DFT Properties(cont.)

•  Parseval's theorem:

With special case for x=y, the "energy balance formula," since the left-hand side then becomes the energy of the signal

•  Symmetry properties:

(a)  Conjugation:

(b) Arguments reversed (modulo reflection through origin):

(c) Real-valued sequences (special case):

By the conjugation property above, applying it to a real-valued sequence , we have the conjugate symmetry property

	(4.2.4)

From this equation, the following four properties follow easily:

i) is even, i.e.,

ii)  Im is odd, i.e.

Im .

iii) is even, i.e.,

iv) arg is odd, i.e.,

arg

These last properties are used for reducing required data storage for the DFT by an approximate factor of ½ in the real-valued image case.