 Spectral common-mode offset rejection

Spectral common-mode offset rejection is cancellation of an additive frequency-axis shift when the same shift enters two spectral estimates with equal sensitivity and the estimates are subtracted. It improves a frequency difference without establishing accurate absolute frequencies. The relevant measurement model must identify what is shared, and over which observations.

For true frequencies ${\omega }_1,{\omega }_2$, write measured estimates asThe unknown common offset $b$ disappears exactly fromFor independent residual errors of standard uncertainty $s$, the difference still has standard uncertainty $\sqrt{2}s$. Cancellation of $b$ is not cancellation of all measurement noise. With correlated residuals, the covariance term must be retained in uncertainty propagation.^[ref]

An original example uses two spectral positions at $1600$ and $1650{\mathrm{cm}}^{-1}$. An additive shift of $3{\mathrm{cm}}^{-1}$ gives measured positions $1603$ and $1653{\mathrm{cm}}^{-1}$, with the same $50{\mathrm{cm}}^{-1}$ separation. A shared gain error is different: $Y_i=1.01{\omega }_i+3$ gives separation $50.5{\mathrm{cm}}^{-1}$. Subtraction rejects the offset but retains the scale error.

The algebra generalizes to a contrast of several spectral positions. If $Z={\sum }_i{w}_i{Y}_i$ and each $Y_i$ contains the same additive $b$, its contribution is $b{\sum }_i{w}_i$. Thus weights satisfying ${\sum }_i{w}_i=0$ reject that offset. Averaging positions with weights summing to one does not. This is an original linear-model derivation of the rejection condition, not a claim that every spectral estimator has this form.

Two failures of the ideal assumption are especially informative. With unequal spectral offset sensitivity, $Y_i={\omega }_i+a_{i}b+{\eta }_i$, subtraction leaves $(a_2-a_1)b$. If the offset drifts between observations, it instead leaves $b(t_2)-b(t_1)$. For a stationary drift with variance $s_b^2$ and correlation ${\rho }_b(\tau )$ at lag $\tau$, this residual has varianceA common instrument or reference does not alone imply perfect cancellation at different times.

An apparatus-level check measures two stable reference features using the same acquisition and fitting pipeline, then deliberately shifts the assigned axis origin. Their separation should stay fixed under an additive reassignment; a scale change should alter it. Repeated reference observations test temporal drift and residual covariance. In two-dimensional infrared spectroscopy, this test concerns frequency-estimation errors; spectral diffusion also describes physical fluctuations and cannot be identified or removed merely by subtracting an instrumental offset.