DFT Properties

The properties of the DFT are similar to those of the discrete Fourier series. The key difference is that the support of the sequence x and of the DFT X is finite. We consider two sequences x and y with the same rectangular support, having DFT transforms X and Y, respectively. We then offer proofs of some of these properties below.

•  Linearity:

when both sequences have the same support

•  Circular convolution :

We define circular convolution for two finite support sequences with the same period as

	(4.2.3)

Using the operator symbol . We then the following transform pair:

.

•  Multiplication:

.

•  Separability:

,

the separable product of a 1-D N₁ -point and N₂ -point DFT.

•  Circular shifting:

where the shift vector is integer valued.