 Product-null spectral contrasts

A product-null spectral contrast is a weighted combination of spectral measurements that cancels a modeled product contribution while retaining sensitivity to the carrier of interest. It is useful when carrier consumption and product formation both change the absorbance recorded in an inhibited autoxidation assay.

After background correction and with path length included in the response coefficients, write a measurement vector asHere $\mathbf{d}$ and $\mathbf{p}$ are known carrier and product spectral response columns. A contrast ${\hat{c}}_D={\mathbf{w}}^{\mathsf{T}}\mathbf{A}$ removes the product and has unit carrier response whenIf the columns are proportional, these constraints are incompatible. More measurements do not help if they add no independent spectral information.

An original three-channel example uses scaled coefficient columns $\mathbf{d}=(3,1,0{)}^{\mathsf{T}}$ and $\mathbf{p}=(1,1,1{)}^{\mathsf{T}}$. The weights $\mathbf{w}=(5,-1,-4{)}^{\mathsf{T}}/14$ satisfy both constraints. Stipulated concentrations $(c_D,c_P)=(0.2,0.1)$ give $\mathbf{A}=(0.7,0.3,0.1{)}^{\mathsf{T}}$ and the contrast returns $0.2$. The coefficient and concentration units have been absorbed into the scaling; these are not measured dye coefficients.

For independent equal-variance channel noise ${\sigma }^2$, the example has variance ${\sigma }^2\Vert \mathbf{w}{\Vert }^2=3{\sigma }^2/14$. These are the minimum-variance weights under the stated noise model: project $\mathbf{d}$ perpendicular to $\mathbf{p}$, then normalize that residual to unit carrier response. The Cauchy-Schwarz inequality shows that any other vector with the same constraints has at least as large a norm. A two-channel difference using just the first two measurements also isolates the carrier, but weights $(1/2,-1/2,0)$ give the larger variance ${\sigma }^2/2$.

For general covariance matrix $\Sigma$, the variance is ${\mathbf{w}}^{\mathsf{T}}\Sigma \mathbf{w}$, so correlated or unequal noise changes the optimal weighting.^[ref] The coupled chemical readout remains the one described by the primary signal-carrier method.^[ref] Additional products or changing protonation require additional response columns; a cancellation calibrated for one product does not remove every unknown spectral contribution.