(prob:processes:wn)=
# White noise

(prob:processes:wn:def)=
## Definition

```{prf:definition} White noise - properties
:label: wn:def:properties

A white noise is a random process with

- zero expected value 

  $$\mathbb{E}[ \xi(t) ] = 0$$

- Dirac delta correlation

  $$\mathbb{E}[ \xi(t) \xi(s) ] = \delta(t-s)$$


```

**todo** *link to math:functional-analysis:distributions*

```{prf:definition} White noise - time derivative of [Wiener process](prob:processes:wiener) $W(t)$ in the sense of distributions
:label: wn:def:derivative

**todo**

```

(prob:processes:wn:external-links)=
## External links

* White noise in [shape filters](https://basics2022.github.io/bbooks-math-miscellanea/ch/system-theory/shape-filter.html) for shaping non-Gaussian processes.
* ...