Linearity of the DTFT

$\displaystyle \zbox {\alpha x_1 + \beta x_2 \;\longleftrightarrow\;\alpha X_1 + \beta X_2}$ (3.9)

or

$\displaystyle \hbox{\sc DTFT}(\alpha x_1 + \beta x_2) = \alpha\cdot \hbox{\sc DTFT}(x_1) + \beta \cdot\hbox{\sc DTFT}(x_2)$ (3.10)

where $ \alpha, \beta$ are any scalars (real or complex numbers), $ x_1$ and $ x_2$ are any two discrete-time signals (real- or complex-valued functions of the integers), and $ X_1, X_2$ are their corresponding continuous-frequency spectra defined over the unit circle in the complex plane.


Proof: We have

\begin{eqnarray*}
\hbox{\sc DTFT}_\omega(\alpha x_1 + \beta x_2)
& \isdef & \sum_{n=-\infty}^{\infty}[\alpha x_1(n) + \beta x_2(n)]e^{-j\omega n}\\
&=& \alpha\sum_{n=-\infty}^{\infty}x_1(n)e^{-j\omega n} + \beta \sum_{n=-\infty}^{\infty}x_2(n)e^{-j\omega n}\\
&\isdef & \alpha X_1(\omega) + \beta X_2(\omega)
\end{eqnarray*}

One way to describe the linearity property is to observe that the Fourier transform ``commutes with mixing.''


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Existence of the Fourier Transform