 Spectral frequency-increment measurement noise

Spectral frequency-increment measurement noise is the contribution of frequency-estimation errors to a statistic comparing a transition frequency at two times. It can create an apparent nonzero spectral-diffusion frequency-increment variance even when the underlying frequency has not changed.

For an explicit measurement model, write $Y_t={\omega }_t+{ϵ}_t$ and define $\Delta \omega ={\omega }_T-{\omega }_0$, $\Delta ϵ={ϵ}_T-{ϵ}_0$. With zero-mean endpoint errors,The last term vanishes only under an appropriate lack-of-correlation assumption between the physical increment and its estimation errors. Merely naming the errors noise does not establish that assumption.

If that cross term vanishes, equal endpoint standard uncertainties $u$ and error correlation coefficient ${\rho }_{ϵ}$ giveIndependent errors give $2u^2$. A perfectly shared offset cancels from the difference, although the absolute frequencies can remain uncertain. This correlated spectral endpoint error is an ordinary application of correlated-input uncertainty propagation.^[ref]

An original example has true root-mean-square increment $0.1\mathrm{rad}\mathrm{p}{\mathrm{s}}^{-1}$ and $u=0.2\mathrm{rad}\mathrm{p}{\mathrm{s}}^{-1}$. Independent endpoint errors give $D_{\mathrm{obs}}=0.01+0.08=0.09$ and apparent root-mean-square increment $0.3\mathrm{rad}\mathrm{p}{\mathrm{s}}^{-1}$. At ${\rho }_{ϵ}=0.75$, the added term is only $0.02$, giving an apparent increment about $0.173\mathrm{rad}\mathrm{p}{\mathrm{s}}^{-1}$.

A noise subtraction can also be uncertain. Observed second moment $0.03$ minus estimated noise contribution $0.08$ gives $-0.05$ in the same squared-frequency units. That negative estimate does not establish a negative physical variance; it flags uncertainty or a mismatch in the assumptions used to separate the contributions.

Repeated observations of a stable reference can test the estimator's noise and temporal correlation. Filtering, common references and reuse of the same data can correlate errors at nearby times. Primary two-dimensional infrared spectroscopy analysis connects spectral geometry to a frequency-correlation model rather than directly recording every molecule's frequency trajectory.^[ref] The additive-error equations apply only when the stated frequency estimator and error model describe the actual measurement.