6 ms·

An Interactive Guide To The Fourier Transform

218 points by lnmx 14y ago

Jun8 14y ago
Very good explanation, the smoothie-recipe decomposition approach is the one I used when I taught this to students (I used drinks, but well). However, this example really works only for teaching Fourier Series, which should be the first step to the FT anyway. If you want to understand this stuff, the order you should approach, I think is the Fourier Series, the FT (ignoring many mathematical difficulties), the discrete-time FT, which has its own quirks.
mark-r 14y ago
This line towards the end has a wonderful meta cleverness about it: "The analogy is flawed, and that's ok: it's a raft to use, and leave behind once we cross the river."
meaty 14y ago
Great article. I did Fourier Transforms in EE maths and could apply them but never properly understand them. Typical calculation versus understanding. The greatest graphic in this entire article is if you ask me: http://betterexplained.com/wp-content/uploads/images/DerivedDFT.png http://betterexplained.com/wp-content/uploads/images/Derived... If all mathematics was explained like this, many less eyeballs would have been gouged out of sockets.
- kalid 14y ago
  Thanks. I love the conversion of equations into "math English" (that particular diagram is from Stuart Riffle).
DoubleMalt 14y ago
This is indeed an awesome explanation. And I'm regularly shocked when people do not explain Fourier Transform as generalization of Fourier Series. This makes it quite intuitive to me (well, at least as far as FT can be intuitive ...)
haberman 14y ago
I have desperately wanted to understand the Fourier Transform and signal processing for a long time. In high school I read every tutorial or book I could get my hands on, hoping that one would finally "click." The very first thing I did when I got to college was approach the CS faculty and ask for recommendations for literature that would help me understand it. My college didn't have a signals and systems class, but a few years later I heard that Richard Lyons's book "Understanding Digital Signal Processing" was supposed to be the friendliest book on the subject so I bought and read it. I say all of this just to impress upon you that I was serious about wanting to understand this stuff. Despite all the effort, the FFT and DFT were never more than opaque blobs of math to me, mysterious boxes that you put inputs into and could extract outputs from (albeit in a very obscure format that seemed to involve complex numbers for no apparent reason). I sort of gave up and forgot about it for several years. Recently I came upon Stuart Riffle's article (http://www.altdevblogaday.com/2011/05/17/understanding-the-fourier-transform/ http://www.altdevblogaday.com/2011/05/17/understanding-the-f...) which explained it in terms of the circular approach that this article adopts. Rotate the waveform about the origin at the frequency of interest and average the samples, simple as that. I am not exaggerating when I say that after 10 minutes of reading that article, I had an understanding that years of highly motivated knowledge-seeking had not given me. I take a few things from this story. One is that, given a specific learner (ie. me), some ways of explaining things are infinitely better than others. I mean that literally, because this new, circle-based explanation gave me an understanding that I literally was not capable of achieving with all of the symbolic and algebraic explanations I had studied previously. Judging from other comments I've read, I'm not alone in this, which says to me that there could be exciting advances ahead of us in the way we learn. Just as the Khan Academy is making learning more accessible, I hope that it could also help discover and widely disseminate the best explanations for things. Sal Khan is very good at explaining things but we can't expect him to think of everything. This fantastic DFT explanation was created by a random systems/graphics programmer; I hope it will percolate into DFT curricula until everyone who is learning about the subject is at least exposed to it. My other takeaway, though, is that hard-core math people really think in a fundamentally different way than I do. I'm a highly intuitive thinker, and formulas are a sea of meaningless symbols to me without an intuitive understanding of what is going on. That someone could understand the DFT without thinking in terms of the circular interpretation is amazing to me. I now know that I am indeed capable of understanding the concept, but only by thinking about it in a different way than most math people do. I suspect, however, that my intuitive way of thinking about it would be more difficult to formalize and make rigorous, so in the end I am dependent on the mathematicians and their way of thinking, even if I can't as easily understand things in their terms. (One other example of this: I think calculus is simpler to understand in terms of infinitesimals rather than limits, but this is another example where the infinitesimals are more difficult to make rigorous).
wglb 14y ago
I studied frequency domain/time domain when I was in eight grade and high school learning the technical aspects of how radios work. An oscilloscope was a big boost for developing this intuition, particularly if hooked to a microphone where you could see how various sounds showed up, then how tones from the speaker of a radio would look. And using radios day-in, day-out, and understanding how the spectrum of an AM radio signal looked. I am glad that this analogy works for many of you here, but I have always been deeply suspicious of using analogies to teach concepts. For one thing, there is always the part that "Ok, what I just told you and you learned is not true in the following ways.." And the famous analogy between water in a pipe and electricity being actually potentially dangerous to a new student. In fact, I am sure that Fred remembers me railing "all analogies are False". And I have had very little success teaching complex topics using analogies. Incidentally, the AM radio spectrum is quite easy to understand. It wasn't until an advanced signals course that I saw the math for the spectrum of an FM signal. Much more complicated.
derleth 14y ago
> music recognition services compare recipes, not individual drops What does this mean? I understand 'recipes' here, but what's 'drop' in this context? A continuation of the food metaphor?
- kalid 14y ago
  I should probably change that. In this case, a "drop" would be something like a single second of audio. Instead of saying "Does this single second of audio show up in other songs?" we should ask "Do the frequency components in this song show up in other songs?" (similar ratios of bass, treble, etc.)
iooi 14y ago
Ended up playing with the animation for a while, this is probably nothing new but just thought I'd share a funny quirk I discovered. If you plug in the fibonacci sequence for the time, you get symmetrical strengths and phases (except for the first term): For (0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610) you get: 99.74 77.77:51.4 53.46:82.8 39.97:104.3 32.41:121.8 27.98:137.3 25.39:152 24.03:166.2 23.62:180 24.03:-166.2 25.39:-152 27.98:-137.3 32.41:-121.8 39.97:-104.3 53.46:-82.8 77.77:-51.4 Formatting magic: 99.74 1 - 77.78 51.4 2 - 53.46 82.8 3 - 39.97 104.3 4 - 32.42 121.8 5 - 27.98 137.3 6 - 25.39 152 7 - 24.03 166.2 8 - 23.63 180 7 - 24.03 -166.2 6 - 25.39 -152 5 - 27.98 -137.3 4 - 32.42 -121.8 3 - 39.97 -104.3 2 - 53.46 -82.8 1 - 77.78 -51.4 It works for all sequence lengths I've tried, although the app starts rounding off when you start getting in the hundreds.. not that it matters in this case. Couldn't find any relationship between the first constant and the pairs, nor any relationship between the ratio of the pairs. Just something interesting.
- napoleond 14y ago
  Without wanting to quell any enthusiasm (or cause too much confusion), it's probably worth pointing out that any real-valued time-series input will result in a similarly symmetric pattern. I can't explain why that's true in a way that makes sense with the analogy used here, but it is.
- kalid 14y ago
  Awesome, glad you were able to explore! As the other reply mentioned, it actually turns out that all real (1d) signals are symmetric. The key reason is in order for a circular path to stay on the real axis, it needs to be combined with another path rotating the opposite way, so their sum will stay on the x-axis. The "opposite" rotation can be a negative frequency (1Hz vs -1Hz) or a very fast positive frequency (if you have a 12-hour clockface, 1 hour backwards = 11 hours forward, aka -1 (mod 12) = 11 (mod 12)). I'd like to explore this in the follow-up, glad you discovered it.
philip1209 14y ago
This amuses me. My friend and I spent part of our time at a bar trying to explain Fourier transforms to her philosophy major boyfriend. My explanation of how optical trapping using Snell's law and light momentum went much more smoothly.

218 points by lnmx 14y ago

Jun8 14y ago

Very good explanation, the smoothie-recipe decomposition approach is the one I used when I taught this to students (I used drinks, but well). However, this example really works only for teaching Fourier Series, which should be the first step to the FT anyway. If you want to understand this stuff, the order you should approach, I think is the Fourier Series, the FT (ignoring many mathematical difficulties), the discrete-time FT, which has its own quirks.

mark-r 14y ago

This line towards the end has a wonderful meta cleverness about it: "The analogy is flawed, and that's ok: it's a raft to use, and leave behind once we cross the river."

meaty 14y ago

Great article. I did Fourier Transforms in EE maths and could apply them but never properly understand them. Typical calculation versus understanding. The greatest graphic in this entire article is if you ask me: http://betterexplained.com/wp-content/uploads/images/DerivedDFT.png http://betterexplained.com/wp-content/uploads/images/Derived... If all mathematics was explained like this, many less eyeballs would have been gouged out of sockets.

kalid 14y ago
Thanks. I love the conversion of equations into "math English" (that particular diagram is from Stuart Riffle).

DoubleMalt 14y ago

This is indeed an awesome explanation. And I'm regularly shocked when people do not explain Fourier Transform as generalization of Fourier Series. This makes it quite intuitive to me (well, at least as far as FT can be intuitive ...)

haberman 14y ago

I have desperately wanted to understand the Fourier Transform and signal processing for a long time. In high school I read every tutorial or book I could get my hands on, hoping that one would finally "click." The very first thing I did when I got to college was approach the CS faculty and ask for recommendations for literature that would help me understand it. My college didn't have a signals and systems class, but a few years later I heard that Richard Lyons's book "Understanding Digital Signal Processing" was supposed to be the friendliest book on the subject so I bought and read it. I say all of this just to impress upon you that I was serious about wanting to understand this stuff. Despite all the effort, the FFT and DFT were never more than opaque blobs of math to me, mysterious boxes that you put inputs into and could extract outputs from (albeit in a very obscure format that seemed to involve complex numbers for no apparent reason). I sort of gave up and forgot about it for several years. Recently I came upon Stuart Riffle's article (http://www.altdevblogaday.com/2011/05/17/understanding-the-fourier-transform/ http://www.altdevblogaday.com/2011/05/17/understanding-the-f...) which explained it in terms of the circular approach that this article adopts. Rotate the waveform about the origin at the frequency of interest and average the samples, simple as that. I am not exaggerating when I say that after 10 minutes of reading that article, I had an understanding that years of highly motivated knowledge-seeking had not given me. I take a few things from this story. One is that, given a specific learner (ie. me), some ways of explaining things are infinitely better than others. I mean that literally, because this new, circle-based explanation gave me an understanding that I literally was not capable of achieving with all of the symbolic and algebraic explanations I had studied previously. Judging from other comments I've read, I'm not alone in this, which says to me that there could be exciting advances ahead of us in the way we learn. Just as the Khan Academy is making learning more accessible, I hope that it could also help discover and widely disseminate the best explanations for things. Sal Khan is very good at explaining things but we can't expect him to think of everything. This fantastic DFT explanation was created by a random systems/graphics programmer; I hope it will percolate into DFT curricula until everyone who is learning about the subject is at least exposed to it. My other takeaway, though, is that hard-core math people really think in a fundamentally different way than I do. I'm a highly intuitive thinker, and formulas are a sea of meaningless symbols to me without an intuitive understanding of what is going on. That someone could understand the DFT without thinking in terms of the circular interpretation is amazing to me. I now know that I am indeed capable of understanding the concept, but only by thinking about it in a different way than most math people do. I suspect, however, that my intuitive way of thinking about it would be more difficult to formalize and make rigorous, so in the end I am dependent on the mathematicians and their way of thinking, even if I can't as easily understand things in their terms. (One other example of this: I think calculus is simpler to understand in terms of infinitesimals rather than limits, but this is another example where the infinitesimals are more difficult to make rigorous).

wglb 14y ago

I studied frequency domain/time domain when I was in eight grade and high school learning the technical aspects of how radios work. An oscilloscope was a big boost for developing this intuition, particularly if hooked to a microphone where you could see how various sounds showed up, then how tones from the speaker of a radio would look. And using radios day-in, day-out, and understanding how the spectrum of an AM radio signal looked. I am glad that this analogy works for many of you here, but I have always been deeply suspicious of using analogies to teach concepts. For one thing, there is always the part that "Ok, what I just told you and you learned is not true in the following ways.." And the famous analogy between water in a pipe and electricity being actually potentially dangerous to a new student. In fact, I am sure that Fred remembers me railing "all analogies are False". And I have had very little success teaching complex topics using analogies. Incidentally, the AM radio spectrum is quite easy to understand. It wasn't until an advanced signals course that I saw the math for the spectrum of an FM signal. Much more complicated.

derleth 14y ago

> music recognition services compare recipes, not individual drops What does this mean? I understand 'recipes' here, but what's 'drop' in this context? A continuation of the food metaphor?

kalid 14y ago
I should probably change that. In this case, a "drop" would be something like a single second of audio. Instead of saying "Does this single second of audio show up in other songs?" we should ask "Do the frequency components in this song show up in other songs?" (similar ratios of bass, treble, etc.)

iooi 14y ago

Ended up playing with the animation for a while, this is probably nothing new but just thought I'd share a funny quirk I discovered. If you plug in the fibonacci sequence for the time, you get symmetrical strengths and phases (except for the first term): For (0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610) you get: 99.74 77.77:51.4 53.46:82.8 39.97:104.3 32.41:121.8 27.98:137.3 25.39:152 24.03:166.2 23.62:180 24.03:-166.2 25.39:-152 27.98:-137.3 32.41:-121.8 39.97:-104.3 53.46:-82.8 77.77:-51.4 Formatting magic: 99.74 1 - 77.78 51.4 2 - 53.46 82.8 3 - 39.97 104.3 4 - 32.42 121.8 5 - 27.98 137.3 6 - 25.39 152 7 - 24.03 166.2 8 - 23.63 180 7 - 24.03 -166.2 6 - 25.39 -152 5 - 27.98 -137.3 4 - 32.42 -121.8 3 - 39.97 -104.3 2 - 53.46 -82.8 1 - 77.78 -51.4 It works for all sequence lengths I've tried, although the app starts rounding off when you start getting in the hundreds.. not that it matters in this case. Couldn't find any relationship between the first constant and the pairs, nor any relationship between the ratio of the pairs. Just something interesting.

napoleond 14y ago
Without wanting to quell any enthusiasm (or cause too much confusion), it's probably worth pointing out that any real-valued time-series input will result in a similarly symmetric pattern. I can't explain why that's true in a way that makes sense with the analogy used here, but it is.
kalid 14y ago
Awesome, glad you were able to explore! As the other reply mentioned, it actually turns out that all real (1d) signals are symmetric. The key reason is in order for a circular path to stay on the real axis, it needs to be combined with another path rotating the opposite way, so their sum will stay on the x-axis. The "opposite" rotation can be a negative frequency (1Hz vs -1Hz) or a very fast positive frequency (if you have a 12-hour clockface, 1 hour backwards = 11 hours forward, aka -1 (mod 12) = 11 (mod 12)). I'd like to explore this in the follow-up, glad you discovered it.

philip1209 14y ago

This amuses me. My friend and I spent part of our time at a bar trying to explain Fourier transforms to her philosophy major boyfriend. My explanation of how optical trapping using Snell's law and light momentum went much more smoothly.