I am trying to understand the estimation of the power spectral density of a continuous time stochastic process from it's samples.
Consider a normal wide-sense stationary white noise process with average power $\sigma^2$. Its autocorrelation is a dirac-delta impulse of height $\sigma^2$. By the Wiener-Khinchin theorem, the power-spectral density of the process is the fourier transform of its autcorrelation function, i.e. constant $\sigma^2$.
Note that I'm avoiding to claim that any $x(t)$ in the process is a normally distributed random variable with variance $\sigma^2$. Apparently this notion is controversial.
In practice, one can estimate the power spectral density from a finite number of samples of the process from the DFT using the periodogram / Bartlett / Welch method.
Trying to generate these samples artificially in MATLAB, I noticed, that I need to scale the variance with the assumed sampling rate $f_s$ to get the right height in my periodogram.
x = sigma * sqrt(fs) * randn(N, 1)
Perhaps I just did not recognize it for what it was, but I never found this detail mentioned anywhere in the literature (except in the documentation on the band-limited white noise block of Simulink). This perhaps makes sense, because one usually samples a real process and does not try to generate these samples artificially, but perhaps someone could explain or point to a reference explaining the relationship between the variance of a set of samples from a stochastic process and the properties of the stochastic process itself. For example, Papoulis says that the autocorrelation of a samples process is the sampled version of the cont. time process' autocorrelation function, but nothing about variance is stated.