Linearity

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

$$\zbox {\alpha x + \beta y \leftrightarrow \alpha X + \beta Y}$$

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:

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