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 .

**Linearity**

**Time Shifting**

**Proof:**

If we let , the above becomes

**Time Reversal**

**Frequency Shifting**

**Differencing**Differencing is the discrete-time counterpart of differentiation.

**Proof:**

**Differentiation in frequency**

**proof:**Differentiating the definition of discrete Fourier transform with respect to , we get

**Convolution Theorems**The convolution theorem states that convolution in time domain corresponds to multiplication in frequency domain and vice versa:

Recall that the convolution of periodic signals and is

Here the convolution of periodic spectra and is similarly defined as

**Proof of (a):**

**Proof of (b):**

**Parseval's Relation**