"One weird trick that made pure mathematicians hate him"
Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
31–40 of 105 posts
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#32This is a great post, but it's a little bit misleading when talking about MP3s and lossy compression and conflates analog fourier analysis with discrete analysis. When you're talking about a digital signal, it is the sample rate that determines the maximum frequency you can represent. It's not MP3s that "throw out the really high notes" -- it's any digital signal. A discrete fourier transform actually is lossless, bu…
Thanks for your feedback. Sure, any digital signal is by definition finite in its resolution (the sample rate or bits). I was trying to address the distinction between wave files of the type stored on audio CDs, and MP3s - both digital signals. I agree that the Fourier transform is in principle lossless, but it's particularly useful to use it in a lossy way, i.e. to throw out the least important (to us) components of…
I think you were kind of hand-wavy, but that's just how it has to be for this kind of article. If you're trying to explain this stuff to a general audience, you can't just jump right into talking about nyquist limits and stuff.
But maybe you could allude to the fact that there's more going on.
Anyways, I liked the article.
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#33> 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…
I don't know much more about Fourier analysis than you, but theoretically it's an infinite series. So to describe an arbitrary smooth curve, you need a potentially infinite number of circles (or sin terms). I'm not sure if it's a countably infinite series or not. I think in principle you could have a component for every number in the real line.
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Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#34Coincidentally, I just had a talk with one of our principle developers about Fourier transforms. He's an audio expert and was trying to explain re-sampling and aliasing to me. I understand the high level steps, but the math is all a blur to me. Recently I've been trying to become much stronger in math, as I eventually want to study aerodynamics and astrophysics. So I've been studying calculus (textbook) and dynamics…
Concerning sampling and aliasing, you might find this video series from xipf interesting: http://www.xiph.org/video/
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#35Hi Hacker News - I'm the author of the piece, also on twitter @aatishb. Look forward to hearing your thoughts. I encourage you to share your thoughts and insights with other readers by leaving a comment on the post, particularly if you know of other interesting applications about the Fourier transform. Cheers!
OK, since you asked: > The sound wave produced by a piano note is a simple sine wave. No, it's not. A piano note is a complicated stack of overtones (some of which are harmonic and some of which aren't) and transients. If it was just a sine wave, it would sound like a sine wave and not like a piano. This is part of why things like Shazam are so difficult: musical notes aren't just a single frequency in the FFT, they…
Auditory masking is more than that. The psychoacoustic model used for MP3s will discard some frequencies during others, like you said, but it will also discard "information" proceeding and following an acoustic event.
While not a perfect analog, it is similar to how vision works. If you are in a bright room and the lights go out, it takes you a little time to readjust to the darker room before you can see things again. While your irises were adjusting, the chair and take were still there, and light reflecting from those objects were still reaching your eyes, but you were unable to see them. It works the other way too. If you are still in that dark room and the lights come on, it will take your eyes a little time to readjust, less time then it took your eyes to adjust to the dark, but still your eyes are effectively discarding that information.
The psychoacoustic model in MP3s is even more bizarre. It turns out that some frequencies, when heard BEFORE another frequency, your brain will throw that first frequency away. It is unintuitive, but that has been proven in laboratory settings. Knowing how the majority of humans discard the same auditory information under different conditions, MP3s are compressed by throwing away the same information that the brain would normally discard. The Fourier transform is a significant part, but it isn't the whole story.
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#36If your into software and you don't know a out fourior transform, your not into software. this is something programmers without math will discover, along with Pythagorean theorem and basic trig. Otherwise, you are an over-hyped semantic duck-taper.
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#37It's interesting to note that Fourier wasn't trying to any of the things that the Fourier Transform is commonly used for today, namely signal processing. He was trying to solve heat transfer equations when he came up with the Fourier series. I'm not even sure if he was that interested in the Transform as such, e.g. looking at a signal in frequency space and then efficiently applying filters before transforming back to time domain.
My dad remembers his professor, sometime in the 40s, posing the question of calculating when a worm buried in the ground would experience the same temperature we'd experience at Christmas (Erdwuermchen's Weihnachten) and the solution had to be calculated with Fourier's heat transfer equations.
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#38Earlier quoted context omitted.
I love the FFT even more than you and enjoyed that it is getting lauded, but would have found more algorithmic details even more interesting. Breaking down DFT, etc. and then showing the performance magic of FFT is a great way to approach discussion of many issues in problem analysis and algorithm design. So, Nice enough article for slipping into the topic - now give me more! harder! faster!
Sparse fast Fourier transform is even more "magical" than fast Fourier transform (FFT). If you assume that the discrete Fourier transform (DFT) has only k non zero coefficients, then, there exists an algorithm to compute it in O(k log(n)). That's right, you do not have to see the entire signal to compute the DFT, which is pretty awesome. If you are interested, see http://groups.csail.mit.edu/netmit/sFFT/ .
I work in this field (though I'm none of the authors). If anybody has any questions about the sFFT, let me know.
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#39It's interesting to note that Fourier wasn't trying to any of the things that the Fourier Transform is commonly used for today, namely signal processing. He was trying to solve heat transfer equations when he came up with the Fourier series. I'm not even sure if he was that interested in the Transform as such, e.g. looking at a signal in frequency space and then efficiently applying filters before transforming back t…
when it first came out most people ridiculed it for being a mere intellectual curiosity.
I would characterize the fourier transform (especially the DFT) as the single most important mathematical innovation that enables the interface between the digital and analog world.
Re: Fourier Transforms – The Math Trick Behind MP3s, JPEGs, and Homer Simpson’s Face
#40This is a great post, but it's a little bit misleading when talking about MP3s and lossy compression and conflates analog fourier analysis with discrete analysis. When you're talking about a digital signal, it is the sample rate that determines the maximum frequency you can represent. It's not MP3s that "throw out the really high notes" -- it's any digital signal. A discrete fourier transform actually is lossless, bu…
of course, analog or discrete, without the fourier (and related) transforms, you can't even do the analysis that allows you to "throw out information".