Minimum time window needed to capture frequencies

In summary, the conversation discusses the problem of detecting and accurately measuring the frequencies of a signal composed of standing wave sinusoids, with a limited time window for sampling. Spectral leakage is mentioned as a potential issue, but may not fully address the concerns. The concept of "picking out" frequencies is clarified as detecting with potential amplitude uncertainty. The term for sampling within a time window is not remembered.
  • #1
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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 [itex]\tau[/itex], so I'm really only detecting [itex]\tau=NTp[/itex] (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
[tex]g(t)=\sum_{N=1}^p \cos(\frac{2\pi \nu t}{N})[/tex]

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.
 
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  • #2
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.
 
  • #3
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.
 

1. What is the significance of the minimum time window in capturing frequencies?

The minimum time window is the shortest period of time needed to accurately capture all frequencies present in a signal. This is important because if the time window is too short, some frequencies may be missed, leading to incomplete or inaccurate data.

2. How is the minimum time window calculated?

The minimum time window is calculated by dividing the number of cycles in the signal by the highest frequency present. This gives the minimum amount of time needed to capture at least one full cycle of each frequency.

3. Can the minimum time window be too long?

Yes, the minimum time window can be too long if it is longer than the duration of the signal being captured. In this case, it would not be necessary to use the entire minimum time window, as all frequencies have already been captured.

4. Does the minimum time window differ for different types of signals?

Yes, the minimum time window can vary depending on the type of signal being captured. For example, a periodic signal with a lower frequency may require a longer minimum time window compared to a non-periodic signal with a higher frequency.

5. What happens if the minimum time window is not met?

If the minimum time window is not met, some frequencies may be missed or distorted in the final data. This can lead to errors and inaccuracies in any analysis or conclusions drawn from the data. It is important to ensure that the minimum time window is met in order to capture all frequencies accurately.

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