Linearity of the DTFT
![]() |
(3.9) |
or
![]() |
(3.10) |
where




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*}](http://www.dsprelated.com/josimages_new/sasp2/img126.png)
One way to describe the linearity property is to observe that the Fourier transform ``commutes with mixing.''
Next Section:
Time Reversal
Previous Section:
Existence of the Fourier Transform