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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.

Projection of shared-scale uncertainty onto a spectral contrast determines which measured spectral features are sensitive to an uncertain common subtraction coefficient. In correlated spectral residuals from a shared subtraction scale, a fixed reference shape $\mathbf{x}$ and scale error $\delta k$ contribute $-\mathbf{x}\delta k$ to the spectrum. For a linear contrast $c={\mathbf{a}}^T\mathbf{z}$,The contrast is insensitive to this particular error when ${\mathbf{a}}^T\mathbf{x}=0$. This is an exact cancellation for a fixed reference shape; it does not require the scale error to have a Gaussian distribution.

Suppose the desired spectral signal has known shape $\mathbf{s}$ and unknown amplitude $\theta$, so that $\mathbf{z}=\theta \mathbf{s}-\delta k\mathbf{x}+\mathbf{ϵ}$. To retain its amplitude while rejecting the scale error, requireFor an original two-channel example, take $\mathbf{x}=(1,1{)}^T$ and $\mathbf{s}=(1,0{)}^T$. The difference $c=z_1-z_2$ has unit signal response and cancels any common scale error. If the two independent reading noises each have standard deviation $3$ in the chosen units, the difference has standard deviation $\sqrt{18}\approx 4.24$. Using channel 1 alone would have reading-noise standard deviation $3$ but would retain the unknown common contribution. Rejection has a noise cost even when the nuisance cancellation is exact.

If $\mathbf{s}$ is proportional to $\mathbf{x}$, the two requirements are incompatible: cancelling that reference direction also cancels the signal. In more channels, a minimum-variance spectral contrast orthogonal to a reference shape can balance detector noise among the remaining directions. The relevant variance is ${\mathbf{a}}^{T}C\mathbf{a}$, including correlations.^[ref] Uncertain reference shape or scale estimates correlated with the measured spectra require their own covariance contributions.