GATE 2023 EE Q.17: solution and answer

GATE 2023 · EEQ.17 · 1 mark · MCQ

A continuous-time system that is initially at rest is described by
$\frac{d y ( t )}{d t} + 3 y (t) = 2 x (t),$
where $x (t)$ is the input voltage and $y (t)$ is the output voltage. The impulse response of the system is

(A)
$3 e^{- 2 t}$
(B)
$\frac{1}{3} e^{- 2 t} u (t)$
(C)
$2 e^{- 3 t} u (t)$
(D)
$2 e^{- 3 t}$

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Answer and solution

ANSWER

The impulse response is $h (t) = 2 e^{- 3 t} u (t)$ , option (C).

Official answer: CSource: Official answer key, GATE 2023 EE (IIT Kanpur)

Laplace transform with zero initial conditions. The system is initially at rest, so $y (0^{-}) = 0$ and $\frac{d y}{d t} \to s Y (s)$ : $s Y (s) + 3 Y (s) = 2 X (s)$
Transfer function. $H (s) = \frac{Y ( s )}{X ( s )} = \frac{2}{s + 3}$
Impulse response = inverse Laplace of $H (s)$ . The pair $\frac{1}{s + a} \leftrightarrow e^{- a t} u (t)$ with $a = 3$ gives $h (t) = 2 e^{- 3 t} u (t)$
Why the $u (t)$ matters. A system described by a linear constant-coefficient differential equation and initially at rest is causal, so $h (t) = 0$ for $t < 0$ . Options (A) and (D) have no $u (t)$ : they are non-zero for $t < 0$ and grow without bound as $t \to - \infty$ , so neither can be the answer. Option (D) has the right exponential and gain but is two-sided; options (A) and (B) have the wrong pole ( $s = - 2$ instead of $s = - 3$ ).
Check by substitution. Differentiate $h (t) = 2 e^{- 3 t} u (t)$ with the product rule, using $e^{- 3 t} δ (t) = δ (t)$ : $\frac{d h}{d t} = - 6 e^{- 3 t} u (t) + 2 δ (t)$ Then $\frac{d h}{d t} + 3 h (t) = - 6 e^{- 3 t} u (t) + 2 δ (t) + 6 e^{- 3 t} u (t) = 2 δ (t),$ which is exactly the right-hand side $2 x (t)$ with $x (t) = δ (t)$ . So the answer is option (C).

KEY IDEA

For a first-order system $\frac{d y}{d t} + a y = b x$ that starts at rest, $H (s) = \frac{b}{s + a}$ , so $h (t) = b e^{- a t} u (t)$ . The coefficient of $y$ sets the decay rate, the coefficient of $x$ sets the gain, and starting at rest (causality) supplies the $u (t)$ .

WATCH OUT FOR

Picking (D), $2 e^{- 3 t}$ , because the exponent and gain are right, and forgetting that a system starting at rest is causal, so its impulse response must carry $u (t)$ . The other common slip is swapping the coefficients, which gives the $e^{- 2 t}$ terms in (A) and (B).

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GATE 2023 EE Q.17: solution and answer

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