Hi 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!

 >> 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 really such a fundamental* aspect of frequencies and sound, it should have been the starting point of the article. "Look how noisy and squiggly this wave is, but with FFT we can see that it really only consists of waves at multiples of the same frequency".* no pun intended
 This is actually really valuable feedback and one of the things I like about the HN audience. I struggle to strike a balance between precision and simplicity, and your comments raised many interesting subtleties that I had overlooked. So thanks for helping me get a better handle on this.
 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.
 > This is actually the exact same number of numbers as in the time-domain series.In a way, it's twice as many, since they have both real and imaginary components.
 Nope - the amount of numbers is in fact identical. The discrete Fourier transform has redundant information in bins above NFFT/2 (where NFFT is the size of the transform/number of samples in the signal) for purely real valued input. This means the "number of numbers" is the same - NFFT/2 bins of necessary complex values after transformation, versus NFFT real values in the original series.
 Ah yeah, forgot about that. Thank you.
 I liked your article, however, it is not so much the Fourier Transform itself that makes possible all these wonderful applications, but rather the Fast Fourier Transform, which you fail to mention even once.The "naive" Fourier Transform has a pleasing mathematical simplicity and makes it very useful for deriving all sorts of theoretical properties for scientific analysis, but in practice it has a computational complexity of O(N^2), that would quickly make all those applications you name in your article utterly infeasible.Instead, only with the discovery of the Fast Fourier Transform algorithm roughly halfway the 20th century[0], lowered the complexity to a very manageable O(N log N). And only then all these applications and properties of the Fourier Transform came within computational reach. Without the FFT, the Fourier Transform would mostly just be useful on paper for functional analysis and math proofs, things like that, but not in practice and we'd be missing out on all those MP3s, JPGs and speech recognition.It's definitely not an overstatement when the WP article says: Fast Fourier transforms have been described as "the most important numerical algorithm[s] of our lifetime".[0] or 1805, depending: https://en.wikipedia.org/wiki/Fast_fourier_transform "The basic ideas were popularized in 1965, but some FFTs had been previously known as early as 1805."
 Awesome article! Here are two things that stuck out for me...1.) In the first part of the article talking about decomposing a periodic signal into sinusoids of different frequencies - it would be more correct to refer to it as the Fourier _Series_, which is different from the transform.2.) The image showing three sinusoids summing together into a non-period signal made me cringe (the sum is also periodic - that's the whole point of the Fourier Series).
 1) ...which when applied to a sampled signal, is the discrete-time Fourier transform (DTFT)2.) It will only be periodic if there exist some frequency for which the frequencies of the sinusoids are integer multiple.

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