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Linearity



Theorem: For any $ x,y\in{\bf C}^N$ and $ \alpha,\beta\in{\bf C}$, the DFT satisfies

$\displaystyle \zbox {\alpha x + \beta y \;\longleftrightarrow\;\alpha X + \beta Y}
$

where $ X\isdeftext \hbox{\sc DFT}(x)$ and $ Y\isdeftext \hbox{\sc DFT}(y)$, as always in this book. Thus, the DFT is a linear operator.



Proof:

\begin{eqnarray*}
\hbox{\sc DFT}_k(\alpha x + \beta y) &\isdef & \sum_{n=0}^{N-1...
...n=0}^{N-1}y(n) e^{-j 2\pi nk/N} \\
&\isdef & \alpha X + \beta Y
\end{eqnarray*}


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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]