December 2005
Intermediate to advanced
475 pages
12h 6m
English
Content preview from Signals and Systems
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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
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
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