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Say that I have a continuous signal. If I sample the signal at sampling rate f_sample, then the highest signal frequency that I can resolve without worrying about aliasing, etc. is the Nyquist rate f_Nyquist = 0.5 * f_sample, so arguably I could interpret this as meaning that the time resolution of my signal is dt_min = 1 / f_Nyquist = 2 / f_sample. Fine.

However, the above is for instantaneous sampling. How does the picture change if each of my samples involves averaging over a given finite time period t_width, with the possibility that successive time periods overlap by a time t_overlap? (See picture. Equation for t_overlap in the picture ignores cases where overlap extends beyond that of nearest neighbors, but it's easy to generalize.)

Signal sampling scheme with averaging and overlapping.

I.e., in the case that t_width = t_overlap = 0, I have f_Nyquist = 0.5 * f_sample, dt_min = 2 / f_sample, as above. For the case that t_width, t_overlap != 0, is there still a simple statement that I can make regarding f_Nyquist, or more importantly some generalization of f_Nyquist which is relevant to the quantity that I am actually interested in, namely the time resolution of my signal t_min? Is there a simple equation for something that is interpretable as t_min that could be used in such a situation?

Extra:

The above is the main thing I am interested in. However, just to add one more generalization in case anyone happens to know it: If the averaging weight function over t_width is not uniform, but is instead something more complicated (e.g., Gaussian), is there still a relationship or theorem that can help me figure out what t_min is?

Thanks for any help!

P.S.

  1. If my interpretation of the Nyquist rate is wrong or anything, let me know and I'll correct it in the post.

  2. I think this is the right StackExchange site for this question, but if there is another that anyone knows of where an answer is more likely, please suggest it.

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  • $\begingroup$ if a continuous-time real signal $x(t)$ has a bandwidth of $W$ Hz, then according to Nyquist theorem, the minimum sampling rate which avoids aliasing is $F_s = 2W$ and this rate $2W$ is called as the Nyquist rate associated with the signal. Given that you sample a signal with an arbitray rate $F_s$, then the maximum frequency that could be represented by the samples is $F_s/2$, called the Nyquist frequency of the sampler. Combining the two, if the bandwidth of a signal is less than the Nyquist frequency of the sampler then u r oversampling the signal otherwise undersampling (aliasing). $\endgroup$ – Fat32 Jan 4 '19 at 22:02
  • $\begingroup$ Do you (if so, how) want to perform overlapping averaging before sampling? $\endgroup$ – Laurent Duval Jan 5 '19 at 16:35
  • $\begingroup$ Thanks for the responses. Laurent, not really - I am working with a data set that has been given to me, not one that I generated, and I am trying to understand it better. (For me, this involves understanding signal analysis in general better than I have had to in the past.) The sampling scheme of the data set has overlapping averaging similar to what I described. $\endgroup$ – user3558855 Jan 5 '19 at 17:35
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The process that you are describing can be broken down into two steps

  1. Filter with a moving average filter with length $t_{width}$
  2. Sampling with an interval $delta\_t$

A moving average filter is a low pass filter, but not a very good with lots of side lobes. It will help reduce aliasing in the sample process, but it won't eliminate it with any type of hard "threshold". The longer $t_{width}$ is and the the smaller $delta\_t$ is the less aliasing you get.

I think you approaching this backwards: Your "time resolution" is simply determined by the highest frequency in the signal. You need to determine this first and work backwards from there.

In practice, there is always aliasing. No real world signal is actually bandwidth limited in the mathematical sense. So you need to determine the frequency range "of interest", figure out how much energy the signal has outside this region, determine how much noise or aliasing you can tolerate, and design a suitable anti-aliasing filter to meet your specific requirements.

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  • $\begingroup$ Thanks Hilmar. You are correct, this is a moving average filter and I did not notice! I think there is also a hidden step before your step 1 - namely, sample the data with an interval delta_t_original < delta_t. Okay, so now my thinking is: when given data collected using a non-trivial sampling scheme, it can be a good idea to think of it as having been generated by a trivial sample scheme (t_width = 0), then filtered until equivalent to the actual sampling scheme. This will help in thinking about how your band limits etc. will be affected. Does this sound reasonable? $\endgroup$ – user3558855 Jan 5 '19 at 17:57

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