Fisher information analysis of digital pulse timing

Abstract

In positron emission tomography (PET), it is possible to obtain useful time-of-flight (TOF) information if the gamma-ray detectors have less than 750 ps timing resolution. Including TOF information to the reconstruction algorithm increases the signal-to-noise ratio (SNR). We obtain timing information by analyzing digital sampled waveforms, where the sampling frequency and number of points acquired affect timing estimation. An efficient data acquisition system acquires the minimum number of samples that contains the most timing information for a desired resolution. We describe a maximum-likelihood (ML) estimation algorithm to assign a time stamp to digital pulses. The method is based on a contracting-grid search algorithm that can be implemented in a field-programmable gate array (FPGA). The Fisher information (FI) matrix that corresponds to the likelihood in the ML estimator quantifies the amount of timing information that can be extracted from the waveforms. Fisher information analyses on different segments of the waveform allow us to determine the amount of data that we need to acquire in order to obtain a desired timing resolution. We present how we simulate waveforms for ML estimation and FI analysis, the ML estimation algorithm and the timing resolution obtained from experimental data using a LaBr 3 crystal and two photomultiplier tubes (PMTs). The results show that for lengthening segments of the pulse, timing resolution approaches a limit. This information will be used to build an efficient DAQ with reduced complexity and cost that nonetheless preserves full timing performance.