As a special case of general Fourier transform, the discrete time transform shares all properties (and their proofs) of the Fourier transform discussed above, except now some of these properties may take different forms. In the following, we always assume and .
Differencing is the discrete-time counterpart of differentiation.
The convolution theorem states that convolution in time domain corresponds to
multiplication in frequency domain and vice versa:
Proof of (a):
Proof of (b):