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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}[\alpha x(n) + \beta y(n)]e^{-j 2\pi nk/N}\\
&=& \sum_{n=0}^{N-1}\alpha x(n)e^{-j 2\pi nk/N} + \sum_{n=0}^{N-1}\beta y(n) e^{-j 2\pi nk/N} \\
&=& \alpha \sum_{n=0}^{N-1}x(n)e^{-j 2\pi nk/N} + \beta \sum_{n=0}^{N-1}y(n) e^{-j 2\pi nk/N} \\
&\isdef & \alpha X + \beta Y
\end{eqnarray*}


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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