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# 11: Discrete Fourier Transforms

The Discrete Fourier Transform (DFT) is a discretized version of the Fourier transform, which is widely used in numerical simulation and analysis. Given a set of $$N$$ numbers $$\{f_0, f_1, \dots, f_{N-1}\}$$, the DFT produces another set of $$N$$ numbers $$N$$ numbers $$\{F_0, F_1, \dots, F_{N-1}\}$$, defined as follows:

$\mathrm{DFT}\Big\{f_0, f_1, \dots, f_{N-1}\Big\} = \Big\{F_0, F_1, \dots, F_{N-1}\Big\} \qquad\mathrm{where}\quad F_n = \sum_{m=0}^{N-1} e^{-2\pi i \frac{mn}{N}}\, f_m.$

The inverse of this transformation is the Inverse Discrete Fourier Transform (IDFT):

$\mathrm{IDFT}\Big\{F_0, F_1, \dots, F_{N-1}\Big\} = \Big\{f_0, f_1, \dots, f_{N-1}\Big\} \qquad\mathrm{where}\quad f_m = \frac{1}{N} \sum_{n=0}^{N-1} e^{2\pi i \frac{mn}{N}}\, F_n.$

The inverse relationship between the DFT and the IDFT is straightforward to prove, by using the identity

$\sum_{m=0}^{N-1} e^{\pm 2\pi i \frac{m(n-n')}{N}} = N \delta_{nn'},$

where $$\delta_{nn'}$$ denotes the Kronecker delta. This identity is derived from the geometric series formula.