5 ms·

I studied math, so I can never understand the insistence that the truth of the fourier transform is that it converts from the time domain into the frequency dom

I studied math, so I can never understand the insistence that the truth of the fourier transform is that it converts from the time domain into the frequency domain. What if you apply the fourier transform twice? Something similar is done in voice processing, and it's perfectly valid. And the origin of the Fourier transform is not in the signal world, but rather in the diffusion world, and there's not necessarily a signal in time involved. I think the concept of the basis of functions, a series of functions which you can combine to approximate any other function is something that engineers should be able to understand. Then you can see the time to frequency as a (very useful) application. BTW, the visualization in the post as to how the transform works is awesome, and can also work with the function basis explanation instead of the frequency one. In fact, it might make even more sense! And the posterior mention of the cosine transform wouldn't need to be hand wavy about real and imaginary parts. In any case, I've seen so many engineers insist in the time to frequency explanation that it must somehow be easier to understand for people. I just lament that the beauty of function spaces is lost in these explanations, as well as the underlying understanding of why Fourier transform is not only useful but deep.

hprotagonist 2y ago
> In any case, I've seen so many engineers insist in the time to frequency explanation that it must somehow be easier to understand for people. Fourier may have used his eponymous transform for working on the heat equation, but for many generations now the primary engineering applications of it have been in electric circuit theory and acoustics, both of which live in constant need for time <-> frequency conversions. Effectively all the literature in fourier analysis in engineering is written by or for someone whose background is in either electronics or acoustics. e.g., the bible for many engineers is Oppenheim, Willsky and Nawab, “Signals and Systems”; all three authors are primarily EE (Nawab is BME, but trained by EEs)
atoav 2y ago
> I studied math, so I can never understand the insistence that the truth of the fourier transform is that it converts from the time domain into the frequency domain. What if you apply the fourier transform twice? Something similar is done in voice processing, and it's perfectly valid. And I never understood the insistence of mathematicians to open with the generalized case when literally 99% of the use cases of a thing involve the more specialized use case. That is like a car mechanic telling you a part can be also used as a paperweigth when that is nearly never what it is used for. Don't get me wrong here – I like to hear about other usecases of something – I also like to hear generalized explainations of a thing – but that isn't how you should start when you explain a mathematical concept. It is nearly always better to start with a common special case in which the gory details don't apply and explain why the concept is important and what it does for us to then branch out than the other way around. Turns out most people first need a motivation why they should invest their brain in a thing and only then they are willing to do it. I could have strangled my maths teacher when they consistently mentioned the application in a side comment after weeks of theory and then did as if that wasn't that important. Yeah if all you do is teaching math or doing math for maths sake, it isn't, but that isn't going to be many people. And the Fourier analysis was famously the solution to a few actual real world problems that were very hard to tackle otherwise – why not tell that story? As I said, function spaces are cool, but maybe it is better to start with something else so people can appreciate it.
- seba_dos1 2y ago
  Are you trying to acquire a tool for your toolbox, or are you trying to understand the concepts that allow you to invent such tools? Teaching math is all about the latter, while some people are only interested in the former and struggle.
  - alejohausner 2y ago
    When I studied group theory, I would have been so much happier if someone told me we were talking about nonzero numbers, or invertible matrices, or functions. A little of bit of concreteness would have helped enormously to motivate the topic. When I took real analysis, I would have been so much happier if the teacher had presented some historical context: what pitfalls had mathematicians fallen into by misunderstanding infinity? That’s just how my brains work. I need some context. I suspect other brains are like mine too.
  - atoav 2y ago
    Your metaphor works both ways. If you teach the kids in kindergarden the theory of intricate japanese wood joints before they know which end of the saw to hold you are wasting your (and their) time. As I said, all things have their place, but some mathematicians have the habbit of starting at the most general (and therefore most abstract and most distant from the layperson) point to then move to more concrete applications. My suggestion wasn't to skip the general perspective, it was to teach it at a point where people already know and maybe even use the thing that is being generalized. That being said, not everybody is a mathematician. For some (and I'd argue: most) people using the fourier transformation as "just" a means to figure out the partials of any given signal is the tool they are looking for.
  - ThrowawayR2 2y ago
    I'm sympathetic to that view but most engineering disciplines are just interested in having the tool for their toolbox. Their interest in understanding the concepts that allow them to invent such tools is focused on their own discipline, which is not mathematics. Motivating students with real world applications first is IMO the only real chance to potentially spark their interest in learning broader concepts.
rsfern 2y ago
I think it’s just that the time/frequency analogy is so strongly rooted in signals processing applications In the physics and crystallography literature the Fourier transform is framed in terms of real space and reciprocal space, which personally I think is much more natural in the context of computing a structure factor or diffraction pattern for example. Many measurements (like diffraction) most directly measure reciprocal space, and one applies the inverse Fourier transform to recover the real space information of interest. Sometimes I think instead of time and frequency it would be more clear to frame signal processing applications in terms of time and reciprocal time just to highlight the symmetry and make it more clear that there’s nothing special about time and frequency vs some other dimension
skybrian 2y ago
Starting with the most abstract explanation may work for mathematicians, but most other people prefer starting with concrete examples.
bigbillheck 2y ago
> What if you apply the fourier transform twice? You're back in the time domain except reversed in time.

hprotagonist 2y ago

> In any case, I've seen so many engineers insist in the time to frequency explanation that it must somehow be easier to understand for people. Fourier may have used his eponymous transform for working on the heat equation, but for many generations now the primary engineering applications of it have been in electric circuit theory and acoustics, both of which live in constant need for time <-> frequency conversions. Effectively all the literature in fourier analysis in engineering is written by or for someone whose background is in either electronics or acoustics. e.g., the bible for many engineers is Oppenheim, Willsky and Nawab, “Signals and Systems”; all three authors are primarily EE (Nawab is BME, but trained by EEs)

atoav 2y ago

> I studied math, so I can never understand the insistence that the truth of the fourier transform is that it converts from the time domain into the frequency domain. What if you apply the fourier transform twice? Something similar is done in voice processing, and it's perfectly valid. And I never understood the insistence of mathematicians to open with the generalized case when literally 99% of the use cases of a thing involve the more specialized use case. That is like a car mechanic telling you a part can be also used as a paperweigth when that is nearly never what it is used for. Don't get me wrong here – I like to hear about other usecases of something – I also like to hear generalized explainations of a thing – but that isn't how you should start when you explain a mathematical concept. It is nearly always better to start with a common special case in which the gory details don't apply and explain why the concept is important and what it does for us to then branch out than the other way around. Turns out most people first need a motivation why they should invest their brain in a thing and only then they are willing to do it. I could have strangled my maths teacher when they consistently mentioned the application in a side comment after weeks of theory and then did as if that wasn't that important. Yeah if all you do is teaching math or doing math for maths sake, it isn't, but that isn't going to be many people. And the Fourier analysis was famously the solution to a few actual real world problems that were very hard to tackle otherwise – why not tell that story? As I said, function spaces are cool, but maybe it is better to start with something else so people can appreciate it.

seba_dos1 2y ago
Are you trying to acquire a tool for your toolbox, or are you trying to understand the concepts that allow you to invent such tools? Teaching math is all about the latter, while some people are only interested in the former and struggle.
- alejohausner 2y ago
  When I studied group theory, I would have been so much happier if someone told me we were talking about nonzero numbers, or invertible matrices, or functions. A little of bit of concreteness would have helped enormously to motivate the topic. When I took real analysis, I would have been so much happier if the teacher had presented some historical context: what pitfalls had mathematicians fallen into by misunderstanding infinity? That’s just how my brains work. I need some context. I suspect other brains are like mine too.
- atoav 2y ago
  Your metaphor works both ways. If you teach the kids in kindergarden the theory of intricate japanese wood joints before they know which end of the saw to hold you are wasting your (and their) time. As I said, all things have their place, but some mathematicians have the habbit of starting at the most general (and therefore most abstract and most distant from the layperson) point to then move to more concrete applications. My suggestion wasn't to skip the general perspective, it was to teach it at a point where people already know and maybe even use the thing that is being generalized. That being said, not everybody is a mathematician. For some (and I'd argue: most) people using the fourier transformation as "just" a means to figure out the partials of any given signal is the tool they are looking for.
- ThrowawayR2 2y ago
  I'm sympathetic to that view but most engineering disciplines are just interested in having the tool for their toolbox. Their interest in understanding the concepts that allow them to invent such tools is focused on their own discipline, which is not mathematics. Motivating students with real world applications first is IMO the only real chance to potentially spark their interest in learning broader concepts.

rsfern 2y ago

I think it’s just that the time/frequency analogy is so strongly rooted in signals processing applications In the physics and crystallography literature the Fourier transform is framed in terms of real space and reciprocal space, which personally I think is much more natural in the context of computing a structure factor or diffraction pattern for example. Many measurements (like diffraction) most directly measure reciprocal space, and one applies the inverse Fourier transform to recover the real space information of interest. Sometimes I think instead of time and frequency it would be more clear to frame signal processing applications in terms of time and reciprocal time just to highlight the symmetry and make it more clear that there’s nothing special about time and frequency vs some other dimension

skybrian 2y ago

Starting with the most abstract explanation may work for mathematicians, but most other people prefer starting with concrete examples.

bigbillheck 2y ago

> What if you apply the fourier transform twice? You're back in the time domain except reversed in time.