Let's say that I have a sampled signal x[n], it is being, in this exact order, up sampled by 2, down sampled by4, up sampled by 4 and down sampled by 2 to produce y[n].

It seems to me that it should be pretty self evident that since we up sampled the signal by 2 and down sampled it by 2, then up sampled it by 4 and down sampled by 4, I should just get the original x[n] back. 

Am I right?

So the real question is, can the various up/down sampling pieces be readily swapped?


Question: Is an ability that can only be activated once per turn still considred activated if it gets countered?
Answer: Yes, the ability was still activated even if it gets countered.

Activating the ability is merely the process of paying its costs and putting it onto the stack, completely independent of what happens after that:


  602.2. To activate an ability is to put it onto the stack and pay its costs, so that it will eventually resolve and have its effect. ...


All that has to be done before anyone gets priority and is able to Stifle it. Countering it merely causes the ability to be removed from the stack without any of its effects occurring; it doesn't rewind the game and undo it being activated in the first place.


  701.5a To counter a spell or ability means to cancel it, removing it from the stack. It doesn’t resolve and none of its effects occur. A countered spell is put into its owner’s graveyard.


Similarly, once you cast a spell, it's been cast, even if it subsequently gets countered.