# Transformation to get rid of distribution multiplets

Are there any well-known distribution transformations that account for doublets, triplets, higher multiplets due to time series independent peaks occasionally being too close to resolve? I feel like this would be a common problem in techniques from single photon detection to flow cytometry where photon energy or cell size are calculated from peak area. The area distribution of doublets would be double the true distribution, the autoconvolution? Multiplet frequencies would follow a Poisson distribution and could be represented as repeated autoconvolutions of the true energy or size distributions. I think the observed distribution (OD) would be a series of these Poisson coefficients (P(n)) and no, double, triple,… autoconvolutions of the true distribution (TD).

$$OD(d) = P(1) TD(d) + P(2)\int_{-\infty}^{\infty}TD(x)TD(d-x)dx + P(3)\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}TD(x)TD(y)TD(d-x-y)dxdy + ...$$

Is there an easy way to solve for the true distribution? All help is appreciated!