If I have a two-dimensional discrete dataset (one space, one time), and I create subsets of this dataset by "sampling" (although the original dataset isn't continuous) it at different rates, should I expect that the same filter performed on different subsets would produce the same field (although sampled at different times)?
To elaborate, I have a discrete dataset that contains a four-times daily sampled field. If I choose every fourth data point in time after the first, I can get a discrete sub-dataset of a once-daily sampled field, although the fields in the 4x and 1x datasets are still the same at the same time. The filtering method is then as follows:
- Choose the sampling rate of the discrete dataset
- Perform an FFT in space and time to get a wavenumber-frequency spectrum
- Filter the spectrum to retrieve signals propagating only in a certain direction
- Invert the time and space transforms to get the field propagating in that direction
Now, the field I get with the 4x rate and that with the 1x rate are wildly different - not just a matter of magnitude or phase (that I can tell), they are almost entirely incomparable. However, my expectation is that signals propagating in one direction in the 4x dataset should still propagate in that direction in the 1x dataset given that they are actually the same dataset! All relevant signals are well below the Nyquist frequency, and I have accounted for scaling by the dataset length.
I have confirmed that the 1x (sampled from the 4x dataset) and a native 1x-daily sampled dataset produce similar results, as I would expect - and I have a physical reason for believing both 1x results are correct. The output is much different when I try to employ the entire 4x dataset.
My only guess is that there is an extra need to account for the dimensional frequency spectrum being different between the 4x and 1x datasets, but I also thought that the FFT shouldn't care what the time step is. Is there any mathematical source of the difference, and if so, how can I correct it?