4 ms·

The Fourier Transform and its Applications

174 points by bsilvereagle 12y ago

j2kun 12y ago
Brad Osgood's lectures are phenomenal. He was trained as a mathematician, so his explanations are crystal clear in terms of definitions and where physicists sweep mathematical formalisms under the rug. After watching his entire lecture for this course, and spending a lot of time reading the course reader, I learned a lot about how to decrypt the kind of mathematical nonsense that physicists tend to say (as well as learning about the real foundations, c.f. tempered distributions and others). That being said, a lot of these lectures can be summarized in a few sentences if you already have a strong foundation in linear algebra. For example, the fact that complex exponentials form an orthonormal basis for periodic functions is the content of the first handful of lectures, and deriving the Fourier transform is only slightly messier to explain in terms of linear algebra (this is not because of the linear algebra but because Fourier transforms are inherently a little messy).
sliverstorm 12y ago
The Fourier transform is possibly my favorite mathematical function, and the closest I've ever been to "Whoa, math is beautiful". Ironically enough, I reached this conclusion in a signal analysis class, not a math class.
- dalacv 12y ago
  Laplace transform is better. So nannynannybooboo
  - zwieback 12y ago
    FT goes from -inf to +inf, LT only 0 to +inf. So, FT is twice as good as LT.
    - flpgr 12y ago
      The Fourier Transform is just a special case of the bilateral LT. Laplace wins =)
      - mjcohen 12y ago
        Come to me my Mellin-choly baby!
jbert 12y ago
I'm interested in understanding more about the use of the transform in discrete signal processing. I wish I had the time to try and absorb 30 50min lectures, but I probably don't. Can anyone recommend a more limited course or tutorial focussing on that, or should I try and extract particular lectures from this course?
- badsock 12y ago
  Maybe try this: http://www.dspguide.com/ch8.htm http://www.dspguide.com/ch8.htm Really great book in any case.
  - jbert 12y ago
    Looks good. Thanks!
- hershel 12y ago
  This is a good source: dsp-book.narod.ru/FFTBB/0270_PDF_C16.pdf
- chipsy 12y ago
  You may enjoy: http://www.katjaas.nl/home/home.html http://www.katjaas.nl/home/home.html
cedias 12y ago
If anyone is looking for a quick intro to FT this was posted on HN a few month ago: http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-h...
- bdevine 12y ago
  Yes, I remember that well. Seeing the animated graphics linked within that article really was an 'ah ha!' moment for me; I'd definitely recommend it as an introductory intuitive motivation.
pling 12y ago
Finest resource of Fourier transforms here: http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ http://betterexplained.com/articles/an-interactive-guide-to-... This is the one that made me get it in the end.
- jonahx 12y ago
  Thanks for the link. His website is terrific. He offers some perspectives for gaining intuition about complex numbers that I'd never heard before, even after all these years. This should be required reading for HS math students.
jokoon 12y ago
isn't DSL internet an application from this theorem ?
quarterwave 12y ago
These are excellent lectures! I saw Chap 4, and it was very satisfying to watch the introduction to convergence, L2 norm, inner product etc. Technically, Fourier transform requires only the L1 norm (the 'Manhattan distance') to be finite, but L2 intersection L1 is a superb way to motivate distributions. These lectures must be made required viewing for all students of engineering and mathematical sciences.
autokad 12y ago
thanks stanford, and thanks ycombinator, i really look forward to going over these lessons
EpiMath 12y ago
Also from Stanford, from longer ago... The Fourier Transform and its Applications by Bracewell. McGraw-Hill. Extremely well written with great examples and some fascinating problems. I highly recommend it. Discrete transforms are a bit of an afterthought in a final chapter, but enough to dig in and see how the FFT works.
juanre 12y ago
I took a class from Stanford with this name many years ago and it was an eye-opener. Bracewell's book of the same name is wonderful as well (http://www.amazon.com/Fourier-Transform-Its-Applications/dp/0070070156/ http://www.amazon.com/Fourier-Transform-Its-Applications/dp/...).
dfan 12y ago
In the same sequence, Stephen Boyd's course Introduction to Linear Dynamical Systems (http://see.stanford.edu/see/courseinfo.aspx?coll=17005383-19c6-49ed-9497-2ba8bfcfe5f6 http://see.stanford.edu/see/courseinfo.aspx?coll=17005383-19...) is also excellent. The title sounds pretty specific but it is really about a wide range of applied linear algebra. Boyd's lecture style is a little idiosyncratic but he's great about constantly emphasizing how to gain an intuitive qualitative sense of what everything means. He also has a couple of courses on convex optimization that cover the theory behind a lot of machine learning.
feelix 12y ago
After spending hours searching, I ended up finding the way I wish I'd seen it explained in the beginning: http://www.altdevblogaday.com/2011/05/17/understanding-the-fourier-transform/ http://www.altdevblogaday.com/2011/05/17/understanding-the-f...

174 points by bsilvereagle 12y ago

j2kun 12y ago

Brad Osgood's lectures are phenomenal. He was trained as a mathematician, so his explanations are crystal clear in terms of definitions and where physicists sweep mathematical formalisms under the rug. After watching his entire lecture for this course, and spending a lot of time reading the course reader, I learned a lot about how to decrypt the kind of mathematical nonsense that physicists tend to say (as well as learning about the real foundations, c.f. tempered distributions and others). That being said, a lot of these lectures can be summarized in a few sentences if you already have a strong foundation in linear algebra. For example, the fact that complex exponentials form an orthonormal basis for periodic functions is the content of the first handful of lectures, and deriving the Fourier transform is only slightly messier to explain in terms of linear algebra (this is not because of the linear algebra but because Fourier transforms are inherently a little messy).

sliverstorm 12y ago

The Fourier transform is possibly my favorite mathematical function, and the closest I've ever been to "Whoa, math is beautiful". Ironically enough, I reached this conclusion in a signal analysis class, not a math class.

dalacv 12y ago
Laplace transform is better. So nannynannybooboo
- zwieback 12y ago
  FT goes from -inf to +inf, LT only 0 to +inf. So, FT is twice as good as LT.
  - flpgr 12y ago
    The Fourier Transform is just a special case of the bilateral LT. Laplace wins =)
    - mjcohen 12y ago
      Come to me my Mellin-choly baby!

jbert 12y ago

I'm interested in understanding more about the use of the transform in discrete signal processing. I wish I had the time to try and absorb 30 50min lectures, but I probably don't. Can anyone recommend a more limited course or tutorial focussing on that, or should I try and extract particular lectures from this course?

badsock 12y ago
Maybe try this: http://www.dspguide.com/ch8.htm http://www.dspguide.com/ch8.htm Really great book in any case.
- jbert 12y ago
  Looks good. Thanks!
hershel 12y ago
This is a good source: dsp-book.narod.ru/FFTBB/0270_PDF_C16.pdf
chipsy 12y ago
You may enjoy: http://www.katjaas.nl/home/home.html http://www.katjaas.nl/home/home.html

cedias 12y ago

If anyone is looking for a quick intro to FT this was posted on HN a few month ago: http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-homer-simpsons-face http://nautil.us/blog/the-math-trick-behind-mp3s-jpegs-and-h...

bdevine 12y ago
Yes, I remember that well. Seeing the animated graphics linked within that article really was an 'ah ha!' moment for me; I'd definitely recommend it as an introductory intuitive motivation.

pling 12y ago

Finest resource of Fourier transforms here: http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ http://betterexplained.com/articles/an-interactive-guide-to-... This is the one that made me get it in the end.

jonahx 12y ago
Thanks for the link. His website is terrific. He offers some perspectives for gaining intuition about complex numbers that I'd never heard before, even after all these years. This should be required reading for HS math students.

jokoon 12y ago

isn't DSL internet an application from this theorem ?

quarterwave 12y ago

These are excellent lectures! I saw Chap 4, and it was very satisfying to watch the introduction to convergence, L2 norm, inner product etc. Technically, Fourier transform requires only the L1 norm (the 'Manhattan distance') to be finite, but L2 intersection L1 is a superb way to motivate distributions. These lectures must be made required viewing for all students of engineering and mathematical sciences.

autokad 12y ago

thanks stanford, and thanks ycombinator, i really look forward to going over these lessons

EpiMath 12y ago

Also from Stanford, from longer ago... The Fourier Transform and its Applications by Bracewell. McGraw-Hill. Extremely well written with great examples and some fascinating problems. I highly recommend it. Discrete transforms are a bit of an afterthought in a final chapter, but enough to dig in and see how the FFT works.

juanre 12y ago

I took a class from Stanford with this name many years ago and it was an eye-opener. Bracewell's book of the same name is wonderful as well (http://www.amazon.com/Fourier-Transform-Its-Applications/dp/0070070156/ http://www.amazon.com/Fourier-Transform-Its-Applications/dp/...).

dfan 12y ago

In the same sequence, Stephen Boyd's course Introduction to Linear Dynamical Systems (http://see.stanford.edu/see/courseinfo.aspx?coll=17005383-19c6-49ed-9497-2ba8bfcfe5f6 http://see.stanford.edu/see/courseinfo.aspx?coll=17005383-19...) is also excellent. The title sounds pretty specific but it is really about a wide range of applied linear algebra. Boyd's lecture style is a little idiosyncratic but he's great about constantly emphasizing how to gain an intuitive qualitative sense of what everything means. He also has a couple of courses on convex optimization that cover the theory behind a lot of machine learning.

feelix 12y ago

After spending hours searching, I ended up finding the way I wish I'd seen it explained in the beginning: http://www.altdevblogaday.com/2011/05/17/understanding-the-fourier-transform/ http://www.altdevblogaday.com/2011/05/17/understanding-the-f...