K-Delta Function in Autocorrelation of Gaussian White Noise

[dora]

hi,
I would like to know why dirak-delta function is used in autocorrelation in a way that the following is true:

<å(t)å(t')>=2Dä(t-t')

where å(t)is Gaussian white noise and D is the strength of the noise.

Dora

 

[Kalimaa23]

I'm not familiar with the application that you are describing.

One thing that you should mind is that the Dirac delta is not a function, but a distribution. It is the limit of a sequence of functions of area one, that are centered around a single point, and who's peak increases in the sequence. It can be seen as a "function" that is zero everywhere except for one point, and whose integral of the entire domain yields 1.

 

[selfAdjoint]

Originally posted by dora
hi,
I would like to know why dirak-delta function is used in autocorrelation in a way that the following is true:

<å(t)å(t')>=2Dä(t-t')

where å(t)is Gaussian white noise and D is the strength of the noise.

Dora

This means that the white noise, being uncorrelated, cancels out between two different times, and only gives a D-value when the times coincide. You have $$\int f(t)\delta(a - t)dt = F(a)$$, where F is the antiderivative of f.