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?
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
Thanks for any help!
If my interpretation of the Nyquist rate is wrong or anything, let me know and I'll correct it in the post.
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.