power spectral density - Variance of White Gaussian Noise

It could seem an easy question and without any doubts it is but I'm trying to calculate the variance of white Gaussian noise without any result.

The power spectral density (PSD) of additive white Gaussian noise (AWGN) is $\frac{N_0}{2}$ while the autocorrelation is $\frac{N_0}{2}\delta(\tau)$, so variance is infinite?

Answer

White Gaussian noise in the continuous-time case is not what is called a second-order process (meaning $E[X^2(t)]$ is finite) and so, yes, the variance is infinite. Fortunately, we can never observe a white noise process (whether Gaussian or not) in nature; it is only observable through some kind of device, e.g. a (BIBO-stable) linear filter with transfer function $H(f)$ in which case what you get is a stationary Gaussian process with power spectral density $\frac{N_0}{2}|H(f)|^2$ and finite variance $$\sigma^2 = \int_{-\infty}^\infty \frac{N_0}{2}|H(f)|^2\,\mathrm df.$$

More than what you probably want to know about white Gaussian noise can be found in the Appendix of this lecture note of mine.