The Fourier transform of discrete time signal *x*(*n*) is denoted by *X*(*ω*) and it is given by

where the frequency range of *ω* is from −*π* to *π* or equivalently 0 to 2*π*. *X*(*ω*) is periodic with period 2*π* which can be verified as follows:

We know that

where both *k* and *n* are integers. Since cos 2*πnk* = 1 and sin 2*πnk* = 0, the value of *e*^{−j2πnk} becomes 1. Hence we can write Eq. (11.2) as

Therefore, the Fourier transform of discrete time signal *x*

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