Your solution is **beautifully typeset** and mathematically correct! Here are a few notes and clarifications, plus a ready-to-use LaTeX document you can copy-paste: --- ## **Review and Minor Suggestions** ### 1. **Average Power of $X(t)$** - Your handling of $R_X(0) = 2\delta(0)$ and $P_X = \infty$ is correct for ideal white noise. ### 2. **PSD of $X(t)$** - $S_X(\omega) = 2$ is correct and properly derived. ### 3. **Output PSD** - Your use of the rectangle function and scaling is correct. - If you wish, you could clarify that $\operatorname{rect}\left(\frac{\omega - \omega_0}{B}\right)$ is 1 for $|\omega-\omega_0|<B/2$ and $0$ elsewhere. ### 4. **Autocorrelation of $Y(t)$** - The inverse FT is handled perfectly. - You could note the use of Euler's formula in the last step. ### 5. **Power Consistency** - The limit evaluation is spot-on. --- ## **Ready-to-Use LaTeX Document** ```latex \documentclass{article} \usepackage{amsmath} \usepackage{amssymb} \begin{document} \section*{Solution Summary} \subsection*{1. Average Power of $X(t)$ Using Autocorrelation} The average power $P_X$ of a random process $X(t)$ is given by: \[ P_X = R_X(0) \] Given: \[ R_X(\tau) = 2\delta(\tau) \implies P_X = R_X(0) = 2\delta(0) \] Since $\delta(0) = \infty$, we have: \[ \boxed{P_X = \infty} \] This indicates that $X(t)$ is \textbf{white noise}, which has infinite power. \subsection*{2. Power Spectral Density (PSD) of $X(t)$} The PSD is the Fourier transform of the autocorrelation function: \[ S_X(\omega) = \mathcal{F}\{R_X(\tau)\} = \mathcal{F}\{2\delta(\tau)\} = 2 \] \[ \boxed{S_X(\omega) = 2} \] This constant PSD confirms that $X(t)$ is a \textbf{white noise process}. \subsection*{3. Average Power of $X(t)$ from PSD} \[ P_X = \frac{1}{2\pi} \int_{-\infty}^{\infty} S_X(\omega) \, d\omega = \frac{1}{2\pi} \int_{-\infty}^{\infty} 2 \, d\omega = \infty \] \[ \boxed{P_X = \infty} \] \subsection*{4. PSD of Output $Y(t)$} Given frequency response: \[ H(\omega) = 3\operatorname{rect}\left(\frac{\omega - 20\pi}{4\pi}\right) + 3\operatorname{rect}\left(\frac{\omega + 20\pi}{4\pi}\right) \] Then: \[ |H(\omega)|^2 = 9\operatorname{rect}\left(\frac{\omega - 20\pi}{4\pi}\right) + 9\operatorname{rect}\left(\frac{\omega + 20\pi}{4\pi}\right) \] So the output PSD is: \[ S_Y(\omega) = S_X(\omega) \cdot |H(\omega)|^2 = 2 \cdot \left[9\operatorname{rect}\left(\frac{\omega - 20\pi}{4\pi}\right) + 9\operatorname{rect}\left(\frac{\omega + 20\pi}{4\pi}\right)\right] \] \[ \boxed{S_Y(\omega) = 18\operatorname{rect}\left(\frac{\omega - 20\pi}{4\pi}\right) + 18\operatorname{rect}\left(\frac{\omega + 20\pi}{4\pi}\right)} \] \subsection*{5. Average Power of $Y(t)$ from PSD} \[ P_Y = \frac{1}{2\pi} \int_{-\infty}^{\infty} S_Y(\omega) \, d\omega \] Each rectangular pulse has height $18$ and width $4\pi$, so: \[ \text{Area per pulse} = 18 \cdot 4\pi = 72\pi \] \[ \text{Total area} = 2 \cdot 72\pi = 144\pi \] \[ P_Y = \frac{1}{2\pi} \cdot 144\pi = \boxed{72} \] \subsection*{6. Autocorrelation Function of $Y(t)$} Use the inverse Fourier transform of $S_Y(\omega)$: \[ R_Y(\tau) = \mathcal{F}^{-1}\left\{18\operatorname{rect}\left(\frac{\omega - 20\pi}{4\pi}\right) + 18\operatorname{rect}\left(\frac{\omega + 20\pi}{4\pi}\right)\right\} \] We use the transform pair: \[ \mathcal{F}^{-1}\left\{A \cdot \operatorname{rect}\left(\frac{\omega - \omega_0}{2W}\right)\right\} = \frac{AW}{\pi} \cdot \text{sinc}\left(\frac{W\tau}{\pi}\right) \cdot e^{j\omega_0\tau} \] Given $A = 18$, $W = 2\pi$, and $\omega_0 = \pm 20\pi$: \[ R_Y(\tau) = 18 \cdot \frac{\sin(2\pi\tau)}{\pi\tau} \left(e^{j20\pi\tau} + e^{-j20\pi\tau} \right) \] \[ = \frac{36 \cos(20\pi\tau) \sin(2\pi\tau)}{\pi\tau} \] \[ \boxed{R_Y(\tau) = \frac{36 \cos(20\pi\tau) \sin(2\pi\tau)}{\pi\tau}} \] \subsection*{7. Average Power of $Y(t)$ from Autocorrelation} \[ P_Y = R_Y(0) = \lim_{\tau \to 0} \frac{36 \cos(20\pi\tau) \sin(2\pi\tau)}{\pi\tau} \] Using limits: \[ \lim_{\tau \to 0} \cos(20\pi\tau) = 1, \quad \lim_{\tau \to 0} \frac{\sin(2\pi\tau)}{\tau} = 2\pi \] \[ P_Y = \frac{36}{\pi} \cdot 2\pi = \boxed{72} \] This matches the power from the PSD, verifying consistency. \end{document} ``` --- **You can copy, paste, and compile this code to get a clean, publication-ready PDF!** If you want a PDF output, just let me know.