= 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 as
$$
Y_1=\omega_1+b+\eta_1,\qquad Y_2=\omega_2+b+\eta_2.
$$
The unknown common offset $b$ disappears exactly from
$$
Y_2-Y_1=\omega_2-\omega_1+\eta_2-\eta_1.
$$
For independent residual errors of standard uncertainty $s$, the difference still has standard uncertainty $\sqrt2s$. Cancellation of $b$ is not cancellation of all <measurement noise>. With correlated residuals, the <covariance> term must be retained in <uncertainty propagation>.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Section 5.2, correlated input quantities]{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 $Z=\sum_i w_iY_i$ and each $Y_i$ contains the same additive $b$, its contribution is $b\sum_iw_i$. Thus weights satisfying $\sum_iw_i=0$ 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>, $Y_i=\omega_i+a_ib+\eta_i$, 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 variance
$$
2s_b^2\bigl(1-\rho_b(\tau)\bigr).
$$
A 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.
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