Noise - HackMD

# Noise - Let $N(u,t)$ be a bandpass WSS random process whose autocorrelation function $R_{N}(\tau)$ is bandpass, i.e. $$ S_{N}(f)=0 \textrm{ for all }\left\lvert f \right\rvert\notin \left[ f_{c}-W,f_{c}+W \right] $$ - Let $N_{l}(u,t)=N_{i}(u,t)+iN_{q}(u,t)$ be the lowpass equivalent of $N(u, t)$ such that $$ \begin{aligned} N(u,t) &= \Re\left\{N_{l}(u,t)e^{i2\pi f_{c}t}\right\}\\ &=N_{i}(u,t)\cos(2\pi f_{c}t)-N_{q}(u,t)\sin(2\pi f_{c}t) \end{aligned} $$ - Our Goal is to show the following result. :::success #### Theorem 1 $$ S_{N}(f)=\frac{1}{4}\left[S_{N_{l}}(f-f_{c})+S_{N_{l}}(-f-f_{c})\right] $$ ::: - Recall that $$ N_{l}(u,t)=\left[N(u,t)+i\hat{N}(u,t)\right]e^{-i2\pi f_{c}t} $$ where $\hat{N}(u,t)=N(u,t)*\frac{1}{\pi t}$ - Since $N(u,t)$ is WSS and $h(t)=\frac{1}{\pi t}$ is LTI, $\hat{N}(u,t)$ is WSS by following theorem :::success #### Theorem 2 Let $X(u,t)$ be a WSS random process and let $\mathcal{H}$ be an LTI system with impulse response $h(t)$; then $$ y(u,t) = \mathcal{H}\left\{x(u,t)\right\}=\int h(t-s)x(u,s)ds $$ $$ R_{yx}(\tau) = h(\tau)*R_{x}(\tau) $$ $$ R_{xy}(\tau) = R_{x}(\tau)*h^{*}(-\tau) $$ $$ R_{y}(\tau) = h(\tau)*R_{x}(\tau)*h^{*}(-\tau) $$ and $$ S_{Y}(f)=\left\lvert H(f)\right\rvert^{2}S_{X}(f) $$ Moreover, $y(u,t)$ exist iff $\left\lvert R_{y}(\tau)\right\rvert<\infty$ ::: Hence :::danger $$ \begin{aligned}N_{+}(u,t)=\frac{1}{2}\left[N(u,t)+i\hat{N}(u,t)\right] \quad\textrm{is WSS}\end{aligned} $$ ::: Next, note $$\begin{aligned} &R_{N_{l}}(t+\tau,t)\\ &=\mathbb{E}\left\{\underbrace{\left[N(u,t+\tau)+i\hat{N}(u,t+\tau)\right]e^{-i2\pi f_{c}(t+\tau)}}_{N_{l}(u,t+\tau)}\times \underbrace{\left[N(u,t)+i\hat{N}(u,t)\right]e^{-i2\pi f_{c}(t)}}_{N_{l}(u,t)}\right\}\\ &=e^{-i2\pi f_{c}\tau}\left[R_{N}(\tau)+R_{\hat{N}}(\tau)-iR_{N\hat{N}}(\tau)+iR_{N\hat{N}}(-\tau)\right] \end{aligned} $$ :::success #### Theorem 3 $$ \begin{aligned}N_{l}(u,t)=N_{i}(u,t)+iN_{q}(u,t) \quad\textrm{is WSS}\end{aligned} $$ ::: ###### tags: `Digital Communication` `Book`