{'fraction_non_alphanumeric': 0.01749271137026239, 'fraction_numerical': 0.0, 'mean_word_length': 5.481132075471698, '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}}
Perfect Reconstruction Filter Banks with Rational Sampling Rates in One and Two Dimensions

Multirate filter banks with integer sampling rate changes are used widely in subband coding and transmultiplexing. A more general scheme is obtained when one allows arbitrary rational sampling rate changes, leading to non-uniform division of the frequency spectrum. This paper shows how to obtain arbitrary non-uniform perfect reconstruction filter banks by building trees of divisions into two (possibly unequal) channels. The construction uses the commutativity of subsampling and upsampling. The commutativity result, which is well known in one dimension, is then extended to two dimensions.