Question

If x(n) is a real sequence, then what is the value of XI(ω)?

(a) \(\sum_{n=-∞}^∞ x(n)sin⁡(ωn)\)

(b) –\(\sum_{n=-∞}^∞ x(n)sin⁡(ωn)\)

(c) \(\sum_{n=-∞}^∞ x(n)cos⁡(ωn)\)

(d) –\(\sum_{n=-∞}^∞ x(n)cos⁡(ωn)\)

The question was asked by my college director while I was bunking the class.

This interesting question is from Properties of Fourier Transform  for Discrete Time Signals in portion Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right choice is (b) –\(\sum_{n=-∞}^∞ x(n)sin⁡(ωn)\)

The best explanation: If the signal x(n) is real, then xI(n)=0

We know that,

XI(ω)=\(\sum_{n=-∞}^∞ x_R (n)sinωn-x_I (n)cosωn\)

Now substitute xI(n)=0 in the above equation=>xR(n)=x(n)

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