Question

If X(ω) is the Fourier transform of the signal x(n), then what is the Fourier transform of the signal x(n-k)?

(a) e^jωk. X(-ω)

(b) e^jωk. X(ω)

(c) e^-jωk. X(-ω)

(d) e^-jωk. X(ω)

This question was posed to me in an international level competition.

My doubt is from Properties of Fourier Transform  for Discrete Time Signals topic in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right choice is (d) e^-jωk. X(ω)

Explanation: Given

F{x(n)}= X(ω)=\(\sum_{n=-∞}^∞ x(n)e^{-jωn}\)

=>F{x(n-k)}=\(\sum_{n=-∞}^∞ x(n-k)e^{-jωn}=e^{-jωk}.\sum_{n=-∞}^∞ x(n-k)e^{-jω(n-k)}\)

=>F{x(n-k)}= e^-jωk. X(ω)