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GATE 2016 EC – Question 42

Networks, Signals and Systems · Time and Frequency Domain Analysis of Linear Circuits · 2 marks · Multiple choice

A first-order low-pass filter of time constant $T$ is excited with different input signals (with zero initial conditions up to $t = 0$). Match the excitation signals X, Y, Z with the corresponding time responses for $t \geq 0$:

X: Impulse

Y: Unit step

Z: Ramp

P: $1 - e^{-t/T}$

Q: $t - T(1 - e^{-t/T})$

R: $e^{-t/T}$

  1. XR, YQ, ZP
  2. XQ, YP, ZR
  3. XR, YP, ZQ
  4. XP, YR, ZQ

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Show answer and explanation

Correct answer: (C) XR, YP, ZQ

Explanation

The filter is $H(s) = \frac{1}{1 + sT}$. Its impulse response is a decaying exponential $e^{-t/T}$ (up to the constant $\frac{1}{T}$), which is R. Its step response is $1 - e^{-t/T}$, which is P. Its ramp response is the integral of the step response, $t - T(1 - e^{-t/T})$, which is Q. So X goes with R, Y with P and Z with Q.