Minimum time window needed to capture frequencies

[Mindscrape]

I'm pretty sure there have to be some theorems out there, but I am probably not putting in the right search terms to be able to find them. Here's the problem:

I have a signal uniquely composed of a finite summation of standing wave sinusoids (well there's some DC and other background, but let's ignore those). Let's say I sample at some rate NT, and that my highest frequency is NT/2 (so I'm good on Nyquist). However, let's also say that I can only watch this signal for some time $\tau$, so I'm really only detecting $\tau=NTp$ (where p is number of samples) time overall.

So let's actually ignore the discrete time samples for a second, in continuous time I would see
$$g(t)=\sum_{N=1}^p \cos(\frac{2\pi \nu t}{N})$$

So on one hand, how much time do I have to sample for to pick out all the correct frequencies. But, additionally, given that I am actually only measuring steps of NT seconds (sample and hold) does this affect the consequences of having a finite time window to measure all these beats correctly. Spectral leakage is pretty close to what I'm looking for, but not quite.

 

[mfb]

You get p equations, so in general you can solve for p unknown variables. As you limit your highest frequency in the right way, I would expect that the system always has a unique solution (even if it might be ugly in terms of numerics). It's a different question if that solution corresponds to your actual signal, that will depend on the type of waves you fit to the data points.

 

[FactChecker]

The phrase "pick out" frequencies is misleading. Assume that all frequencies are below the Nyquist frequency. Given an infinite time to sample, the amplitude errors for the frequencies will be 0. Given less time, the amplitude uncertainty will be larger. So "pick out" is not the right way to say it. They will be detected, but with amplitude uncertainty. I can not remember the name for sampling within a time window. Maybe someone can help.