{'fraction_non_alphanumeric': 0.0225, 'fraction_numerical': 0.0, 'mean_word_length': 4.977611940298507, '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}}
Rate-distortion analysis of Delta-Sigma modulators

Delta-Sigma modulators are often utilized to convert analog signals into digital signals. The quantization error of a Delta-Sigma modulator can be reduced by oversampling. However, oversampling increases the number of bits per time if the same number of bits are assigned to each output of the quantizer. Due to limited bandwidth, the rate-distortion relation is necessary to balance the rate and the distortion. In this paper, we analyze the relationship between the rate and the distortion of an optimal scalar Delta-Sigma modulator that minimizes the variance of the error in the output of the system connected to the Delta-Sigma modulator. Numerical examples are provided to show rate-distortion relations of the optimal Delta-Sigma modulators.