Signal Averaging

Note I also asked this in math stackexchange I was reading a wiki page about signal averaging specifically in the section Signal power for sampled signals. At least at first glance the information seems to be quite wrong.

Passage from wikipedia follows:

Considering $$n$$vectors $$V_i,\,i \in\{1,\ldots,n\}$$ of signal samples of length $$T$$ $$V_i = \left[s_{i,1}, \ldots, s_{i,T}\right],\quad s_{i,k} \in \mathbb K^T$$,

the power $$P_i$$ of such a vector simply is

$$P_i = \sum_{k=1}^T {s_{i,k}^2} = \left| V_i \right|^2$$.

Again, averaging the $$n$$ vectors $$V_{i},\,i=1,\ldots,n$$, yields the following averaged vector

$$V_\text{avg} = \frac 1n \sum_{k=1}^T\sum_{i=1}^n s_{i,k}= \frac 1n \sum_{i=1}^n \sum_{k=1}^T s_{i,k}$$.

In the case where $$V_n \equiv V_m \forall m,n \in \{ 1,\ldots, n\}$$, we see that $$V_\text{avg}$$ reaches a maximum of

$$V_\text{avg, identical signals} = P_i$$.

In this case, the ratio of signal to noise also reaches a maximum,

$$\text{SNR}_\text{avg, identical signals} = \frac{V_\text{avg, identical signals}}{N_\text{avg}} = n \text{SNR}$$.

This is the oversampling case, where the observed signal is correlated (because oversampling implies that the signal observations are strongly correlated).

Firstly, doesn't $$V_i = \left[s_{i,1}, \ldots, s_{i,T}\right],\quad s_{i,k} \in \mathbb K^T$$

Imply that $$V_i$$ is actually a vector of functions, $$f:T\rightarrow K$$ which is not what I think the intended meaning is. Shouldn't $$V_i \in K^T$$ and $$s_{i,k} \in K$$? Secondly, for $$V_\text{avg} = \frac 1n \sum_{k=1}^T\sum_{i=1}^n s_{i,k}= \frac 1n \sum_{i=1}^n \sum_{k=1}^T s_{i,k}$$ Isn't this just a scalar since they summed all the vectors and over the vectors? How is this the average vector? It may be that I don't understand something here but I feel like this section is full of mistakes and poorly explains whats going on. Am I correct that this is a mistake in the page or is there something I don't understand?

• additionally, if this is indeed incorrect and someone wants to take what I have above and fix it to explain signal averaging for signal noise ratio that would be greatly appreciated – Colin Hicks Oct 2 '19 at 20:42
• I asked this question in math stackexchange as well since a bit time sensitive. I have a lab due tommorow and from what it seems either this page has problems or my lab does. So a better understanding or confirmation would be appreciated – Colin Hicks Oct 3 '19 at 0:17
• The section seems to me full of wrongly-written stuff, from indics to non-consistent equations, like $V_\textrm{avg}=P_i$ (unit and square of the unit on each side) – Laurent Duval Oct 3 '19 at 8:24