If x(n)=cosω0n and W(ω) is the Fourier transform of the rectangular

Question

If x(n)=cosω0n and W(ω) is the Fourier transform of the rectangular signal w(n), then what is the Fourier transform of the signal x(n).w(n)? A. 1/2[W(ω-ω0)- W(ω+ω0)] B. 1/2[W(ω-ω0)+ W(ω+ω0)] C. [W(ω-ω0)+ W(ω+ω0)] D. [W(ω-ω0)- W(ω+ω0)] This question was addressed to me in a national level competition. The origin of the question is Frequency Analysis of Signals Using DFT in section Discrete Fourier Transform – Properties and Applications of Digital Signal Processing

fahmy · Accepted Answer

Correct answer is B. 1/2[W(ω-ω0)+ W(ω+ω0)] Best explanation: According to the exponential properties of Fourier transform, we get Fourier transform of x(n).w(n)=  1/2[W(ω-ω0)+ W(ω+ω0)]