Editing Discrete Fourier transform (section)

==Example==
<!-- This illustration needs a better description understand: [[Image:Dft visualization rev2 n0008 trimmed nobox.svg|thumb|300px|Depiction of the matrix of the DFT for N=8. Each element is represented by a picture of its location in the complex plane in relation to the unit circle.]] -->

This example demonstrates how to apply the DFT to a sequence of length <math>N = 4</math> and the input vector

<math display="block">\mathbf{x} =
\begin{pmatrix}
x_0 \\ x_1 \\ x_2 \\ x_3
\end{pmatrix}
=
\begin{pmatrix}
1 \\ 2-i \\ -i \\ -1+2i
\end{pmatrix}.
</math>

Calculating the DFT of <math>\mathbf{x}</math> using {{EquationNote|Eq.1}}

<math display="block">\begin{align}
X_0 &= e^{-i 2 \pi 0 \cdot 0 / 4} \cdot 1 + e^{-i 2 \pi 0 \cdot 1 / 4} \cdot (2-i) + e^{-i 2 \pi 0 \cdot 2 / 4} \cdot (-i) + e^{-i 2 \pi 0 \cdot 3 / 4} \cdot (-1+2i) = 2 \\
X_1 &= e^{-i 2 \pi 1 \cdot 0 / 4} \cdot 1 + e^{-i 2 \pi 1 \cdot 1 / 4} \cdot (2-i) + e^{-i 2 \pi 1 \cdot 2 / 4} \cdot (-i) + e^{-i 2 \pi 1 \cdot 3 / 4} \cdot (-1+2i) = -2-2i \\
X_2 &= e^{-i 2 \pi 2 \cdot 0 / 4} \cdot 1 + e^{-i 2 \pi 2 \cdot 1 / 4} \cdot (2-i) + e^{-i 2 \pi 2 \cdot 2 / 4} \cdot (-i) + e^{-i 2 \pi 2 \cdot 3 / 4} \cdot (-1+2i) = -2i \\
X_3 &= e^{-i 2 \pi 3 \cdot 0 / 4} \cdot 1 + e^{-i 2 \pi 3 \cdot 1 / 4} \cdot (2-i) + e^{-i 2 \pi 3 \cdot 2 / 4} \cdot (-i) + e^{-i 2 \pi 3 \cdot 3 / 4} \cdot (-1+2i) = 4+4i
\end{align}</math>

results in
<math display="block">\mathbf{X} =
\begin{pmatrix}
X_0 \\
X_1 \\
X_2 \\
X_3
\end{pmatrix}
=
\begin{pmatrix}
2 \\
-2-2i \\
-2i \\
4+4i
\end{pmatrix}.
</math>