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11: Discrete Fourier Transforms

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  • 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.

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