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Raman spectral rebinning with conserved integrated counts redistributes finite-bin measurements onto a new spectral grid while preserving their summed quantity over the same covered range. It is different from relabeling bin centres or interpolating peak heights. The parent spectral-density conversion between wavelength and Raman shift derives the continuous Jacobian; finite bins require a stated assumption about unresolved variation inside each bin.

For an original piecewise-constant wavelength-density model, let old bin $I_i$ have width $\Delta {\lambda }_i$ and count $N_i$. Invert the new Raman-shift edges to wavelength intervals $J_j$. Define the overlap matrixWhen the new bins partition the old covered range, ${\sum }_j{A}_{ji}=1$, hence ${\sum }_j{N}_j^{\prime }={\sum }_i{N}_i$. Clipping the new range instead excludes the corresponding counts and must be reported.

For original old wavelength bins $[600,602]$ and $[602,604]\mathrm{nm}$ with counts $100$ and $300$, choose new edges corresponding to wavelengths $600$, $601$, $603$ and $604\mathrm{nm}$. ThenThe total remains $400$. A different within-bin model can distribute that total differently; the original measurements do not resolve every sub-bin feature.

Covariance of count-conserving rebinned Raman channels transforms as $C^{\prime }=AC{A}^T$ for this fixed linear operation. Fractional rebinned estimates are therefore not automatically independent Poisson photon counts. Splitting one measured bin among neighbours creates shared uncertainty; finer output pixels do not create additional spectral resolution.

The primary FSRS instrument records a dispersed spectrum on an array and calibrates its Raman-shift axis.^[ref] The overlap algorithm above is an original illustrative reconstruction, not a claim about that instrument's published processing. Apply radiometric response and background corrections consistently with the quantity whose integral is being conserved.

Rank deficiency from deterministic spectral upsampling occurs because a deterministic map cannot manufacture independent measurements. For $\mathbf{y}=A\mathbf{n}$ and input covariance matrix $C$,When the output has more channels than the input, its covariance matrix is necessarily singular. Covariance of count-conserving rebinned Raman channels gives the conserved-count example and its full propagated uncertainty.^[ref]

Using that parent's two-to-three channel map, the output always satisfies $y_1-y_2+y_3=0$. If the input readings are $20$ and $30$, the rebinned vector is $(10,25,15{)}^T$. Recovering the original readings as $n_1=2y_1$ and $n_2=2y_3$ shows that the third displayed number is not a third independent observation.

For an original fitting example, suppose both input means equal an unknown $\theta$ and their independent measurement-error variances are known to be $25$. Fitting in the original channels gives $\hat{\theta }=(20+30)/2=25$, with variance $25/2=12.5$. A correct generalized least squares fit after rebinning gives the same result and uncertainty. It must use the singular covariance on its supported range, for example through the Moore-Penrose inverse, together with the exact linear constraints.

The range restriction in singular-covariance spectral fitting matters if a proposed output model fails those constraints. A residual along $(1,-1,1{)}^T$ cannot arise from this rebinning model at all. Merely inserting a pseudoinverse into a quadratic objective would assign zero penalty to that impossible direction. Either enforce the range condition or supply a physically justified additional noise model; adding a small diagonal term solely to make the matrix invertible changes the statistical assumptions.

The motivating Raman apparatus measures detector-array spectra before any subsequent numerical grid change.^[ref] Working on the original grid often avoids this singular fitting problem while retaining the actual measurement information.