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Honestly, it's not so bad. It's easy to pick any such attempt apart. This is close to my favorite pithy way of explaining it, too, which is to break it down com

by tel 4y ago

Honestly, it's not so bad. It's easy to pick any such attempt apart. This is close to my favorite pithy way of explaining it, too, which is to break it down component-wise using the idea of filter banks. It's not a single sentence, but here's what I tend to say: Any signal—like sounds or electrical signals, or even images—can be thought of as having a certain amount of 'energy' at any choice of frequency. This makes the most sense in music where we might thing of a 3-note chord as having 3 distinct packets of energy at 3 different frequencies. For any given frequency, we can compute the amount of energy a signal contains at that frequency by comparing the signal with a test signal, a "pure tone" at that frequency. Pure tones are signals that have the unique property of putting all of the energy at exactly one frequency. The Fourier Transform is an equation which packages this idea up, showing us how to represent all of these measurements of energy at all frequencies. The natural idea of a "pure tone" might be a sine wave. This is what we think of when we think of a musical pure tone and it certainly exists only at a single frequency. But sine waves make for bad comparison signals due to the problem of "phase": two sine waves played together can perfectly support one another and become twice as loud, or they can perfectly interrupt one another and become silence. This happens because sine waves oscillate between positive values and negative values and positive things can cancel out negative things. When you look at the equation for the Fourier transform you'll see an exponent of a complex number. This is an improved version of a 'pure tone' which avoids the phasing issues of a sine wave. It does this by spinning like a clock hand in two dimensions, remaining always at the same length. This extra dimension lets us preserve enough information so that things never cancel out like with sine waves.

sam_lowry_ 4y ago
I like your explanation in that it gives reason to the rotation.
glitchc 4y ago
That's better than the blog post actually (ECE background here).
deleted 4y ago
[deleted]
teolandon 4y ago
This is good, but I think it's cyclical (heh). We want to compare our signal with a "pure tone". What's a pure tone? A sine wave. Why is a sine wave a pure tone? Because when we compare it with a pure tone, it's identical.
- tel 4y ago
  I appreciate the feedback. I’m person I talk about pure tones more, mostly because I love how pretty complex spirals are. Here, I just tried to define them “Pure tones are signals that have the unique property of putting all of the energy at exactly one frequency” and then use sine as an example. Truly, complex rotations are a much better definition, but it’s somewhat reasonable to expect people to be familiar with sine waves.
- lathyrus_long 4y ago
  That's a question of why Fourier transforms are important though, not just how they're defined and computed. The next-level answer is presumably that sinusoids (or complex exponentials in general) are the eigenfunctions of general linear time-invariant systems, i.e. that if the input to an LTI system is exp(j*w*t), then its output will be A*exp(j*w*t) for some complex constant A. Some other comments here already alluded to that, noting that sinusoids are good for solving linear differential equations (which are LTI systems), or that the sum of two sinusoids of the same frequency shifted in time (which is an LTI operation, since addition and time shift are both LTI) is another sinusoid of that same frequency. LTI systems closely model many practical systems, including the tuning forks and flutes that give our intuition of what a "pure tone" means. I guess there's a level after that noting that conservation laws lead to LTI systems. I guess there's further levels too, but I'm not a physicist. That eigenfunction property means we can describe the response of any LTI system to a sinusoid (again, complex exponential in general) at a given frequency by a single complex scalar, whose magnitude represents a gain and whose phase represents a phase shift. No other set of basis functions has this property, thus the special importance of a Fourier transform. We could write a perfectly meaningful transform using any set of basis functions, not just sinusoids (and e.g. the graphics people often do, and call them wavelets). But if we place one of those non-sinusoidal basis functions at the input of an LTI system, then the output will in general be the sum of infinitely many different basis functions, not describable by any finite number of scalars. This makes those non-sinusoidal basis functions much less useful in modeling LTI systems.
stevebmark 4y ago
As someone who still doesn't understand the Fourier Transform, this explanation doesn't help, nor does the article's. As a complete noob, your explanation shows off what you know but doesn't help newcomers learn it.
- deleted 4y ago
  [deleted]
- SV_BubbleTime 4y ago
  Imagine you have a voice recording. You are looking at it a long squiggly line with no uniformity. Zoom way in. If you go far enough it’ll just look like a curved line. That curved line can be estimated down into a sin wave, or more accurately, a few sin waves that combine to make almost the same wave you have. FFT is a way to take a complex wave and reduce it down to the sin wave components that would all combine to make it up. To practically apply this, your voice recording is very high resolution and has a lot of bits at a high sample rate. If you instead used sin waves to represent the data, it could be almost as good sounding while being a lot less data to store. … Watch a video. It’s something many people need to see to get.
  - AwaAwa 4y ago
    Great explanation! I'm finally on the first step of understanding. Now to read more!
    - jaypinho 4y ago
      I once read an explanation of the Fourier transformation as akin to looking at a fully blended smoothie and being able to calculate exactly how much of each ingredient went into it. I don't profess to be a DSP expert whatsoever, but the more familiar I've become with Fourier transformations, the more apt that analogy seems. Once you grasp that all sound is just a large addition problem of many, many sine waves, the ability to distinguish between them to a fairly high degree of fidelity feels almost like magic.
- gehwartzen 4y ago
prof-dr-ir 4y ago
To be honest I think your explanation misses the point in two important ways. First, the Fourier transform does not just measure the 'energy' or amplitude of a single wave; it must also take into account its phase. Second, complex exponentials are in no way an essential ingredient for defining the Fourier transform. We just work with them for computational simplicity, essentially because exp(ix) exp(iy) = exp(i(x+y)). I also have trouble assigning meaning to your last paragraph. After all, complex exponentials can cancel just as much as sine waves (proof: sine waves are a combination of complex exponentials).
- canadianfella 4y ago
  [dead]
- tel 4y ago
  I agree with these criticisms. Really, my goal when I talk about Fourier Transforms is to avoid talking about phase. It's important, clearly, but it's both less intuitive and less practically meaningful. For a lot of applications, the phase data is just discarded anyway. And if you're in a situation where it matters, you're probably reading more than 3 short paragraphs. I'd love other thoughts on how to handle the ideas in the last paragraph. I'm not being technical, but am trying to allude to how the complex exponentials can capture more information more conveniently... and honestly waving my hands a lot. Really, I just want to explain why they're there instead of a more recognizable sine or cosine functions. I've also tried to show the Fourier transform as the sum of a sine and a cosine transform, but that's too much, I think.

by tel 4y ago

sam_lowry_ 4y ago

I like your explanation in that it gives reason to the rotation.

glitchc 4y ago

That's better than the blog post actually (ECE background here).

deleted 4y ago

[deleted]

teolandon 4y ago

This is good, but I think it's cyclical (heh). We want to compare our signal with a "pure tone". What's a pure tone? A sine wave. Why is a sine wave a pure tone? Because when we compare it with a pure tone, it's identical.

tel 4y ago
I appreciate the feedback. I’m person I talk about pure tones more, mostly because I love how pretty complex spirals are. Here, I just tried to define them “Pure tones are signals that have the unique property of putting all of the energy at exactly one frequency” and then use sine as an example. Truly, complex rotations are a much better definition, but it’s somewhat reasonable to expect people to be familiar with sine waves.
lathyrus_long 4y ago
That's a question of why Fourier transforms are important though, not just how they're defined and computed. The next-level answer is presumably that sinusoids (or complex exponentials in general) are the eigenfunctions of general linear time-invariant systems, i.e. that if the input to an LTI system is exp(j*w*t), then its output will be A*exp(j*w*t) for some complex constant A. Some other comments here already alluded to that, noting that sinusoids are good for solving linear differential equations (which are LTI systems), or that the sum of two sinusoids of the same frequency shifted in time (which is an LTI operation, since addition and time shift are both LTI) is another sinusoid of that same frequency. LTI systems closely model many practical systems, including the tuning forks and flutes that give our intuition of what a "pure tone" means. I guess there's a level after that noting that conservation laws lead to LTI systems. I guess there's further levels too, but I'm not a physicist. That eigenfunction property means we can describe the response of any LTI system to a sinusoid (again, complex exponential in general) at a given frequency by a single complex scalar, whose magnitude represents a gain and whose phase represents a phase shift. No other set of basis functions has this property, thus the special importance of a Fourier transform. We could write a perfectly meaningful transform using any set of basis functions, not just sinusoids (and e.g. the graphics people often do, and call them wavelets). But if we place one of those non-sinusoidal basis functions at the input of an LTI system, then the output will in general be the sum of infinitely many different basis functions, not describable by any finite number of scalars. This makes those non-sinusoidal basis functions much less useful in modeling LTI systems.

stevebmark 4y ago

As someone who still doesn't understand the Fourier Transform, this explanation doesn't help, nor does the article's. As a complete noob, your explanation shows off what you know but doesn't help newcomers learn it.

deleted 4y ago
[deleted]
SV_BubbleTime 4y ago
Imagine you have a voice recording. You are looking at it a long squiggly line with no uniformity. Zoom way in. If you go far enough it’ll just look like a curved line. That curved line can be estimated down into a sin wave, or more accurately, a few sin waves that combine to make almost the same wave you have. FFT is a way to take a complex wave and reduce it down to the sin wave components that would all combine to make it up. To practically apply this, your voice recording is very high resolution and has a lot of bits at a high sample rate. If you instead used sin waves to represent the data, it could be almost as good sounding while being a lot less data to store. … Watch a video. It’s something many people need to see to get.
- AwaAwa 4y ago
  Great explanation! I'm finally on the first step of understanding. Now to read more!
  - jaypinho 4y ago
    I once read an explanation of the Fourier transformation as akin to looking at a fully blended smoothie and being able to calculate exactly how much of each ingredient went into it. I don't profess to be a DSP expert whatsoever, but the more familiar I've become with Fourier transformations, the more apt that analogy seems. Once you grasp that all sound is just a large addition problem of many, many sine waves, the ability to distinguish between them to a fairly high degree of fidelity feels almost like magic.
gehwartzen 4y ago

prof-dr-ir 4y ago

To be honest I think your explanation misses the point in two important ways. First, the Fourier transform does not just measure the 'energy' or amplitude of a single wave; it must also take into account its phase. Second, complex exponentials are in no way an essential ingredient for defining the Fourier transform. We just work with them for computational simplicity, essentially because exp(ix) exp(iy) = exp(i(x+y)). I also have trouble assigning meaning to your last paragraph. After all, complex exponentials can cancel just as much as sine waves (proof: sine waves are a combination of complex exponentials).

canadianfella 4y ago
[dead]
tel 4y ago
I agree with these criticisms. Really, my goal when I talk about Fourier Transforms is to avoid talking about phase. It's important, clearly, but it's both less intuitive and less practically meaningful. For a lot of applications, the phase data is just discarded anyway. And if you're in a situation where it matters, you're probably reading more than 3 short paragraphs. I'd love other thoughts on how to handle the ideas in the last paragraph. I'm not being technical, but am trying to allude to how the complex exponentials can capture more information more conveniently... and honestly waving my hands a lot. Really, I just want to explain why they're there instead of a more recognizable sine or cosine functions. I've also tried to show the Fourier transform as the sum of a sine and a cosine transform, but that's too much, I think.