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.

  • $\begingroup$ 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! $\endgroup$ Commented Dec 19, 2021 at 15:00
  • $\begingroup$ Okay, maybe that is where my confusion lies. Why do you say that it have infinite energy from the construction? $\endgroup$ Commented Dec 19, 2021 at 15:07
  • $\begingroup$ 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. $\endgroup$ Commented Dec 19, 2021 at 15:08
  • $\begingroup$ 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). $\endgroup$ Commented Dec 19, 2021 at 15:12
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    $\begingroup$ Does this answer your question? Is white noise WSS by nature or not? $\endgroup$ Commented Dec 19, 2021 at 17:45


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