Naming convenction of random variables in spectral representation

[mbvoyager]

Samples of stochastic processes generated using spectral representation are now all labeled by the symbol :gm.

This results in the naming of the collection of generated signals (stochastic processes) as gm and the random phase angles as gm_1, gm_2, gm_3, .... This is confusing. Since at least for spectral representation only the phase angles ϕ are the random variables, I think it would be better to name and number them as ϕ_1, ϕ_2, ϕ_3, ...

[FriesischScott]

Since we need a reference to the name of the spectral representation gm_ϕ_1 would be the obvious choice.

I have a few more ideas on how we can potentially improve the PSD and SRE code:

• Don't discretize the PSD but keep it as a proper function that could be called as PSD(ω)
  • This could a candidate for memoization so we can cache repeated calls for the same ω
• Instead of setting ω as a Vector we could define the cutoff frequency and number of discretization points N in the SpectralRepresentation and then get the vector from that whenever needed. Might be cleanier for the struct.

A question about N. You need a relatively large N for the stochastic process to be Gaussian. In one of the tests we're using 100 which is insufficient. Is there any recommendation or rule of thumb for how large N should be?

Thoughts @mbvoyager, @marcobehrendt?

[mbvoyager]

Good points!

I think gm_ϕ_1 is a good choice. For me that would work.

It is also a good idea to just keep the function, I do not understand the second point regarding "memorization", I guess it's because if we keep it as a function, we need to evaluate it multiple times?

The last two points are connected.

There is a rule of thumb and this is connected to the Nyquist frequency. Following two equations must be regarded, then the guassianity of the generated signals should be ensured.

$\Delta \omega = \omega_u / N$ $\Delta t \leq 2\pi/2\omega_u$

$N$: summation terms, $\Delta \omega$: frequency increment, $\omega_u$: cut-off frequency, $\Delta t$: time increment