Paired with this is the following expression, called the inverse discrete Fourier transform (IDFT)

                          (11.28)

These transformations map the discrete signal of a finite number on the time axis to the discrete spectra of a finite number on the frequency axis, or vice versa. These expressions using complex numbers are called complex discrete Fourier transform and the complex inverse discrete Fourier transform. We also have transformation using only real numbers. One is the real discrete Fourier transform, given by

                  ( n = 0, 1, 2, …, N -1)                        (11.29)