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A minimum-variance spectral contrast orthogonal to a reference shape estimates a desired signal amplitude while exactly rejecting one fixed nuisance shape. Let $\mathbf{z}=\theta \mathbf{s}+\beta \mathbf{x}+\mathbf{ϵ}$, with known shapes $\mathbf{s},\mathbf{x}$ and a positive-definite noise covariance matrix $C$. Choose $\hat{\theta }={\mathbf{a}}^T\mathbf{z}$ to minimize ${\mathbf{a}}^{T}C\mathbf{a}$, subject toThe constraints retain unit signal response and reject the reference amplitude, as in projection of shared-scale uncertainty onto a spectral contrast. The quadratic variance includes the full noise covariance.^[ref]

Remove the reference component in the inverse-covariance geometry:For nonzero $\mathbf{x}$ and $D>0$, the minimizing contrast and its variance areFor a short proof, every other feasible contrast is $\mathbf{a}+\mathbf{v}$ with ${\mathbf{v}}^T\mathbf{s}={\mathbf{v}}^T\mathbf{x}=0$. Since $C\mathbf{a}$ is a linear combination of those shapes, the cross term vanishes, leaving the added nonnegative variance ${\mathbf{v}}^{T}C\mathbf{v}$. This is a constrained best linear unbiased estimator.

In an original three-channel example, take $\mathbf{s}=(1,0,0{)}^T$, $\mathbf{x}=(1,1,1{)}^T$ and $C=\mathrm{diag}(1,4,1)$. The optimum is $\mathbf{a}=(1,-1/5,-4/5{)}^T$, with variance $1.8$. Equal subtraction of the two reference channels, $(1,-1/2,-1/2{)}^T$, has variance $2.25$. The quieter third channel receives more of the subtraction.

If the desired and reference shapes are proportional, $D=0$ and no unit-response rejecting contrast exists. The noise penalty from nearly parallel signal and reference shapes already becomes large as $D$ approaches zero. This construction assumes fixed known shapes and covariance; estimating them from the same noisy spectra requires additional uncertainty terms.

The noise-whitened angle between spectral signal and nuisance shapes measures their separation in units of measurement noise rather than raw plotting units. For nonzero shapes $\mathbf{s},\mathbf{x}$ and a positive-definite covariance $C$, choose a square whitening transformation $W$ with $WC{W}^T=I$. The angle isIt is the ordinary Euclidean angle after whitening. Both parallel and antiparallel shapes are indistinguishable up to an unknown nuisance amplitude; the precision penalty depends on ${\sin }^2\varphi$, as in noise penalty from nearly parallel signal and reference shapes.

For an original example, take raw shapes $\mathbf{s}=(1,1{)}^T$, $\mathbf{x}=(1,-1{)}^T$ and $C=\mathrm{diag}(1,100)$. The plotted shapes are orthogonal, but whitening gives $(1,0.1{)}^T$ and $(1,-0.1{)}^T$. Thus $\cos \varphi =0.99/1.01\approx 0.98020$ and $\varphi \approx 11.42^{\circ }$. The second channel, which separates their signs, is noisy. Rejecting the nuisance multiplies the optimum variance by $1/{\sin }^2\varphi =25.5025$, or standard uncertainty by $5.05$.

The invariance of noise-whitened spectral angles to the whitening factor follows from $W^{T}W=C^{-1}$. Different valid square whitening factors can rotate the whitened coordinates without changing the angle. Likewise an invertible change of measurement units transforms both shapes and covariance, leaving the inverse-covariance expression unchanged. Rescaling only the displayed curves while leaving their noise model fixed is a different operation.

The covariance must describe the actual measured channels and shared errors, using the full propagation rule.^[ref] If $C$ is singular, first restrict to its physically allowed range; an unrestricted inverse whitening factor does not exist.

The primary stimulated-Raman experiment described systematic spectral noise and subtraction of a pure-solvent reference.^[ref] The angle above assesses which desired patterns remain distinguishable after such nuisance rejection under a stated covariance. It does not establish that an estimated reference shape or noise model is itself accurate.