@Whippy: Thanks for the comments! Glad you’re enjoying the site :).
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. However, the exposition still lacks something that has always bothered me about discussions of Fourier’s ideas: Specifically, it doesn’t actually explain why the transform works at all. 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. The result of this effort was an essay, with graphs and mathematics, which I originally called “Fourier for Dummies” - apeing the interminable string of “XXXYouNameIt for Dummies” books.While I left this original title in the link on my web page, I actually named the piece more descriptively as “A look at Fourier from a High School Math Perspective: Fourier and AC Signal Processing”. You can read the essay and/or download a PDF from my web server at:
http://linuxbio.med.buffalo.edu/Fourier/AC_Signal_Processing.html
You may hate it or you may love it but it may be worth a look.
Wonderful and revelatory stuff - you make learning it a delightful experience with all the visual metaphors and animations gradually building to the abstract formulas. what a rarity - maths taught in a human way! I’ve been trying to grok the fourier transform for months with little success outside of the basic concept that it decomposes a signal into frequencies. The formulas themselves just confused me. Now i really feel I have a handle on it.
Hi Niko, great question. Each animation is over the course of 1 second. If you are specifying a 0Hz (constant) component, then a single value is fine, since it’ll be the same throughout.
If you are analyzing a 1Hz signal inside that interval, you just need a measurement at the beginning and halfway (at 0.0 and 0.5 seconds) to make a determination of its strength. If you only had the measurement at the beginning (0.0 seconds) you wouldn’t know how strong the 1Hz signal was halfway, when it was completing its cycle.
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 0.0, 0.333, and 0.666 seconds) to specify its behavior. Yep, this is related to the sampling theorems, I’ll need a follow-up on that and build up my own deeper intuition :).
After years of trawling the net…;this is one place where I truly understood fourier transform
@Purushottam: Awesome, glad it helped!
Seriously, can we have you education minister, for the whole world? okthnx
a) I love you.
b) I basically hate practically all my math teachers, for the destruction they have spread in mine, and everybody else’s mind.
@Anonymous: 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.
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.
The metaphors /analogies were excellent while the animations are superb !
Would you please be kind enough and consider doing the same magic and explain the concept of (Claude) Shannon Entropy ? That’s another painful concept to grasp.
Thanks
A. Scarlat MD
Is this presentation targeted for persons already familiar with Fourier Series?
Think With Circles, Not Just Sinusoids:
What is a circle? Just a circle itself? A sinusoid?, a Complex exponential? or vice versa. 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.
Very complicated explanation.
The section where you introduce the animations needs to be clarified. The way it is written confuses me. I don’t understand what is happening to the animation when you change the values in the Cycles and Time box. When you change one it automatically changes the other, why? Why is there always a 0? What does a sentence like “Nothing at 0Hz, 1Hz of strength 1, 2Hz of strength 1” even mean?
Confused.
This is a work of a an extremely talented and gifted person!!!
Question - i am trying to understand what probability density functions have to do with “cyclicality”?? 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…
Thank you!!!
What a shame that some feel the need to squelch the brilliance of others, presumably to bolster their own inadequacies.
The ability to recognize an analogy like this does not demonstrate a lack of insight, it proves the author’s ability to understand the concept at an intuitive level-- a task which is much more challenging and profound than merely memorizing equations.
To this end, I am confounded by your own statement that the smoothie is not an apt analogy. We can both agree that a smoothie is a ‘whole’ while the ingredients are ‘parts’. 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: berries, which are themselves a whole. 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.
My next concern is that you object that the author does not address the concept of ‘change of variable’: “The key idea of FT – change of variable – is not emphasized at all.” However, the author very clearly mentions this in the very first section after the introduction “From Smoothie to Recipe”:
“A math transformation is a change of perspective. We change our notion of quantity from single items”
In the very next sentence of your comment you mention that “Only well informed people should be allowed to author such articles.” Judging by the fact that you clearly didn’t read (or worse, didn’t understand) the author’s mention of change of variables, I think I will modify your assertion: “Only people who read (and are capable of understanding) the article should be allowed to post incindiary remarks regarding the article’s validity”.
I applaud the author’s work in compiling this article, as I think it does a very good job at laying down the key ideas to newcomers of Fourier Analysis, and manages to motivate the equations intuitively instead of simply asserting them.
On that note, I am saddened that this article did not meet its mark for you.
What is the practical use of this circle view approach in solving practical problems?
For example what is the “circle” output of a linear system for any periodic “circle” input? How we represent the “circle” amplitude response and the “circle” phase response?
What is the “circle” transfer function of a linear system?
Thank you very much for this work sharing your insights, there has been very practice for my math career, keep going!
Kudos! 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!!!
Hi Matija, glad it helped
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Actually, the amplitudes don’t need to be 1.0 (for example: http://imgur.com/11VKmdJ)
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The time slot values is the strength of the signal. For example, seeing [1 2 3 4] on the time side means “The signal starts with strength 1. At the next intervals it has strength 2, then 3, then 4”. So if you double the amplitude of all the components, you’ll double the amplitudes of all the time slots.
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Yep :). Think of it like this: N frequencies and N time samples convey the same information about the signal (it’s like changing coordinates from the time-domain to the frequency-domain… but either way, you need the same amount of data to represent the signal).
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If you want fractional frequencies (4.5Hz) and therefore fractional time measurements (1.5 intervals), you need the continuous version of the Fourier transform [not the discrete one]. In math class, when working with analytic functions, you’ll learn the continuous one. But for engineering applications (with quantized time measurements) you’ll use the discrete one, since computers are storing individual data points, not an analytic function.
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. It’s a spiral of theory, practice, theory, practice…