 Noise penalty from nearly parallel signal and reference shapes

The 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 correlationThe Cauchy-Schwarz inequality gives $|r|\le 1$. The minimum-variance spectral contrast orthogonal to a reference shape has varianceIf the nuisance amplitude were known and subtracted without uncertainty, the corresponding optimum variance would be $V_{\mathrm{known}}=1/({\mathbf{s}}^T{C}^{-1}\mathbf{s})$. 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 $C={\sigma }^{2}I$, $\mathbf{x}=(1,0{)}^T$ 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.^[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.^[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.