White Gaussian Noise

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Continous: We define continuous white noise as the derivative of the Wiener Process

$$n(t)=\underset{\Delta \to 0}{\lim }\frac{b(t+\Delta )-b(t)}{\Delta }$$

• What is Var$\left[{\int }_0^{s}n(t)dt\right]$ and Var$[n(t)]$?

We start by approximating the integral as

$$S=\sum _{k=0}^{t/\Delta }\Delta n(k\Delta )\approx {\int }_0^{t}n(\tau )d\tau$$

The variance of $S$ is then given by

$$\begin{array}{rl}Var[S]& =Var\left[\sum _{k=0}^{t/\Delta }\Delta n(k\Delta )\right]={\Delta }^{2}Var\left[\sum _{k=0}^{t/\Delta }n(k\Delta )\right]\\ & ={\Delta }^2\left(\frac{t}{\Delta }\right)Var[n(t)]\end{array}$$

Solving for $Var[n(t)]$ finally yields

$$Var[n(t)]=\frac{1}{t\Delta }Var[S]$$

If $S$ is a Wiener Process, then its variance is given by $t$ when $\Delta \to 0$. However this implies that the variance of $n$ would go to infinity.

$$Var[n(t)]=\underset{\Delta \to 0}{\lim }\frac{1}{t\Delta }t=\underset{\Delta \to 0}{\lim }\frac{1}{\Delta }=\infty$$

Discrete: Easy to define, simply a sequence consisting of iid Gaussian RVs.