= Spectral frequency-increment measurement noise
\b[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 $Y_t=\omega_t+\epsilon_t$ and define $\Delta\omega=\omega_T-\omega_0$, $\Delta\epsilon=\epsilon_T-\epsilon_0$. With zero-mean endpoint errors,
$$
\mathbb E[(Y_T-Y_0)^2]=\mathbb E[(\Delta\omega)^2]+\operatorname{Var}(\Delta\epsilon)+2\operatorname{Cov}(\Delta\omega,\Delta\epsilon).
$$
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> $\rho_\epsilon$ give
$$
D_{\rm obs}(T)=D_{\rm true}(T)+2u^2(1-\rho_\epsilon).
$$
Independent 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>.<Evaluation of measurement data — Guide to the expression of uncertainty in measurement by JCGM>[Section 5.2, covariance terms]{ref}
An original example has true root-mean-square increment $0.1\,\mathrm{rad\,ps^{-1}}$ and $u=0.2\,\mathrm{rad\,ps^{-1}}$. Independent endpoint errors give $D_{\rm obs}=0.01+0.08=0.09$ and apparent root-mean-square increment $0.3\,\mathrm{rad\,ps^{-1}}$. At $\rho_\epsilon=0.75$, the added term is only $0.02$, giving an apparent increment about $0.173\,\mathrm{rad\,ps^{-1}}$.
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.<Frequency-frequency correlation functions and apodization in two-dimensional infrared vibrational echo spectroscopy: A new approach by Kyungwon Kwak and others>[Introduction and Section II, spectroscopic observables and frequency-frequency correlation function]{ref} The additive-error equations apply only when the stated frequency estimator and error model describe the actual measurement.
Back to article page
