Linearity of the DTFT

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

or

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