From this point on it is assumed that any problem amenable to solution with the aid of the Discrete Fourier Transform (or DFT) will in fact be treated computationally with a fast routine (or FFT), and that the data sets are of length N = 2ⁿ, unless noted otherwise. The DFT pair will be written

In this formulation, Y₀ represents the average value of the data, the so-called DC term. In addition, |Y_n|² represents the contribution that frequencies near n/N make to total data variance.

(Note: Some FFT routines require that the data first be made complex.)

Experiment with some versions of the FFT available to you, first using a digitised sinusoid as input. Then attempt to duplicate the figure below, comment on variance normalisation.