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The Fourier Transform and its Applications

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Re: The Fourier Transform and its Applications

#51

Post by gajjanag »

yagyu wrote: post #17

There is this hard question that is often glossed over in a first course on Fourier analysis: We learn that the Fourier transform of f(t) is an integral of exp(iwt)f(t) over all t. But when we look at a table of Fourier transform pairs, we find things like the Fourier transform of a constant function (is the Dirac delta). Integral does not converge? WTF? I really think Prof. Osgood hits a sweet spot of not sweeping t…

As someone who has gone down the rabbit hole of measure theory, functional analysis, and the like, I completely agree with the wonderful balance that Prof. Osgood strikes.

For a concrete instance, he brings up the distributional viewpoint of Laurent Schwartz and talks about the class of decaying functions for which one can operate nicely. This was something I next saw primarily in mathematics texts, with very few engineering books caring to talk about this even though it is not very hard to introduce and gives significant clarity to the process IMHO.

Studying Prof. Osgood's stuff off Stanford SEE during my summer vacation at the end of high school was one of the best learning experiences I have had till date.

Re: The Fourier Transform and its Applications

#52

Post by JadeNB »

Xcelerate wrote: post #40
JadeNB wrote: post #36

Earlier quoted context omitted.

> the Fourier transform over compact groups … is arguably the most generalized/abstract formulation of the concept of a Fourier transform. A good-spirited correction: anyone who knows enough to know this much generality knows that calling anything mathematical the "most generalised" version of a concept is begging to be corrected. Source: I do harmonic analysis on non-compact groups, and suspect that the folks over a…

Haha, fair point (although in my defense, I did qualify my statement with "arguably"). That's interesting that you do harmonic analysis on non-compact groups; I've been meaning to learn more about that actually. I've been working on group synchronization problems related to the ideas in this paper ( http://arxiv.org/pdf/1505.03840.pdf ), which briefly mentions how one might extend the concept to non-compact groups, b…

Sorry for the delayed response; I'm "submitting too fast".

> I did qualify my statement with "arguably"

I always tell my students—and, though it's a joke, I think that it's more true than casual consideration might suggest—that the correct answer (though not one that they're allowed to use on tests!) to any question in mathematics is "it depends." The more elementary the question, the more profound it sounds that you can see angles from which different answers can be correct. (Of course, backing this up with actual profound re-interpretations is the hard part. :-) )

> That's interesting that you do harmonic analysis on non-compact groups; I've been meaning to learn more about that actually. I've been working on group synchronization problems related to the ideas in this paper (http://arxiv.org/pdf/1505.03840.pdf), which briefly mentions how one might extend the concept to non-compact groups, but doesn't elaborate further.

As a decidedly pure mathematician, that's outside my wheelhouse, although I did have the pleasure of having some of the applications explained to me by one of the authors of one of the papers on cryo-electron microscopy cited in the linked article. I'm afraid that the idea of embedding a non-compact group in a larger, compact group is foreign to my way of thinking about things, so (in case you were implicitly asking) unfortunately I can't shed any light there.

Re: The Fourier Transform and its Applications

#53

Post by fgimenez »

drostie wrote: post #48

One of the fun ones, and this may or may not be covered by the link, is generating the "cumulants" for probability density functions. The basic idea in (basic, continuous) probability theory is that you have these things called "random variables" X,Y,Z which now need to be thought of taking on values with various probabilities; and the way we do that is to use calculus (where the d- in "dx" is a special notation comi…

Thanks for this cool writeup. My undergrad probability professor had us prove the CLT using cumulants, but I never kept the notes and had trouble finding the same proof later in life. Do you know of any reference that goes in depth about cumulants/FT/CLT?

Re: The Fourier Transform and its Applications

#55

Post by jomamaxx »

Omnipresent wrote: post #38
jomamaxx wrote: post #9

I 'took' the stanford NLP course online it was great. Doing this one now. Funny - I did Fourier in Comp. Eng. - but it was jammed together with so many other things, I felt overwhelmed by all of it, I did enough to do decently on the exam, but I feel a lot of it was lost on me, and I forgot much of it. Using fourier in most computer applications is very rare! I'm excited to have a project I need to use it for - and w…

Which NLP course are you referring to? Did it have any videos available?

Stanford NLP - Professor Dan Jurafsky & Chris Manning

https://www.youtube.com/watch?v=nfoudtpBV68

It's on youtube.

Re: The Fourier Transform and its Applications

#56

Post by drostie »

fgimenez wrote: post #53
drostie wrote: post #48

One of the fun ones, and this may or may not be covered by the link, is generating the "cumulants" for probability density functions. The basic idea in (basic, continuous) probability theory is that you have these things called "random variables" X,Y,Z which now need to be thought of taking on values with various probabilities; and the way we do that is to use calculus (where the d- in "dx" is a special notation comi…

Thanks for this cool writeup. My undergrad probability professor had us prove the CLT using cumulants, but I never kept the notes and had trouble finding the same proof later in life. Do you know of any reference that goes in depth about cumulants/FT/CLT?

Actually probably one of the best simple ones is Scholarpedia's article: in particular it covers one of the things that are kind of at the periphery of a non-applied statistician's mind when using cumulants: how the log-normal distribution serves as a handy counterexample to just about everything you'd like to prove about them (e.g. with some tweaking you can prove that the log-normal distribution's cumulant expansion is not unique, therefore you have an injection from PDFs to cumulants, not a surjection).

Something I've never seen rigorously explored: Cumulants appeared in my Master's work in condensed matter, where they at times seemed to have a mysterious tie to Feynman diagrams; that is, there seems to be some sort of analogy between how the Nth cumulant "subtracts out" all of the components of the Nth moment which are just products of the previous N-1 moments to obtain something "additive", and a Feynman diagram containing a vertex joining N particles. In particular Feynman diagrams for pairwise interactions seemed to be eerily similar to cumulant expansions which did not have any 3rd or higher moments. I confess I totally forgot this until I Googled today and saw a question on Math Overflow about it, but it was too sketchy to show me a real satisfying resolution of my mental curiosity. Similarly I remember seeing Young tableaux in some lecture notes on cumulants and then the day after I was working with Marcin Dukalski on some crazy approach he was taking in his thesis, and he spontaneously started talking about "hm, what do the Young tableaux look like for that?" and I was only able to help at all because of some limited understanding from the previous day's reading! So... there's definitely some mathematical resonances when you get to quantum theory.

Re: The Fourier Transform and its Applications

#57

Post by jholman »

jsprogrammer wrote: post #14
tptacek wrote: post #2

In case this is useful to anyone else: There's probably an easier way to do it, but if you want to get faster playback (2x is about as fast as I can comprehend this lecturer at), this line seems to do the trick: document.getElementById("myElement").querySelector("video").playbackRate = 2

document.getElementsByTagName('video')[0].playbackRate = 2; Not tested.

Here's my general-purpose speed-adjustment bookmarklet:

    javascript:void%20function(){document.querySelector(%22video%22).playbackRate=parseFloat(prompt(%22Set%20the%20playback%20rate%22))}();
And I find that about +70% is the speed I most-frequently want, so here's my one-click "x1.7":

    javascript:void%20function(){document.querySelector(%22video%22).playbackRate=1.7}();
Both of these adjust the speed of all non-iframe-embedded HTML5 video elements on the current page.

Of course, because of XSS-prevention rules, they won't help with iframe-embedded players. For that I guess you're gonna need a plugin.

Re: The Fourier Transform and its Applications

#58

Post by Zombieball »

notalaser wrote: post #22
Zombieball wrote: post #16

Earlier quoted context omitted.

> electrical engineering, physics and computer science books mostly relegated the topics to a few pages in an appendix. Out of curiosity, is this really true? Perhaps I misunderstand you, but I have always been under the impression that Fourier Transform was a fundamental building block covered extensively in all EE curriculum? I took Computer Engineering with some classes shared between Comp. Eng. and EE students. I…

I don't know about physics, but it's definitely not true about electrical engineering. The Fourier transform is one of those things that is absolutely fundamental to any branch of electrical engineering, you run into it when doing anything, from microelectronics to power distribution. And there's typically no way you get an EE diploma without a soul-wrenching course on signals & systems. As a personal anecdote: I end…

Thanks! This definitely parallels my experience in university.

Re: The Fourier Transform and its Applications

#59

Post by Zombieball »

notalaser wrote: post #22
Zombieball wrote: post #16

Earlier quoted context omitted.

> electrical engineering, physics and computer science books mostly relegated the topics to a few pages in an appendix. Out of curiosity, is this really true? Perhaps I misunderstand you, but I have always been under the impression that Fourier Transform was a fundamental building block covered extensively in all EE curriculum? I took Computer Engineering with some classes shared between Comp. Eng. and EE students. I…

I don't know about physics, but it's definitely not true about electrical engineering. The Fourier transform is one of those things that is absolutely fundamental to any branch of electrical engineering, you run into it when doing anything, from microelectronics to power distribution. And there's typically no way you get an EE diploma without a soul-wrenching course on signals & systems. As a personal anecdote: I end…

Thanks! This definitely parallels my experience in university.

Re: The Fourier Transform and its Applications

#60

Post by jsprogrammer »

jholman wrote: post #57
jsprogrammer wrote: post #14

Earlier quoted context omitted.

document.getElementsByTagName('video')[0].playbackRate = 2; Not tested.

Here's my general-purpose speed-adjustment bookmarklet: javascript:void%20function(){document.querySelector(%22video%22).playbackRate=parseFloat(prompt(%22Set%20the%20playback%20rate%22))}(); And I find that about +70% is the speed I most-frequently want, so here's my one-click "x1.7": javascript:void%20function(){document.querySelector(%22video%22).playbackRate=1.7}(); Both of these adjust the speed of all non-ifram…

`queySelector` only returns the first matching element (according to Mozilla docs).
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