I am studying White Noise. But I am really beginner level, so I have a confusion with its construction. White Noise is usually defined as a Wide Sense Stationary process $N=\{N_t\}_{t\in T}$ (for $T$ a time index set), that have a constant PSD, say $S_{NN}(f)=\sigma^2$. Since Correlation function $R_{NN}(t)=\mathcal{F}[S_{NN}(f)]$, we have also that $R_{NN}(t)=\sigma^2\delta(t)$ where $\delta$ is the Dirac delta. The thing is that $\mathbb{E}\{|N_t|^2\}=R_{NN}(0)=\sigma^2\delta(0)$. Indeed, if we assume that $N$ is ergodic we have that Power is also the Variance of the Process [$Var(N)=R_{NN}(0)-\mu_N(t)=R_{NN}(0)=\lim_{\tau\to\infty}\int_{-\infty}^{\infty}\frac{\mathbb{E}\{|\hat{N}^\tau(f)|^2\}}{2\tau}df$, where $\hat{N}^\tau$ is the Fourier transform of windowing of $N$ on a interval of lenght $2\tau$]. So, what I do not understand is how $N$ can be WSS if it has an infinite power and variance; because $\int_{-\infty}^{\infty}\sigma^2df=\infty.$
As you can see, I am requiring a process to have finite second moment for being WSS. Also, I have that this WN has identical distribution with zero mean and variance $\sigma^2$.
In short, my confusion lies on the fact that for a WN, $\sigma^2\delta(0)=R_{NN}(0)=\mathbb{E}\{N_t^2\}$ for every $t$. By hypothesis, $\sigma^2=\mathbb{E}\{N_t^2\}<\infty$. So, I see a contradiction where we have $\sigma^2\delta(0)=\sigma^2$.
I feel that I have a big misconception, but I do not know where the problem is. I really appreciate any help.
