GATE 2023 EC – Question 59
Let $X(t)$ be a white Gaussian noise with power spectral density $\frac12$ W/Hz. If $X(t)$ is input to an LTI system with impulse response $e^{-t}u(t)$. The average power of the system output is ________ W (rounded off to two decimal places).
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 0.24 to 0.26
Explanation
$H(f)=\dfrac1{1+j2\pi f}$ and $\int_{-\infty}^\infty|H(f)|^2df=\int_{-\infty}^\infty\dfrac{df}{1+4\pi^2f^2}=\dfrac12$. The output power is $S_0\int|H|^2df=\dfrac12\times\dfrac12=0.25$ W. (Equivalently $\frac12\int_0^\infty e^{-2t}dt\cdot$ with unit-normalisation of the two-sided PSD.)