{'fraction_non_alphanumeric': 0.01907032181168057, 'fraction_numerical': 0.0, 'mean_word_length': 5.412213740458015, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Estimating the Wigner distribution of linear time-invariant dynamical systems

An estimation algorithm for the Wigner distribution (time-frequency representation) of the unmeasured states of a linear time-invariant system is presented. Given that the inputs and outputs are measured, the algorithm involves designing a Luenberger-like observer for each frequency of interest. Under noise-free conditions, we show that the es- timates converge to the true Wigner distribution under a detectability assumption on the time-frequency representation. The estimation algorithm provides estimates which converge to a neighbourhood of the true Wigner distribution where its norm is dependent on the norm of the measurement noise. We also illustrate the efficacy of the estimation algorithm on an academic example and a model of neuron populations.