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Signals and Systems by Smarajit Ghosh

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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

 

image

 

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:

 

image

 

We know that

 

ej2πnk = cos 2πnkjsin 2πnk     (11.3)

 

where both k and n are integers. Since cos 2πnk = 1 and sin 2πnk = 0, the value of ej2πnk becomes 1. Hence we can write Eq. (11.2) as

 

 

Therefore, the Fourier transform of discrete time signal x

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