Question

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

(a) ωc.π

(b) -ωc/π

(c) ωc/π

(d) none of the mentioned

I had been asked this question during an interview.

I'm obligated to ask this question of Frequency Analysis of Discrete Time Signal in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct option is (c) ωc/π

For explanation: We know that, x(n)=\(\frac{1}{2\pi} \int_{-\pi}^{\pi}X(\omega)e^{j\omega n} dω\)

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

At n=0,

x(n)=x(0)=\(\int_{-ω_c}^{ω_c}1 dω=\frac{1}{2\pi}(2 ω_c)=\frac{ω_c}{\pi_ω}\)

Therefore, the value of the signal x(n) at n=0 is ωc/π.