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 xI(n)?

(a) \(\frac{1}{2π} \int_0^{2π}\)[XR(ω) sinωn+ XI(ω) cosωn] dω

(b) \(\int_0^{2π}\)[XR(ω) sinωn+ XI(ω) cosωn] dω

(c) \(\frac{1}{2π} \int_0^{2π}\)[XR(ω) sinωn – XI(ω) cosωn] dω

(d) None of the mentioned

This question was addressed to me at a job interview.

My question comes from Properties of Fourier Transform  for Discrete Time Signals topic in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right option is (a) \(\frac{1}{2π} \int_0^{2π}\)[XR(ω) sinωn+ XI(ω) cosωn] dω

Explanation: We know that the inverse transform or the synthesis equation of a signal x(n) is given as

x(n)=\(\frac{1}{2π} \int_0^{2π}\) X(ω)e^jωn dω

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

xI(n)=\(\frac{1}{2π} \int_0^{2π}\)[XR(ω) sinωn+ XI(ω) cosωn] dω