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Auto-correlation of white noise

Started by mr.t 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