On the continuity of the stationary state distribution of DPCM

Continuity and singularity properties of the stationary state distribution of differential pulse code modulation (DPCM) are explored. Two-level DPCM (i.e. delta modulation) operating on a first-order autoregressive source is considered, and it is shown that, when the magnitude of the DPCM prediction coefficient is between zero and one-half, the stationary state distribution is singularly continuous; i.e. it is not discrete but concentrates on an uncountable set with a Lebesgue measure of zero. Consequently, it cannot be represented with a probability density function. For prediction coefficients with magnitude greater than or equal to one-half, the distribution is pure, i.e. either absolutely continuous and representable with a density function, or singular. This problem is compared to the well-known and still substantially unsolved problem of symmetric Bernoulli convolutions