White noise is usually defined as a wide sense stationary (WSS) process $N=\{N_t|t\in T\}$ (for $T$ a time index set), that has a constant power spectral density, say $S_{NN}(f)=\sigma^2$. Since the 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 function. 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:
$$\begin{align}
\mathbb{V}(N)
&= R_{NN}(0) – \mu_N(t) \\[14pt]
&= R_{NN}(0) \\[6pt]
&= \lim_{\tau\to\infty} \int \limits_{-\infty}^{\infty}\frac{\mathbb{E}\{|\hat{N}^\tau(f)|^2\}}{2\tau} \ df, \\[6pt]
\end{align}$$
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:
$$\int_{-\infty}^{\infty}\sigma^2 \ df=\infty.$$
As you can see, I am requiring a process to have finite second moment for being WSS. Also, I have that this white noise has identical distribution with zero mean and variance $\sigma^2$.
I feel that I have a big misconception, but I do not know where the problem is. I really appreciate any help.
Best Answer
Yes, strictly speaking a continuous white noise is not a WSS process because its variance is not finite, and is actually not even defined (the discrete white noise is instead WSS).
There are two alternative ways to solve this issue: the first is to consider that in any real system excited by white noise the bandwidth is finite, such that the variance of the filtered process is finite; the second is to enlarge the class of stochastic processes in the same way we enlarge the class of functions to include generalized functions (distributions) like the delta "function".
The former approach is common in the engineering literature, where white noise is typically used to model thermal noise in electronic and communication circuits, which indeed have a finite bandwidth. This approach, which is a bit of handwaving, is the simplest but not mathematically sound.
The latter approach can be implemented by considering stochastic processes as linear functionals associating a random variable to any function in a certain class of test functions. The correlation function for these generalized processes is defined as a bilinear functional on the class of test functions, and it has a finite value for all test functions in the case of continuous white noise. The details of this approach can be found in section 24.1 of the following book:
A. M. Yaglom, Correlation theory of stationary and random functions, vol. I; Basic results, Springer Series in Statistics, Springer-Verlag, 1987.