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GATE 2024 EC – Question 20

Communications · Random Processes · 1 mark · Multiple choice

A white Gaussian noise $w(t)$ with zero mean and power spectral density $\dfrac{N_0}{2}$, when applied to a first-order RC low pass filter produces an output $n(t)$. At a particular time $t=t_k$, the variance of the random variable $n(t_k)$ is ________.

  1. $\dfrac{N_0}{4RC}$
  2. $\dfrac{N_0}{2RC}$
  3. $\dfrac{N_0}{RC}$
  4. $\dfrac{2N_0}{RC}$

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Correct answer: (A) $\dfrac{N_0}{4RC}$

Explanation

$|H(f)|^2=\dfrac1{1+(2\pi fRC)^2}$ and $\int_{-\infty}^{\infty}|H(f)|^2df=\dfrac1{2RC}$. The output variance is $\dfrac{N_0}{2}\cdot\dfrac1{2RC}=\dfrac{N_0}{4RC}$.