One of my favorites: http://www.altdevblogaday.com/2011/05/17/understanding-the-f...
Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
81–90 of 105 posts
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#82ImageMagick has a nice tutorial on the use of Fourier transforms in image processing, if you want to get a more intuitive feel on the subject: http://www.imagemagick.org/Usage/fourier/
But why do they use holocaust_tn.gif as an example filename?
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#83Earlier quoted context omitted.
This needs to be upvoted rather than downvoted.
I disagree. It's needlessly inflammatory and misspelt throughout. (Some people just have trouble spelling, and that's fair enough. But writing "a out" instead of "about" and not fixing it is just lazy and disrespectful to readers. AsymetricCom would probably have got a different response had s/he written something like this instead: "If you're really into software and want to be more than a semantic duck-taper, you n…
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Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#84I would love to know more about FT's, along with FFT's and how they help with for example signal processing or finding a signal when looking at a sample or multiple samples of a SDR. Are there any good books/papers/web articles on this topic that are accessible? I often find myself reading papers where some of the math goes over my head. Something with examples/code (code makes me understand math so much easier!) wou…
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#85ImageMagick has a nice tutorial on the use of Fourier transforms in image processing, if you want to get a more intuitive feel on the subject: http://www.imagemagick.org/Usage/fourier/
But why do they use holocaust_tn.gif as an example filename?
Holocaust Memorial: http://en.wikipedia.org/wiki/Memorial_to_the_Murdered_Jews_o...
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#86> You could just tell them a handful of numbers—the sizes of the different circles in the picture above. Maybe I'm crazy and just missing something, but this feels a little too good to be true. This would put the set of smooth curves in 1-1 correspondence with the set of finite sets (since each curve is being specified completely by a finite set of numbers). But the set of finite sets is countably infinite since it's…
First of all, there are four kinds of "Fourier transforms" in common use, so the cardinality problem that you have observed cannot really be discussed until we pick one to talk about. For instance, the FT maps uncountable to uncountable and the DFT maps finite to finite (and can be understood with intro Linear Algebra as a change of basis). The things we are calling countable and uncountable are actually dimensionali…
A related point of entry is the "Gibbs Phenomenon" -- http://en.wikipedia.org/wiki/Gibbs_phenomenon .
If I remember my "history of the Fourier transform" correctly, before the difference between the sup-norm and L2 norms were fully understood by everyone (e.g., by the late 1800s), the sense of convergence of a Fourier series to a square wave was confusing. In what sense did the Fourier series represent the square wave? When was it a good model and when would it fail?
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#87Ah such a shame. Tried to resist but I can't help pointing out you missed a chance to explain how when Fourier discovered the expansion, it really was an example of "one weird trick all the Mathematical Physicists will hate him for".
It was during his time in Grenoble that Fourier did his important mathematical work on the theory of heat. His work on the topic began around 1804 and by 1807 he had completed his important memoir On the Propagation of Heat in Solid Bodies. The memoir was read to the Paris Institute on 21 December 1807 and a committee consisting of Lagrange, Laplace, Monge and Lacroix was set up to report on the work. Now this memoir is very highly regarded but at the time it caused controversy.
There were two reasons for the committee to feel unhappy with the work. The first objection, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what we now call Fourier series. Further clarification by Fourier still failed to convince them. As is pointed out in [4]:-
"All these are written with such exemplary clarity - from a logical as opposed to calligraphic point of view - that their inability to persuade Laplace and Lagrange ... provides a good index of the originality of Fourier's views"
http://www-history.mcs.st-and.ac.uk/Biographies/Fourier.html
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#88For those of you seeking an intuitive understanding of the Fourier transform, checkout http://betterexplained.com/articles/an-interactive-guide-to-...
For more details, check out Steven Smith's Digital Signal Processing. The entire book is available to read online, and has an excellent treatment of DSP algorithms.
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#89I would add to your list of reasons why Fourier Transforms are awesome.
By complete coincidence, Bragg's law, used to do everything from X-Ray Diffraction to particle scattering, just happens to be a fourier transform. Every time we bombard a tiny thing with light or radiation in order to understand the structure, what we literally get out of it is emission dots that correspond to the periodicity of the lattice -- literally the 2D fourier transform of the scattering cross section. When I heard that in Quantum III, it blew my mind. It's straight out of quantum scattering theory.
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#90This article was interesting but didn't really tell me anything I don't already know. Does anyone know where I can find a good article that actually explains the mathematics of performing a Fourier transformation? I thought that is what this article was going to be about.
We learned about Fourier Series in my diffyQ course. The basic idea is that any periodic function( like say y=sinx) can be approximated by an infinite sum of sines and cosines.