Z-transform of white noise

Question

Let's say we have a discrete-time white noise with variance $\sigma ^2$ that is we have a zero-mean Gaussian random variable with variance $\sigma ^2$. What is the $Z$ transform of this noise? Can we define it?

Answer 1

The noise should be defined as a random process $N[n]$ instead of R.V.

If the mean of noise is stationary along time and the autocorrelation of noise can represent in terms of time difference. The random process of noise defined as W.S.S.

And $N[n=i]$ is independent of $N[n\neq i]$

In this case, the autocorrelation of noise random process can be defined as:

$R_{n}[\tau=n_{1}-n_{2}]=\sigma^2\delta[\tau]$ and $m_{n}=0$

And the power spectral density (PSD) of $N[n]$ is Z-transform of $R_{n}[\tau]$:

$S_{n}(z)=\sigma^2$