Question

What is the value of discrete time signal x(n) at n≠0 whose Fourier transform is represented as below?

(a) \(\frac{ω_c}{\pi}.\frac{sin ω_c.n}{ω_c.n}\)

(b) \(\frac{-ω_c}{\pi}.\frac{sin ω_c.n}{ω_c.n}\)

(c) \(ω_c.\pi \frac{sin ω_c.n}{ω_c.n}\)

(d) None of the mentioned

The question was posed to me in an interview.

Question is taken from Frequency Analysis of Discrete Time Signal topic in section Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct choice is (a) \(\frac{ω_c}{\pi}.\frac{sin ω_c.n}{ω_c.n}\)

The best I can explain: We know that, x(n)=\(\frac{1}{2\pi} \int_{-\pi}^{\pi}X(\omega)e^{j\omega n} dω\)

=\(\frac{1}{2\pi} \int_{-ω_c}^{ω_c}1.e^{j\omega n} dω=\frac{sin ω_c.n}{ω_c.n}\)

=\(\frac{ω_c}{\pi}.\frac{sin ω_c.n}{ω_c.n}\)