Mathematical Considerations in Energy Spectrum Recovery for Digital Energy Spectrometers

Energy spectra of particles allow people to observe underlying matter composition and properties in particle physics applications by depicting the probability density function of particle energy. As the most widely used method to estimate energy spectra, energy histograms discretely represent them with precision depending on adopted bin widths. Till recently, the range of bin widths for the nondistortion recovery of energy spectra is still indefinite, especially when digital spectrometers are involved. This can lead to the miss of potential discoveries in studies relying on spectral analysis. Here we propose a novel method that recovers energy spectra from histograms without distortion and the corresponding quantitative method of selecting bin widths with the bandwidths of energy spectra. To do so, we model histograms as equally spaced samples of energy spectra and measure their bandwidths with the frequency spectra of the histograms. By the sampling theorem, we choose bin widths quantitatively and recover energy spectra without distortion by interpolating histograms, showing that our method could be used in the accurate analysis of particle energy data and reliable evaluation of energy spectrometers. For digital spectrometers, we also show that bin widths should be chosen as multiples of energy quantization errors (EQEs) caused by the limited analog-to-digit resolution of digital spectrometers to avoid periodic artifacts in energy histograms.