 Unequal spectral offset sensitivity

Unequal spectral offset sensitivity occurs when a shared instrumental or reference offset influences two spectral estimates by different amounts. It limits spectral common-mode offset rejection: subtraction cancels a common disturbance only to the extent that its two response coefficients match.

For frequencies ${\omega }_1,{\omega }_2$, write an explicit measurement modelwhere $b$ is the shared offset, $a_i$ are dimensionless sensitivities and ${\eta }_i$ are residual errors. ThenThus even perfectly shared $b$ leaves a residual when $a_1\ne a_2$.

An original uncertainty example uses $a_1=0.98$, $a_2=1.02$ and standard uncertainty $u_b=0.50{\mathrm{cm}}^{-1}$. The offset contribution to the difference is $0.04u_b=0.020{\mathrm{cm}}^{-1}$. With independent residual standard uncertainties of $0.010{\mathrm{cm}}^{-1}$ each and no correlation with $b$, the combined uncertainty isA general calculation retains the relevant covariances; the independent-error simplification is an assumption, not a consequence of a shared instrument.^[ref]

Reweighting also changes what is measured. Subtracting $Y_1/a_1$ from $Y_2/a_2$ cancels $b$ when both sensitivities are nonzero, but estimates ${\omega }_2/a_2-{\omega }_1/a_1$, not generally the original frequency difference. Achieving cancellation is insufficient unless the resulting estimator still answers the intended physical question.

A controlled reference perturbation can estimate the slopes $a_1$ and $a_2$ while the true spectral features are held stable. Their difference predicts leakage. Spectral offset identifiability asks whether the available observations can distinguish a real change of frequency separation from this leakage; one observed difference alone cannot determine both an unknown physical change and an unknown offset. Reference features, controlled perturbations or independently established response coefficients supply additional information.