= Noise penalty from nearly parallel signal and reference shapes
{title2=$V_{\rm reject}/V_{\rm known}=1/(1-r^2)$}
The \b[noise penalty from nearly parallel signal and reference shapes] is the precision lost when a spectral estimator must reject a nuisance pattern resembling its desired signal. For nonzero shapes $\mathbf s,\mathbf x$ and positive-definite noise <covariance> $C$, define their inverse-<covariance> correlation
$$
r=\frac{\mathbf s^TC^{-1}\mathbf x}
{\sqrt{(\mathbf s^TC^{-1}\mathbf s)(\mathbf x^TC^{-1}\mathbf x)}}.
$$
The <Cauchy-Schwarz inequality> gives $|r|\leq1$. The <minimum-variance spectral contrast orthogonal to a reference shape> has <variance>
$$
V_{\rm reject}=\frac1{(\mathbf s^TC^{-1}\mathbf s)(1-r^2)}.
$$
If the nuisance amplitude were known and subtracted without uncertainty, the corresponding optimum <variance> would be $V_{\rm known}=1/(\mathbf s^TC^{-1}\mathbf s)$. Exact nuisance rejection therefore multiplies <variance> by $1/(1-r^2)$ and standard uncertainty by $1/\sqrt{1-r^2}$.
The relevant angle is the <noise-whitened angle between spectral signal and nuisance shapes>: it is measured after a <whitening transformation>, not generally from the raw plotted spectra. A channel with large noise contributes little to separating the shapes even if their plotted curves differ there.
For an original example, let $C=\sigma^2I$, $\mathbf x=(1,0)^T$ and $\mathbf s=(1,0.1)^T$. Rejecting the reference forces the contrast to $(0,10)^T$, with <variance> $100\sigma^2$. With known nuisance amplitude, its <variance> would be $\sigma^2/1.01$. The rejection penalty is therefore $101$ in <variance>, about $10.05$ in standard uncertainty. Only the weak second-channel component distinguishes signal from nuisance.
The primary stimulated-Raman experiment reported systematic spectral noise above its calculated shot-noise limit and described pure-solvent subtraction of a common pattern.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Results and Discussion, theoretical shot-noise limit versus observed noise and pure-solvent subtraction]{ref} A reference can remove that shared pattern while also making an overlapping desired signal hard to estimate. Retain full <covariance> in the <variance> calculation.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref} At exact proportionality, no unit-response rejecting contrast exists; a low-noise result obtained by suppressing the signal as well is not a valid amplitude estimate.
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