= Raman spectral rebinning with conserved integrated counts
{c}
{title2=$\mathbf N^{\prime}=A\mathbf N$}
\b[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 matrix
$$
A_{ji}=\frac{|I_i\cap J_j|}{\Delta\lambda_i},\qquad
N'_j=\sum_iA_{ji}N_i.
$$
When the new bins partition the old covered range, $\sum_jA_{ji}=1$, hence $\sum_jN'_j=\sum_iN_i$. 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 $604\,\mathrm{nm}$. Then
$$
A=\begin{pmatrix}1/2&0\\1/2&1/2\\0&1/2\end{pmatrix},\qquad
\mathbf N'=\begin{pmatrix}50\\200\\150\end{pmatrix}.
$$
The 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 $C'=ACA^T$ 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.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Instrumentation/Figure 2 and absolute-shift calibration; Data Collection]{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.
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