DAC Spectrum of Binary Sources with Equally-Likely Symbols

Though distributed arithmetic coding (DAC) is an effective implementation of Slepian-Wolf coding, its performance, which is closely linked with its decoding complexity, has not received a thorough analysis. With binary sources with equally-likely symbols as the research object, this paper develops the DAC spectrum and makes use of it as a tool to answer the complexity problem of the ideal DAC decoder. Based on an in-depth analysis on DAC decoding process, we define the DAC spectrum and propose to find it by a recursive formulation. Firstly, the initial DAC spectrum is constrained by a functional equation and the Fourier transform is utilized to obtain its general explicit form. Secondly, an equation is given through which stage-(i+1) DAC spectrum can be deduced from stage-i DAC spectrum. A numerical method is also proposed for calculating DAC spectrum, whose convergency is proved. To measure the complexity of the ideal DAC decoder, we define the expansion factor and relate it to DAC spectrum. It is proved that if binary symbols 0 and 1 are mapped onto partially overlapped intervals [0, q) and [1-q, 1) respectively, the expansion factor will converge to 2q, i.e., the complexity of the ideal rate-α DAC decoder is approximately O(2^{n(1-α)}).