Linearity of the DTFT
(3.9) |
or
(3.10) |
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.''
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Existence of the Fourier Transform