I have two data records $R_1$ and $R_2$ with sampling periods $T_1$ and $T_2$, where $T_1$ < $T_2$. These records arise from sampling and filtering two signals to remove any noise (including aliasing noise) from signals with shorter periods. This implies that the data in $R_1$ and $R_2$ are not instantaneous observations but represent average states on time scales of $T_1$ and $T_2$, respectively.
I want to fit a model $R_2 = f(R_1)$ and, thus, I want to match the two records as if they would've been sampled simultaneously, i.e., as if they would have the same sampling period.
I tried this by
- (A) Upsampling $R_2$ to $T_1$;
- (B) Downsampling $R_1$ to $T_2$;
- (C) Downsampling both, $R_1$ and $R_2$ to $2 \cdot T_2$.
Afterwards I evaluated the performance of the model with cross-validation and found that the best match was obtained with (C).
I believe this is a consequence of the Nyquist-Shannon theorem, which says that $R_2$ contains complete information only from signals with periods of $2 \cdot T_2$ or longer, while events with shorter periods (for instance, $T_1$) are not resolved by $R_2$.
This means that I cannot determine accurately the relation $R_2 = f(R_1)$ at periods shorter than $2 \cdot T_2$. Therefore, the Nyquist-Shannon theorem seem to have obvious implications on how to match two records:
For instance, to upsample $R_2$ as to match $T_1$ would not be correct, since we have inaccurate information about $R_2$ on the time scale $T_1$. On the contrary, observing the Nyquist-Shannon theorem implies matching the two signals by downsampling both of them to a period of $2 \cdot T_2$ (or longer), a time scale where both records would contain accurate information. My questions are:
- Is this notion correct, i.e., is the downsampling of both records to $2 \cdot T_2$ the correct way of matching the sampling period of $R_1$ and $R_2$?
- If yes, could somebody provide a quote for this from a text book or a paper, as well as some prominent examples in the literature where two records or signals are matched like this, for instance from signal analysis, acoustics, electronics, meteorology, etc?