Distributed Arithmetic Coding (DAC) proves to be an effective implementation of Slepian-Wolf Coding (SWC), especially for short data blocks. To study the property of DAC codewords, the author has proposed the concept of DAC codeword spectrum. For equiprobable binary sources, the problem was formatted as solving a system of functional equations. Then, to calculate DAC codeword spectrum in general cases, three approximation methods have been proposed. In this paper, the author makes use of DAC codeword spectrum as a tool to answer an important question: how many (including proper and wrong) paths will be created during the DAC decoding, if no path is pruned? The author introduces the concept of another kind of DAC codeword spectrum, i.e. time spectrum, while the originally-proposed DAC codeword spectrum is called path spectrum from now on. To measure how fast the number of decoding paths increases, the author introduces the concept of expansion factor which is defined as the ratio of path numbers between two consecutive decoding stages. The author reveals the relation between expansion factor and path/time spectrum, and proves that the number of decoding paths of any DAC codeword increases exponentially as the decoding proceeds. Specifically, when symbols `0 and `1 are mapped onto intervals [0, q) and [(1-q), 1), where 0.5<q<1, the author proves that expansion factor converges to 2q as the decoding proceeds.
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