| The complex plane may be defined as the graph of all complex numbers extit{z} = extit{x} + extit{j y} formed by using the real part extit{x} as the horizontal coordinate and the imaginary part extit{y} as the vertical coordinate on the graph. In other words, each complex number extit{z} = extit{x} + extit{j y} corresponds to a point in the complex plane located at Cartesian coordinate ( extit{x,y}). — Click for https://ccrma.stanford.edu/~jos/mdft/Complex_Plane.html |
|
are any scalars (real or complex
numbers), 
and 
are any two discrete-time signals
(real- or complex-valued functions of the integers), and 
are their corresponding continuous-frequency spectra defined
over the unit circle in the complex plane.
![\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*}](img124.png)
