The FFT or Fast Fourier Transform was described by Cooley & Tukey in their 1965 paper “An Algorithm For The Machine Calculation of Complex Fourier Series” (Ref.1). The method has earlier roots in the works of Carl Gauss and others (Ref.2). The purpose of the FFT is to reduce the calculation complexity of the DFT or Discrete Fourier Transform.
On a practical level, I remember my first exposure to a Spectrum Analyzer in 1976 was the venerable HP141T. This was an amazing instrument that had various plug-ins depending on the band of interest. Only experienced senior engineers were allowed to use it. It had two hefty handles on the front and had a special carrying case that weighed a ton when loaded. Fast forward to 2018 and I now use lightweight portable usb powered instruments that plug into my laptop. All of this is due to the digital revolution and in particular the FFT that makes all of this possible.