When I was interviewed for a position by Hewlett-Packard in the late 1970s they were still a major scientific instruments company with a tagline of if it produces a signal, we can measure it! They had manufacturing facilities in Scotland, or "just down the road" according to Kim, my American interlocutor, a couple of field offices and they were hiring for an R&D facility they were planning in Pinewood, Wokingham.
They had very view dedicated conference rooms in the Winnersh, Wokingham field office and so I was interviewed in their Fourier Analyzer room. I sat alongside a six foot tall, imposing rack machine that included a real-time computer, a Digital to Analog convertor, an Analog to Digital convertor and a lovely HP Oscilloscope. This was an HP digital Fourier Analyzer and it was the first one I had seen.
The purpose of the digital Fourier Analyzer was to take in a complicated continuous signal from the real world, something that was hard to work with like a vibration signal, and break it down into a finite number of manageable sine and cosine functions with their magnitude and phase relationships. This work was based on the development by Jean-Baptiste Fourier, more than a hundred years ago, of his eponymous infinite series. The amazing feat performed by this machine was, however, made possible even with a fast computer in a reasonable amount of time, only by the development of the cunning Cooley-Tukey FFT Algorithm in 1965.
The interactive codepen below gives us an idea of the way that a simple periodic function, like a square wave, or a sawtooth curve, can actually be simulated to a reasonable degree of accuracy with only a limited number of terms.
Interactive Codepen: http://goo.gl/qURwrA
Before digital computers, there were analog devices for Fourier Analysis. If you have a bit more time and you haven't seen them yet, you might enjoy these videos (and the e/book) by +Bill Hammack. He and team restored one such machine. This analog computer was originally developed by Albert Michelson (of Michelson-Morley fame). It uses gears, springs and levers to add sines and cosines.
(1/4) Intro/History: Introducing a 100-year-old mechanical computer: https://goo.gl/YFowTo
(2/4) Synthesis: A machine that uses gears, springs and levers to add sines and cosines: https://goo.gl/y4ZXdH
(3/4) Analysis: Explaining Fourier analysis with a machine: https://goo.gl/O6xIGl
(4/4) Operation: The details of setting up the Harmonic Analyzer: https://goo.gl/fJxIPc
Book (free pdf or buy printed): http://goo.gl/9oy9yS
HP Journal 1970/06: https://goo.gl/W0F0Sw
30 Years of FFT: https://goo.gl/qkPUm0
Fourier Series (Wikip): https://goo.gl/bEfHiI
Fourier Transform (Wikip): https://goo.gl/osR3Cf
Fast Fourier Transform: https://goo.gl/t9ezSe
Fourier Analysis (Wikip): https://goo.gl/B8zC1w
Harmonic Analysis: (Wikip): https://goo.gl/p2U1ur
Image courtesy of Computer History Museum: http://goo.gl/OaoDqy
I guess I should be happy enough with what we have now - 20-30 years from now I'll probably look back at it with the same nostalgia :)