Question

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

(a) π

(b) 1

(c) Non-periodic

(d) 2π

This 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 1

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.