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Linearity

$\displaystyle \zbox{\alpha x_1 + \beta x_2 \leftrightarrow \alpha X_1 + \beta X_2}
$

or

\begin{eqnarray*}
\hbox{\sc DFT}(\alpha x_1 + \beta x_2) & = & \alpha\cdot \hbox{\sc DFT}(x_1) + \beta \cdot\hbox{\sc DFT}(x_2) \\
\alpha, \beta & \in & \mathbb{C}\\
x_1, x_2, X_1, X_2 & \in & \mathbb{C}^N
\end{eqnarray*}

The Fourier Transform ``commutes with mixing.''


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``Review of the Discrete Fourier Transform (DFT)'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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