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

Answer 1

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)]