Many years ago I had a consulting gig that required me to analyze the sound made by ball bearings in (very expensive) rotating machinery. After a bit of research, it became obvious that an algorithm known as the Fast Fourier Transform (FFT) was the way to go. On the processors of the day (in the '70s), computing the FFT was a lengthy process – one run of the final firmware I wrote could take almost an hour.
Recently four researchers at MIT came up with a much, much faster way to compute FFTs when the signal has structure of some kind. That was certainly true of the signal I was analyzing! With just a software change, that piece of gear I built could have been much faster. Dang!
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