# Solution {-}

Given the following spectral density function:

\begin{equation*}
  S(j\omega)=\delta(\omega)+\frac{1}{2}\delta(\omega-\omega_0)+\frac{1}{2}\delta(\omega+\omega_0)+\frac{2}{\omega^2+1}
\end{equation*}

1. Find the autocorrelation function

\begin{align*}
  R(\tau)=&\frac{1}{2\pi}+\frac{1}{4\pi}e^{j\omega_0\tau}+\frac{1}{4\pi}e^{-j\omega_0\tau}+e^{-|\tau|}\\
         =&\frac{1}{2\pi}+\frac{1}{4\pi}\cos \omega_0\tau+e^{-|\tau|}
\end{align*}


```python
from sympy import symbols, inverse_fourier_transform, DiracDelta, pi

t, w,  w0 = symbols('tau omega omega_0')

S = DiracDelta(w) + 1/2*DiracDelta(w - w0) + 1/2*DiracDelta(w + w0) + 2/(w**2 + 1)
S
```




$\displaystyle \delta\left(\omega\right) + 0.5 \delta\left(\omega - \omega_{0}\right) + 0.5 \delta\left(\omega + \omega_{0}\right) + \frac{2}{\omega^{2} + 1}$




```python
R = 1/(2*pi)*inverse_fourier_transform(S, w, t)
R
```




$\displaystyle \frac{0.5 e^{2 i \pi \omega_{0} \tau} + 1 + 0.5 e^{- 2 i \pi \omega_{0} \tau} + 2 \pi e^{- 2 \pi \tau}}{2 \pi}$



facf827cd5d13a95672550b66bce27d216f23b41
ipynb
Jupyter Notebook
Problem 2.3.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
2021-09-07T09:36:36.000Z
2021-09-07T09:36:36.000Z
Problem 2.3.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
Problem 2.3.ipynb
mfkiwl/GMPE340
3602b8ba859a2c7db2cab96862472597dc1ac793
MIT
2021-09-20T18:48:20.000Z
2021-09-20T18:48:20.000Z
Qwen/Qwen-72B
1. YES
2. YES