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On the Wikipedia page of Fast Fourier transform it mentions that

[John] Tukey came up with the idea during a meeting of President Kennedy's Science Advisory Committee where a discussion topic involved detecting nuclear tests by the Soviet Union by setting up sensors to surround the country from outside. To analyze the output of these sensors, an FFT algorithm would be needed.

I gather that by ‘output of these sensors’ this refers to seismographs. If so, what exactly is the procedure by which they intended to detect nuclear tests from this graph so that it must require computing a discrete Fourier transform (DFT) first?

Also, it seems that since a seismograph is a continuous graph and not discretely sampled data, would it not have been sufficient to compute continuous Fourier transforms by means of mechanical devices like this that would have been well known at the time because of, for example, their use in predicting tides (which was the original motivation for it’s invention)?

Why was it established that a DFT must be computed instead so that it became of interest to do it efficiently?

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    $\begingroup$ By mid 1960-s, they wanted to process signals digitally, as Cooley describes in his historical note on FFT:"Digital signal processing (DSP) gave much better accuracy, it removed the constraint that the input-output relation was limited to what could be built with physical devices, and it permitted one to alter and optimize the input-output relation during processing. Then, with digital signal processing, came the need for numerical algorithms the most important of which were algorithms for Fourier transform and convolutions." $\endgroup$
    – Conifold
    Commented Apr 30 at 0:14
  • $\begingroup$ @Conifold I see, so a DFT was necessary for accuracy. It remains to answer why a Fourier transform is required in the first place. I feel like the answer to that might be obvious but I can’t come up with it $\endgroup$
    – anonymous
    Commented Apr 30 at 0:38
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    $\begingroup$ To filter signals from the noise and distinguish different types of them, see Seismic Applications for the FFT:"One of the original uses for the FFT was to distinguish between natural seismic events and nuclear test explosions because they generate different frequency spectra". Similar approach is used to distinguish earthquakes from mining explosions as well. $\endgroup$
    – Conifold
    Commented Apr 30 at 3:51
  • $\begingroup$ @Conifold ah, that’s why. For some reason I had been imagining that there will be some sort of spike in the seismograph corresponding to a nuke which is how it’s detected. Seems that the frequency spectra would have been more distinct and identifiable than a vague spike that’s probably pretty small anyway if the recording station was far away from the detonation. $\endgroup$
    – anonymous
    Commented Apr 30 at 7:14

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On the first question I can answer in general terms. Any analysis of wave phenomena is based on Fourier transform. In our case, the data are obtained from seismographers, and the goal is to find the characteristics of the source of the wave, like its location, duration and energy. This class of problems are called inverse problems (inverse to the "direct problems" of solving a PDE with given initial and boundary conditions). As most problems with linear or almost linear PDE they are solved using Fourier transform.

For technical details, I refer to the book

Victor Isakov, Inverse source problems

On the second question: why discrete FT was used. Indeed, by that time very sophisticated analog harmonic analyzers existed. Like every analog device they were a) very expensive, b) slow (because the input of the data took considerable effort), c) less reliable, and mainly d) very specialized; each of these complicated devices was designed for performing one special task, essentially for solving one problem. These devices usually also require some discretization step: the input is a finite set of numbers rather than a function.

The original paper by Lanczos and Danielson where the FFT was introduced (for solving some problem of optics) cites reason a):

If a modern mechanical analyzer is available, the evaluation of a Fourier integral presents no difficulty. It is our purpose to show that, for occasional analyses at least, one need not depend upon such costly instruments, even when the required number of coefficients is very large.

With the advent of fast digital electronic computers it became clear that they are much more efficient than analog devices, and this is the reason why we do not use these analog computing devices nowadays at all. So there is nothing special about nuclear tests, this was just a part of general development displacing analog computers by digital ones. I recall that the re-invention of FFT by Cooley and Tukey happened at IBM.

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