GATE EC 2024 Set 1 — Question 20
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Communications → Random Processes → Filtering Through LTI Systems
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Question
A white Gaussian noise with zero mean and power spectral density , when applied to a first-order RC low pass filter produces an output . At a particular time , the variance of the random variable is
Correct answer
(A) (N₀)/(4RC)
Solution
The power spectral density (PSD) of the white Gaussian noise is given as .The transfer function of a first-order RC low pass filter is given by:
The magnitude squared of the transfer function is:
The PSD of the output noise is :
The variance of the random variable at any time is equal to the total average power of the output noise, which is the integral of its PSD over all frequencies:
Let . Then, .
When , .
When , .Substitute these into the integral:
The integral is a standard integral:
Substitute the value of the integral back into the expression for :
Thus, the variance of the random variable is .The final answer is
The magnitude squared of the transfer function is:
The PSD of the output noise is :
The variance of the random variable at any time is equal to the total average power of the output noise, which is the integral of its PSD over all frequencies:
Let . Then, .
When , .
When , .Substitute these into the integral:
The integral is a standard integral:
Substitute the value of the integral back into the expression for :
Thus, the variance of the random variable is .The final answer is
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