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A random noise X(t) having a power spectrum

Is applied to a differentiator that has a transfer function H(ω) = jω. The output is applied to a network for which h(t) = t2 e-7t u(t) 
The power spectrum of Y(t) is
  • a)
  • b)
  • c)
  • d)
    None of the above
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
A random noise X(t) having a power spectrumIs applied to a differentia...
For a system with given input power spectral density, and impulse response, the output spectral density is given by:
SY(ω) = SX(ω) H(ω)2
SX(ω) = Input spectral density
SY(ω) = Output spectral density
H(ω) = Impulse response of the system
Analysis:
Given the power spectrum of input as:

As this passes through the differentiator the output PSD becomes:

The transfer function for the second system is:

So, we obtain the transfer function in the frequency domain as:

Therefore, the final output spectrum is given by:
SY(ω) = SY1(ω) H(ω)2
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A random noise X(t) having a power spectrumIs applied to a differentiator that has a transfer function H(ω) = jω. The output is applied to a network for which h(t) = t2 e-7t u(t)The power spectrum of Y(t) isa)b)c)d)None of the aboveCorrect answer is option 'B'. Can you explain this answer?
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