What is the period of the Fourier transform X(ω) of the signal x(n)?(a) π(b) 1(c) Non-periodic(d) 2πThis question was posed to me by my school principal while I was bunking the class.My enquiry is from Frequency Analysis of Discrete Time Signal topic in section Frequency Analysis of Signals and Systems of Digital Signal Processing

What is the period of the Fourier transform X(ω) of the signal x(n)?(

1.	What is the period of the Fourier transform X(ω) of the signal x(n)?(a) π(b) 1(c) Non-periodic(d) 2πThis question was posed to me by my school principal while I was bunking the class.My enquiry is from Frequency Analysis of Discrete Time Signal topic in section Frequency Analysis of Signals and Systems of Digital Signal Processing
Answer» The correct choice is (d) 2π The best I can explain: Let X(ω) be the FOURIER transform of a discrete time signal x(N) which is GIVEN as X(ω)=\(\sum_{n=-∞}^∞x(n)e^{-jωn}\) Now X(ω+2πk)=\(\sum_{n=-∞}^∞ x(n)e^{-j(ω+2πk)n}\) =\(\sum_{n=-∞}^∞ x(n)e^{-jωn}e^{-j2πkn}\) =\(\sum_{n=-∞}^∞ x(n)e^{-jωn}= X(ω)\) So, the Fourier transform of a discrete time FINITE energy signal is periodic with period 2π.

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