Editing Discrete Fourier transform (section)

=== Expressing the inverse DFT in terms of the DFT ===
A useful property of the DFT is that the inverse DFT can be easily expressed in terms of the (forward) DFT, via several well-known "tricks".  (For example, in computations, it is often convenient to only implement a fast Fourier transform corresponding to one transform direction and then to get the other transform direction from the first.)

First, we can compute the inverse DFT by reversing all but one of the inputs (Duhamel ''et al.'', 1988):

:<math>\mathcal{F}^{-1}(\{x_n\}) = \frac{1}{N}\mathcal{F}(\{x_{N - n}\})</math>

(As usual, the subscripts are interpreted [[modular arithmetic|modulo]] ''N''; thus, for <math>n = 0</math>, we have <math>x_{N-0} = x_0</math>.)

Second, one can also conjugate the inputs and outputs:

:<math>\mathcal{F}^{-1}(\mathbf{x}) = \frac{1}{N}\mathcal{F}\left(\mathbf{x}^*\right)^*</math>

Third, a variant of this conjugation trick, which is sometimes preferable because it requires no modification of the data values, involves swapping real and imaginary parts (which can be done on a computer simply by modifying [[pointer (computer programming)|pointer]]s). Define <math display="inline">\operatorname{swap}(x_n)</math> as <math>x_n</math> with its real and imaginary parts swapped—that is, if <math>x_n = a + b i</math> then <math display="inline">\operatorname{swap}(x_n)</math> is <math>b + a i</math>.  Equivalently, <math display="inline">\operatorname{swap}(x_n)</math> equals <math>i x_n^*</math>.  Then

:<math>\mathcal{F}^{-1}(\mathbf{x}) = \frac{1}{N}\operatorname{swap}(\mathcal{F}(\operatorname{swap}(\mathbf{x})))</math>

That is, the inverse transform is the same as the forward transform with the real and imaginary parts swapped for both input and output, up to a normalization (Duhamel ''et al.'', 1988).

The conjugation trick can also be used to define a new transform, closely related to the DFT, that is [[Involution (mathematics)|involutory]]—that is, which is its own inverse.  In particular, <math>T(\mathbf{x}) = \mathcal{F}\left(\mathbf{x}^*\right) / \sqrt{N}</math> is clearly its own inverse: <math>T(T(\mathbf{x})) = \mathbf{x}</math>.  A closely related involutory transformation (by a factor of <math display=inline>\frac{1 + i}{\sqrt{2}}</math>) is <math>H(\mathbf{x}) = \mathcal{F}\left((1 + i) \mathbf{x}^*\right) / \sqrt{2N}</math>, since the <math>(1 + i)</math> factors in <math>H(H(\mathbf{x}))</math> cancel the 2.  For real inputs <math>\mathbf{x}</math>, the real part of <math>H(\mathbf{x})</math> is none other than the [[discrete Hartley transform]], which is also involutory.