Editing Discrete Fourier transform (section)

==Some discrete Fourier transform pairs==

{| class="wikitable" style="text-align: center;"
|+ '''Some DFT pairs'''
|-
! <math>x_n = \frac{1}{N}\sum_{k=0}^{N-1}X_k  e^{i 2 \pi kn/N} </math>
! <math>X_k = \sum_{n=0}^{N-1}x_n  e^{-i 2 \pi kn/N} </math>
! Note
|-
| <math>x_n e^{i 2 \pi n\ell/N} \,</math>
| <math>X_{k-\ell}\,</math>
| Frequency shift theorem
|-
| <math>x_{n-\ell}\,</math>
| <math>X_k  e^{-i 2 \pi k\ell/N} \,</math>
| Time shift theorem
|-
| <math>x_n \in \mathbb{R}</math>
| <math>X_k=X_{N-k}^*\,</math>
| Real DFT
|-
| <math>a^n\,</math>
| <math>\left\{ \begin{matrix}
                   N & \mbox{if } a = e^{i 2 \pi k/N} \\
                   \frac{1-a^N}{1-a \, e^{-i 2 \pi k/N} } & \mbox{otherwise}
                \end{matrix} \right. </math>
| from the [[geometric progression]] formula
|-
| <math>{N-1 \choose n}\,</math>
| <math>\left(1+e^{-i 2 \pi k/N} \right)^{N-1}\,</math>
| from the [[binomial theorem]]
|-
| <math>\left\{ \begin{matrix}
                        \frac{1}{W} & \mbox{if } 2n < W \mbox{ or } 2(N-n) < W \\
                        0 & \mbox{otherwise}
                      \end{matrix} \right. </math>
| <math>\left\{ \begin{matrix}
              1 & \mbox{if } k = 0 \\
              \frac{\sin\left(\frac{\pi W k}{N}\right)}
                   {W \sin\left(\frac{\pi k}{N}\right)} & \mbox{otherwise}
                      \end{matrix} \right. </math>
| <math>x_n</math> is a rectangular [[window function]] of ''W'' points centered on ''n''=0, where ''W'' is an [[odd integer]], and <math>X_k</math> is a [[sinc]]-like function (specifically, <math>X_k</math> is a [[Dirichlet kernel]])
|-
| <math>\sum_{j\in\mathbb{Z}} \exp\left(-\frac{\pi}{cN}\cdot(n+N\cdot j)^2\right)</math>
| <math>\sqrt{cN} \cdot \sum_{j\in\mathbb{Z}} \exp\left(-\frac{\pi c}{N}\cdot(k+N\cdot j)^2\right)</math>
| [[Discretization]] and [[periodic summation]] of the scaled [[Gaussian function]]s for <math>c>0</math>. Since either <math>c</math> or <math>\frac{1}{c}</math> is larger than one and thus warrants fast convergence of one of the two series, for large <math>c</math> you may choose to compute the frequency spectrum and convert to the time domain using the discrete Fourier transform.
|}