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

if-x-n-coso0n-and-w-o-is-the-fourier-transform-of-the-rectangular

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

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)]