Signals and Systems

11.1 FOURIER TRANSFORM OF DISCRETE TIME SIGNALS

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 2π. X(ω) is periodic with period 2π which can be verified as follows:

 

 

We know that

 

e^–j2πnk = cos 2πnk – jsin 2πnk     (11.3)

 

where both k and n are integers. Since cos 2πnk = 1 and sin 2πnk = 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