# Integral of White noise

Started by November 22, 2005
```What happens when you integrate white noise over time? Can anyone give a
mathematical representation of this thing?

```
```"junas125" <junaid_giki@yahoo.com> wrote in message
news:NdidncDTKJsf9B7enZ2dnUVZ_sudnZ2d@giganews.com...
> What happens when you integrate white noise over time? Can anyone give a
> mathematical representation of this thing?
>
>
I believe this is known as a random walk but I am not sure. In engineering
terms it is just coloured noise of a sort.
It's not good to try since and slight dc and it saturates the integrator.

McC

```
```"junas125" <junaid_giki@yahoo.com> wrote in message
news:NdidncDTKJsf9B7enZ2dnUVZ_sudnZ2d@giganews.com...
> What happens when you integrate white noise over time? Can anyone give a
> mathematical representation of this thing?
>
>

It's spectrum is defined as

S(w) = sigma^2/w^2

where sigma is the sd of the white noise and w is freq in rads/s.

McC

```
```in article NdidncDTKJsf9B7enZ2dnUVZ_sudnZ2d@giganews.com, junas125 at
junaid_giki@yahoo.com wrote on 11/22/2005 13:54:

> What happens when you integrate white noise over time? Can anyone give a
> mathematical representation of this thing?

it's called "Brownian motion".  i'm gonna sidestep doing any real work with
the "mathematical representation of this thing" and recommend Google.

--

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

```
```junas125 wrote:
> What happens when you integrate white noise over time? Can anyone give a
> mathematical representation of this thing?

Integrate any signal with a flat spectrum and the result is a spectrum
that tilts down at 20 dB/decade as frequency increases. You can also say
that the spectrum tilts up at 20 dB/decade as frequency decreases, which
implies infinite gain at DC.

The noise you ask about is called "brown noise", after the statistics of
Brownian motion. http://www.ptpart.co.uk/colors.htm

Jerry
--
Engineering is the art of making what you want from things you can get.
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```
```junas125 wrote:

> What happens when you integrate white noise over time? Can anyone give a
> mathematical representation of this thing?
>
>
While you're googling look for "Wiener Process".

White noise is stationary, meaning that knowing what time you're looking
at it doesn't tell you anything about the noise.

Integrated Gaussian white noise has a Gaussian distribution who's
standard deviation goes up with time as the integration period is
increased, hence "random walk".

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
```