## Solution {-}

\begin{equation}
  S_X(j\omega) = \frac{1}{(\omega^2 + 1)^2}
\end{equation}

Rewrite PSD in s-domain:

\begin{equation}
  S_X(s) = \frac{1}{(-s^2 + 1)^2}
\end{equation}

PSD can be written as:

\begin{equation}
  S_X = \frac{As + B}{(s+1)^2} + \frac{Cs + D}{(-s + 1)^2}
\end{equation}

Rewrite PSD again as partial fractions:

\begin{equation}
  S_X = \frac{C_1}{(s+1)^2} + \frac{C_2}{(s + 1)} + \frac{C_3}{(-s + 1)^2} + \frac{C_4}{(-s + 1)}
\end{equation}

Solve positive time part, i.e. $C_1$ and $C_2$:

\begin{align}
  C_1 =& \frac{1}{(s + 1)^2 (-s + 1)^2} (s + 1)^2 = \frac{1}{4}\\
  C_2 =& \frac{d}{ds} [\frac{1}{(s + 1)^2 (-s + 1)^2} (s + 1)^2] = \frac{1}{4}\\
\end{align}

Autocorrelation function $X(t)$:

\begin{equation}
  R_X(\tau) = \frac{1}{4}|\tau| e^{-|\tau|} + \frac{1}{4} e^{-|\tau|}
\end{equation}


```python
from sympy import symbols, inverse_fourier_transform

w, t = symbols('omega tau')

# Given Spectral Density Function
Sx = 1/(1 + w**2)**2

Rx = inverse_fourier_transform(Sx, w, t)
Rx
```




$\displaystyle - \frac{\pi \left(2 \pi \tau \sinh{\left(2 \pi \tau \right)} - 2 \pi \tau \cosh{\left(2 \pi \tau \right)} + \sinh{\left(2 \pi \tau \right)} - \cosh{\left(2 \pi \tau \right)}\right)}{2}$



4852cee7a6b676cf61ef30af113940a8dbf31ad7
ipynb
Jupyter Notebook
Problem 2.16.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
2021-09-07T09:36:36.000Z
2021-09-07T09:36:36.000Z
Problem 2.16.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
Problem 2.16.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
2021-09-20T18:48:20.000Z
2021-09-20T18:48:20.000Z
Qwen/Qwen-72B
1. YES
2. YES