# Effective Nyquist frequency / signal resolution for overlapping time averaged signal samples

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.)

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.

• 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). – Fat32 Jan 4 at 22:02
• Do you (if so, how) want to perform overlapping averaging before sampling? – Laurent Duval Jan 5 at 16:35
• 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. – user3558855 Jan 5 at 17:35

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.