your site. Yep, part of writing is getting in the head of the reader and gently going down the path, vs. I love hearing about other people doing their explanations. Everyone has a different style, so looking forward to checking out yours. Hoping to do some more material on Calculus, Trig, etc.

It takes gr8 effort and flair to be consistent and maintain this network of better explained!! 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 analogy is rather out-of-place, confusing and seems like an unnecessary detour.

Most important of all, the author seems to lack primary insight on the subject himself. Smoothie is a whole and its ingredients are parts. It is not the same with signals at all. They are all whole in their own domain. The analogy is totally misplaced and the information provided only partly correct. The key idea of FT — change of variable — is not emphasized at all. Only well informed people should be allowed to author such articles. That said, there are a couple of good insights for new learners — 1.

What a shame that some feel the need to squelch the brilliance of others, presumably to bolster their own inadequacies. To this end, I am confounded by your own statement that the smoothie is not an apt analogy. But that does not preclude the ingredients themselves from being whole on their own. Before berries are thrown in the blender, they are just that: Regardless, this distinction is primarily one of taste— the important observation is that a signal can be represented as a sum of Fourier modes in the same way that smoothies can be represented as a union of ingredients.

The features of this analogy carry through quite naturally, and the aspects that do not are clearly addressed by the author. I feel we might hear more of him in the future. I hope you will hear more of me in the future. I will keep this article as an example of how intellectual work should be done.

Hi Simon, thanks for the note — hope your nephew enjoys it: Really appreciate the kind words, I hope the strategy of finding specific examples to illuminate abstract concepts gets more traction.

The time values [1 -1] shows the amplitude at these equally-spaced intervals. Like for a 1 Hz signal why are you measuring at 2 points, for a 2 Hz signal at 3 points, for a 3 Hz signal at 4 points and so on? Does this have something to do with the Nyquist-Shannon sampling theorem? Hi Niko, great question. Each animation is over the course of 1 second.

If you are analyzing a 1Hz signal inside that interval, you just need a measurement at the beginning and halfway at 0. If you only had the measurement at the beginning 0. If you are trying to measure a 2Hz cycle which goes up and down twice during the period , then you need at least 2 measurements beyond the starting one so at 0.

Understanding Fourier Transform cika. Kalid, could you possibly tell us what software you use to create these animations? How do you accomplish these? You can open http: The details of how to do web programming will probably need a few more articles though!

I was looking through your material on fourier transform and its by far the best explanation I have found anywhere. I spent a few days reading this and I understand everything except for one hiccup.

Can you please lead me on the right track? There are several versions of the Fourier Transform, they key is realizing you need to average the strengths somewhere along the way when you apply the forward transform and then the reverse. If f x is the forward transform and F x is the reverse, then you can have:. Either way, after doing f F x or F f x , i. If you have a single instant a spike in time like 1 0 0 0 0 0 … , then its magnitude should be shared among every possible frequency which can claim it?

Hey Kalid great work on this one. I started learning about EEG signal analysis and the Fourier tranformation comes up constantly. I never heard of it before and for such a beginner this is a great, very helpful, article. However to better understand everything about what you said I have a couple questions which I hope you can answer:.

In reality, is the signal not comprised of waves of varying amplitude? Think of it like this: I repeat, this article is basically flawed because the analogy does not capture the true essence of the Fourier transform.

No doubt, the animations are nice and the article is well-written, but I have problem with the content, and anyone who understands Fourier transform well would have the same problem. It is unfortunate that some people are unable to accept healthy criticism without descending to provocative language. Well explanation is very good but i m stuck at some points. A strength of 0 means that cycle ingredient is not present in the signal. I am a retired mathematics teacher and I have to say that one of the most inspired pieces of teaching that I have seen.

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.

Probably in a real application the overall time interval would not be 1 second, and therefore the frequencies would change accordingly. If the time interval were 2 seconds then you would actually have 0Hz, 0. I think there is a mistake where you introduce the formulas in the end.

The formula for timepoint and frequency are swapped. Anyhow, thank you so much for this! I was just trying to produce the same results as in the article. Hi Thomas, no problem! I should probably clarify that point. The section where you introduce the animations needs to be clarified. The way it is written confuses me. When you change one it automatically changes the other, why? Why is there always a 0?

Not good at all. I was confused and you added spices into it. The animation controller are the worst. Isolating the individual frequencies is tricky. Let me expand on the analogy in the post. Imagine you have a bunch of toy cars, racing around a circular track. We could have an East-West position and North-South position over time. If this treadmill is going 10mph, then cars going exactly that speed will stay still.

The other cars are going either faster or slower, and will continue to circle around the track over time, their average contribution will be nothing. Only cars matching the speed of 10mph will stick around, and can be measured. Maybe we see 3 cars going that speed. The Fourier Transform takes the notion that any signal really has a bunch of spinning circular paths inside.

That is just the treadmill: Btw, appreciate the support. Basically, we have two ways to describe a signal: 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. Or do something equivalent, but probably a bit more efficient maybe. There are little hairs cilia in you ears which vibrate at specific and different frequencies. Because of this, you can distinguish sounds of various pitches!

So, our ear is setup in a way that each hair is tuned to react at different frequencies. Nature usually has ways to do everything in parallel, while our computers manually crunch through.

Quick notes sticky post synthtech. I dint know what Fourier transform is, one hour ago so this may be stupid question.. Fourier transform has all positive values then how can it give back a signal with negative values??

Similar question shateesh had asked about but ur answer dint satisfy me: Can you add at least one graph of sample Fourier transform for people like me: Hi Great article, especially for somebody like me with no previous Fourier experience. I have one question that is still confusing for me and it would be great if you could help: I am expecting this value to be 1 and not 2.

What am I missing? The Fourier Transform is based on circular paths, which start at an angle of 0 [neutral], go positive [90 degrees], back to zero [ degrees], negative [ degrees], and back to neutral []. By aligning and delaying various circular paths, you can reach the negative numbers. In general, you can modify a positive signal by starting each cycle at the opposite side to make it negative. In the calculator linked, try entering [1 0 0 0 ].

The result is [1 1 1 1], which appears to have magnitude 4, even though the input signal had magnitude 1. I am a junior in DFT, actually i just heard of fourier transformation for the first time shame on me, i know , and tried the wikipedia explanation. I cant say i understand everything just yet, ill need to work a lot harder for that.

But i did have a very clear theoretical and practical idea of what im about to study now. Dnx a million, you make science sound like less science. Nice job with the graphs, and good idea the challenge to try 0, 0, 4, 0.

That was the point when i finally really understood. Awesome, glad it clicked for you. Thanks for letting me know the examples helped. Thanks, you are correct. I was very loose with my terminology, to avoid the need for decimals. If a signal had 4 data points a b c d , I wanted to imagine scaling it up so it took 4 seconds of time to complete. Similarly, something which completed half the cycle each step.

This is a mental conversion I was running in my head, and I need to clarify this part, thanks! New mathematical techniques might allow for X-ray nanocrystallography National Academy of Sciences. Quick notes , September to November synthtech.

Suddenly I discovered the meaning of your site to make money isnt it? My goal is to help people grok the ideas I struggled with. Next time we would take permission from you whether to write blog or not. I just made a 2D fft filtering tool on my website, you can mask off regions of the spectrum as a filter and see the effects by performing an iFFT on the spectrum. And i must say you did the best. Thank you again to explain it so clearly.