# Auto-correlation of white noise

Started by August 18, 2008
```Im a beginner to signal processing so have that in mind.

I know that the auto-covariance, c_v(k) of a white process, v(n),
is:

c_v(k) = \$_v^2 * D(k)

Where \$ is small-sigma and D is diracs deltafunction.
The auto-correlation in this case i believe is the same as
the auto-covariance(?)

r_v(k) = \$_v^2 * D(k)

Is this always true for white processes?

```
```On 18 Aug, 16:44, "mr.t" <tow...@gmail.com> wrote:
> Im a beginner to signal processing so have that in mind.
>
> I know that the auto-covariance, c_v(k) of a white process, v(n),
> is:
>
> c_v(k) = \$_v^2 * D(k)
>
> Where \$ is small-sigma and D is diracs deltafunction.
> The auto-correlation in this case i believe is the same as
> the auto-covariance(?)
>
> r_v(k) = \$_v^2 * D(k)
>
> Is this always true for white processes?

It is true iff the mean of the process is 0.

Rune
```
```On Aug 18, 9:44 am, "mr.t" <tow...@gmail.com> wrote:
> Im a beginner to signal processing so have that in mind.
>
> I know that the auto-covariance, c_v(k) of a white process, v(n),
> is:
>
> c_v(k) = \$_v^2 * D(k)
>
> Where \$ is small-sigma and D is diracs deltafunction.
> The auto-correlation in this case i believe is the same as
> the auto-covariance(?)
>
> r_v(k) = \$_v^2 * D(k)
>
> Is this always true for white processes?

The answer depends on what properties you think white noise
should have.  The power spectral density (PSD) of a random
process is usually defined as the Fourier Transform of its
autocorrelation function; *not* the Fourier Transform of its
autocovariance function.  Your definition of a white noise
process seems to be a process for which the autocovariance
function is an impulse.  Other people think that a random
process should be called a white noise process if and only if
its PSD has constant value for all frequencies.  (This is
equivalent to the requirement that the autocorrelation function
be an impulse.)  Since the PSD of a process with nonzero mean
includes an impulse at zero frequency (i.e., at DC) and thus
cannot be said to have "constant value for all frequencies", these
people's definition of white noise implicitly includes the
requirement
that the mean be zero.  Rune Allnor has already pointed out
that the autocorrelation function equals the autocovariance function
if and only if the mean is zero.

So, if your definition of a white noise process is a process whose
autocovariance function is an impulse, then your white noise process
need not have zero mean.  The autocorrelation function of your
white noise process will not necessarily be an impulse at the origin
and nothing else; it will have value m^2 for all nonzero offsets where
m
is the mean of your process; and the PSD will include an impulse of
magnitude m^2 at zero frequency (but have constant value at all other
frequencies).  Not everyone in this group will agree that what you
have
is a white noise process (unless m = 0 so that the autocorrelation
function equals the autocovariance function and is also an impulse),
but then, as has been noted in another recent and continuing thread,
arguments about semantic distinctions are the basis of many extended
discussions in this newsgroup.
```
```See http://en.wikipedia.org/wiki/Autocovariance#Normalization
```