The development of fast Fourier transform is attributed to Cooley & Tukey, both have written a lot about it is historical development. However, I am searching early publications which showed how people made the discrete frequency axis after FFT.

For example, if we have N data points (N is an even or odd number), the FFT output also consists of N data points. The specific question is:

(i) Who came up with the equation or the proof that the frequency spacing in discrete Fourier transform is = k/L, where k =0,1,2,3,..., N-1 and L is the total duration of time for which the signal was acquired.

(ii) I am sure the concept of folding or aliasing must be in the same work.

Here is one earlier example by Tukey himself from 1950s

The measurement of power spectra


  • $\begingroup$ You probably mean what is variously called the sampling theorem, Kotelnikov theorem or Shannon-Nyquist theorem (erroneously), see Bissell's remarks to his translation of Kotelnikov's 1933 paper. The term "aliasing" was borrowed from radio engineering, and predates even Nyquist's 1928 paper. $\endgroup$ – Conifold Jul 10 '19 at 7:54
  • $\begingroup$ It is interesting, I was reading Hamming's Numerical Methods and he attributed aliasing to Tukey (the Cooley Tukey pair). Most likely aliasing is older as you said. I got more details here: mathoverflow.net/questions/335829/… $\endgroup$ – M. Farooq Jul 10 '19 at 14:06

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