The GATE Grind

GATE 2022 EE – Question 46

Signals and Systems · Linear time invariant and causal systems · 2 marks · Multiple choice

Let a causal LTI system be governed by the following differential equation

$$y(t)+\frac14\frac{dy}{dt}=2x(t),$$

where $x(t)$ and $y(t)$ are the input and output respectively. Its impulse response is

  1. $2e^{-\frac14t}u(t)$
  2. $2e^{-4t}u(t)$
  3. $8e^{-\frac14t}u(t)$
  4. $8e^{-4t}u(t)$

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Show answer and explanation

Correct answer: (D) $8e^{-4t}u(t)$

Explanation

Multiplying by 4 gives $\frac{dy}{dt}+4y=8x$, so $H(s)=\frac{8}{s+4}$ and the causal impulse response is $h(t)=8e^{-4t}u(t)$.