<i>UltraStat</i>: Ultrafast Spectroscopy\nbeyond the Fourier Limit Using Bayesian Inference

The discrete Fourier transform (dFT) plays a central\nrole in many\nultrafast experiments, allowing the recovery of spectroscopic observables\nfrom time-domain measurements. In resonant experiments when population\nrelaxation and coherence components of the signal coexist, the dFT\nis usually preceded by multiexponential fitting to remove the large\npopulation term. However, this procedure results in errors in both\nthe recovered decay rates and the line shapes of the coherence spectral\ncomponents. While other methods such as linear prediction singular\nvalue decomposition fit both terms simultaneously, they are limited\nto specific models that may not represent the true signal. These methods\ndo not allow for systematic noise analysis or error estimation and\nrequire <i>a priori</i> knowledge of the signal rank. Here,\nwe describe a general approach to parameter estimation in ultrafast\nspectroscopy<i>UltraStat</i>grounded in\nBayesian analysis without the limitations set by Fourier theory. Using\nsimulated, but realistic data, we demonstrate in a statistical sense\nhow <i>UltraStat</i> provides accurate parameter estimation\nin the presence of many experimental constraints: noise, signal truncation,\nlimited photon budget, and nonuniform sampling. <i>UltraStat</i> provides superior resolution compared to the dFT, up to an order\nof magnitude in cases where the line shapes are well-approximated.\nIn these cases, we establish that primarily noise, not sampling, limits\nspectral resolution. Moreover, we show that subsampling may reduce\nthe number of acquired points by 90% compared to the Nyquist–Shannon\ncriteria. <i>UltraStat</i> greatly improves parameter estimation\nby providing statistically bound spectral and dynamics analysis, pushing\nthe limits of ultrafast science.