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.


\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

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