Question

If x(n) is a real and odd sequence, then what is the expression for x(n)?

(a) \(\frac{1}{π} \int_0^π\)[XI(ω) sinωn] dω

(b) –\(\frac{1}{π} \int_0^π\)[XI(ω) sinωn] dω

(c) \(\frac{1}{π} \int_0^π\)[XI(ω) cosωn] dω

(d) –\(\frac{1}{π} \int_0^π\)[XI(ω) cosωn] dω

I got this question during an online interview.

My query is from Properties of Fourier Transform  for Discrete Time Signals in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

The correct choice is (b) –\(\frac{1}{π} \int_0^π\)[XI(ω) sinωn] dω

The explanation: If x(n) is real and odd then, x(n)cosωn is odd and x(n) sinωn is even. Consequently

XR(ω)=0

XI(ω)=\(-2\sum_{n=1}^∞ x(n) sin⁡ωn\)

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