Question

An input x(t) = exp(-2t) u(t) + δ(1 - 6) is applied to an LTI system with impulse response h(t) = u(t). The output is

1.

[1 - exp(-2t) u(t)] + u(t + 6)

2.

[1 - exp(-2t)] u(t) + u(t + 6)

3.

0.5[1 - exp(-2t)] u(t) + u(t + 6)

4.

0.5[1 - exp(-2t)] u(t) + u(t - 6)

5.

Answer 1

Correct Answer - Option 4 :

0.5[1 - exp(-2t)] u(t) + u(t - 6)

\(X(s) = \frac{1}{s + 2} + e^{-6s} \text {and H}(s) = \frac{1}{s}\)

\(Y(s) = H(s) \times (s) = \frac{1}{s(s+2)}+\frac{e^{-6s}}{s}=\frac{1}{2}\frac{1}{s}-\frac{1}{2(s+2)}+\frac{e^{-6s}}{s}\)

⇒ y(t) = 0.5(1 - e^-2t) u(t) + u(t - 6)