Question

Which of the following relations are true if x(n) is real?

(a) X(ω)=X(-ω)

(b) X(ω)=-X(-ω)

(c) X*(ω)=X(ω)

(d) X*(ω)=X(-ω)

I got this question in an interview.

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

Answer 1

Correct choice is (d) X*(ω)=X(-ω)

For explanation: We know that, if x(n) is a real sequence

XR(ω)=\(\sum_{n=-∞}^∞\) x(n)cosωn=>XR(-ω)= XR(ω)

XI(ω)=-\(\sum_{n=-∞}^∞\) x(n)sin⁡(ωn)=>XI(-ω)=-XI(ω)

If we combine the above two equations, we get

X*(ω)=X(-ω)