Linearity of the DTFT
where 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.
Proof: We have
One way to describe the linearity property is to observe that the Fourier transform ``commutes with mixing.''
Existence of the Fourier Transform