Question

If x(n) is a real signal, then x(n)=\(\frac{1}{π}\int_0^π\)[XR(ω) cosωn- XI(ω) sinωn] dω.

(a) True

(b) False

I had been asked this question during a job interview.

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

Answer 1

Right answer is (a) True

To explain I would say: We know that if x(n) is a real signal, then xI(n)=0 and xR(n)=x(n)

We know that, xR(n)=x(n)=\(\frac{1}{2π}\int_0^{2π}\)[XR(ω) cosωn- XI(ω) sinωn] dω

Since both XR(ω) cosωn and XI(ω) sinωn are even, x(n) is also even

=> x(n)=\(\frac{1}{π} \int_0^π\)[XR(ω) cosωn- XI(ω) sinωn] dω