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Consider an LTI system with impulse response h(t) = e⁻⁵t u(t). If the output of the system is y(t) = e⁻³t u(t) - e⁻⁵t u(t) then the input, x(t), is given by

1. e⁻³t u(t)
2. 2 e⁻³t u(t)
3. e⁻⁵t u(t)
4. 2 e⁻⁵t u(t)

Correct answer: 2 e⁻³t u(t)

Solution

H(s) = 1/(s+5). Y(s) = 1/(s+3) - 1/(s+5) = 2/[(s+3)(s+5)]. So X(s) = Y(s)/H(s) = 2(s+5)/[(s+3)(s+5)] = 2/(s+3), giving x(t) = 2 e^(-3t) u(t).

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