 Fixed-phase sinusoidal least squares

When frequency and phase are known, sinusoidal amplitude and offset fitting is a two-parameter linear regression. For equal known errors $\sigma$, profiling out the offset gives $K=\sum (s_i-\bar{s})(y_i-\bar{y})/\sum (s_i-\bar{s}{)}^2$ and $\mathrm{Var}(K)={\sigma }^2/\sum (s_i-\bar{s}{)}^2$. Identical phases cannot separate amplitude from offset. Unknown phase instead requires independent sine and cosine coefficients.