Question

If x(n)=xR(n)+jxI(n) is a complex sequence whose Fourier transform is given as X(ω)=XR(ω)+jXI(ω), then what is the value of XR(ω)?

(a) \(\sum_{n=0}^∞\)xR (n)cosωn-xI (n)sinωn

(b) \(\sum_{n=0}^∞\)xR (n)cosωn+xI (n)sinωn

(c) \(\sum_{n=-∞}^∞\)xR (n)cosωn+xI (n)sinωn

(d) \(\sum_{n=-∞}^∞\)xR (n)cosωn-xI (n)sinωn

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

I would like to ask this question from Properties of Fourier Transform  for Discrete Time Signals in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct choice is (c) \(\sum_{n=-∞}^∞\)xR (n)cosωn+xI (n)sinωn

Explanation: We know that X(ω)=\(\sum_{n=-∞}^∞\) x(n)e^-jωn

By substituting e^-jω = cosω – jsinω in the above equation and separating the real and imaginary parts we get

XR(ω)=\(\sum_{n=-∞}^∞\)xR (n)cosωn+xI (n)sinωn