Stability of DPCM Coders for Television

In digital pulse-code modulation (DPCM) transmission of pictures, often a two-dimensional (2-D) predictor is considered for use in the feedback loop. This paper considers the problem of stability of this 2-D recursive filter. It is assumed the predictor is designed to minimize the mean-square error over the space of all linear predictors. For positive definite correlation arrays it is shown that the first-order recursive filter which uses the previous element in the same row and column can be unstable. It is also shown that the bilinear recursive filter which uses the three nearest neighbors to the left and above may be unstable. When the correlation function of the data is isotropic, it is shown that stability follows for these two classes of filters.