---
# System prepended metadata

title: Noise
tags: [Book, Digital Communication]

---

# Noise

- Let <font color=#FF0000>$N(u,t)$ be a bandpass WSS random process</font> whose autocorrelation function <font color=#FF0000>$R_{N}(\tau)$ is bandpass</font>, i.e.
$$
S_{N}(f)=0    \textrm{ for all }\left\lvert f \right\rvert\notin \left[ f_{c}-W,f_{c}+W \right]
$$
- Let <font color=#FF0000>$N_{l}(u,t)=N_{i}(u,t)+iN_{q}(u,t)$</font> 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`
