White noise does not contradicts Wide Sense Stationarity?

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.

• but it doesn't have infinite variance nor infinite power. It has infinite energy, but finite variance $\sigma^2$. Maybe you want to spell out why you think it has infinite energy! Dec 19, 2021 at 15:00
• Okay, maybe that is where my confusion lies. Why do you say that it have infinite energy from the construction? Dec 19, 2021 at 15:07
• well, if it has nonzero constant power and you integrate that (over infinite time) or nonzeroconstant power spectral density and you integrate that (over infinite bandwidth), then you get an infinite value. Dec 19, 2021 at 15:08
• Exactly, but having a nonzero constante PSD means that power is the total area over infinite interval, thus having an infinite power. Indeed, in various text I have read that for a WN the power is infinite (that's the reason why they are not more than an ideal process). Dec 19, 2021 at 15:12
• Does this answer your question? Is white noise WSS by nature or not? Dec 19, 2021 at 17:45