A key insight for someone interested in this sort of thing not touched on in the article is the relationship between the fourier transform and a discrete consine transform.<p>JPEG uses DCT in particular because it has the nice property that the "top left" corner of the block will contain the DC offset (since cosine of 0 is 1) and the coefficients near the top corner correspond to half-wave and full cycles which gets you most of the way to simple gradients of color across the block with the right coefficients. So for most areas of an image only the top left coefficients will be significant. By using a zig-zag pattern for each block we are grouping the largest values to the front and zeroes to the back, which when coupled with RLE makes the rows of zero in each block a very compact, further-compressable representation.<p>Meanwhile, a fourier transform gives you imaginary magnitudes for frequencies which corresponds to the phase shift that is most appropriate for that frequency to match most strongly (as opposed to be aligned at the corner/beginning of the integral window). Not useful in an image format where you won't get the transformed magnitudes all nice and grouped for you. This is useful in audio compression where we care to find the location of transients that correspond to note attacks, percussion strikes, etc. Note that even in MP3 this is only used to drive the psychoacoustical model that decides the frame type and where to allocate the bits; the audio data itself is processed out of the time domain by overlapped DCT just like Ogg Vorbis.
Digital music could absolutely exist without the Fourier Transform.<p>The author should familiarize himself with the history of digital audio, and its milestones such as the development of the compact disc format: long before MP3, Ogg and the popularization of the Internet, and its use for media streaming.<p>At its bare bones, digital audio requires time domain sampling and reconstruction, sandwiched between some filters that can be analog. It requires understanding of the Nyquist limit, which can be purely in time domain terms (sampling frequently enough to avoid an temporal aliasing ambiguity in the reconstruction).<p>Digital <i>synthesis</i> of music can also be as simple as playing recorded samples in loops, and scaling them in the time domain for various pitches (or changing the sample rate, or both), which doesn't require Fourier.
DCT, FFT, close enough I guess. No mention of Shannon or Nyquist?<p>Ugh, I see a trend starting here:<p>"This is the first in a new experimental series called Favored Equation. Each month, we’ll dive into a piece of math which makes your life easier in some way without you even realizing."<p>This is a spin on <a href="http://objectsobjectsobjects.com/" rel="nofollow">http://objectsobjectsobjects.com/</a> by The Atlantic:<p>"Object lessons: An ongoing series about the hidden lives of ordinary things."<p>I'm a fan of this type of writing. But when Sagan and Feynman did it -- hell when pornographers did it with OMNI Magazine -- it wasn't quite so rough around the edges.<p>I'm now a month into arguing with some ex-Gawker hack at <i>The Atlantic</i> over quotes like "New effects can change a guitarist’s playing ability completely" and a declaration that the transistor was invented in the 1960s. No corrections or retractions imminent.<p>No interest in battling the newer, younger, even-less-experienced Gawker editor too.
...they could at least have used a proper example. Their "squiggly sine lines" don't make sense at all:<p>"""But add them together, and that pleasant sounding chord actually looks altogether more messy, like this:"""<p>No, it doesn't look like this, at all.
Or almost all forms of recorded audio for that matter. It is basically miraculous that a single speaker can create the impression of an orchestra; it obviously relies on an implicit instantiation of this same principle.
certainly could, and did. What enabled digitization and reproduction of digital audio were the works of Nyquist and Shannon, work that showed how it would work. FFT is an elaboration useful for filtering and spectrum contouring, and for compression. But "Digital Audio" is not a synonym for "Compressed Digital Audio". And digitization had to be a precursor for applying an FFT filter to the digitized result. FFT is _not_ used to implement the digitization itself, and is useless on its own with the digitzed sample stream to work on.
What about Wavelet transforms: <a href="http://en.wikipedia.org/wiki/Wavelet_transform" rel="nofollow">http://en.wikipedia.org/wiki/Wavelet_transform</a> ?<p>I think it's just a total different approach to compression that doesn't use Fourier.