SINR averaging in communication system
Signal-to-interference-plus-noise ratio (SINR) is by definition 


$SINR = \frac{P}{I+N}$, 


where $P$ - is useful signal power, $I$ - interference signal power and $N$ - noise power in the band of interest. Look 

http://en.wikipedia.org/wiki/Signal-to-interference-plus-noise_ratio


for example. 

So the first formula you've presented seems to be rigth one. Time span of calculation $SINR$ depends on your needs and interference nature. The common way is using duration of $x$. In general you have to distinguish both $x$ and $i$ at first, because $i$ is non-stationary random process so you can't think it as receiver noise. So in most common case in real life you can estimate $SINR$ only if you are able to combat such an interference in you system.

If inteference appears as short pulses in time you can do measurement with relatively short sliding window to estimate $SINR$ in dynamics. If signal power is quite stationary in short averaging you can use previously estimated $P$ in formula above to find out $SINR$ even if demodulator fails to defeat it. 

You can combine measurement of $SINR$ at the input of demodulator, $MSE$ at its output and $BER$ at the decoder's output to make some sophisticated modem quality control.