Estimating variance in arbitrary, periodic signal
Do a cross correlation with your known $s$; you will get peaks of height $\|s\|A_k$ at the positions of your shifts.

You just sum up these peaks, call that sum $P$, calculate the overall energy in your signal (i.e. $\|x\|$), and calculate $\text{SNR}=\frac P{\|x\|-P}$ if you need the SNR. If you just need the noise: $\|x\|-P$ (quite directly). Done!