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GATE 2017 EE – Question 40

Signals and Systems · Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform · 2 marks · Multiple choice

Let a causal LTI system be characterized by the following differential equation, with initial rest condition

$$\frac{d^2y}{dt^2} + 7\frac{dy}{dt} + 10y(t) = 4x(t) + 5\frac{dx(t)}{dt}$$

where, $x(t)$ and $y(t)$ are the input and output respectively. The impulse response of the system is ($u(t)$ is the unit step function)

  1. $2e^{-2t}u(t) - 7e^{-5t}u(t)$
  2. $-2e^{-2t}u(t) + 7e^{-5t}u(t)$
  3. $7e^{-2t}u(t) - 2e^{-5t}u(t)$
  4. $-7e^{-2t}u(t) + 2e^{-5t}u(t)$

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Correct answer: (B) $-2e^{-2t}u(t) + 7e^{-5t}u(t)$

Explanation

The transfer function is $H(s) = \frac{5s + 4}{s^2 + 7s + 10} = \frac{5s + 4}{(s + 2)(s + 5)}$. The residue at $s = -2$ is $\frac{-10 + 4}{3} = -2$, and at $s = -5$ it is $\frac{-25 + 4}{-3} = 7$. So $h(t) = -2e^{-2t}u(t) + 7e^{-5t}u(t)$.