Source Coding for Multiple Descriptions II: A Binary Source

Let be a sequence of independent drawings of the binary random variable X, where Pr{X = 0} = Pr{X = 1} = 1/2. Assume that this sequence appears at a rate of 1 symbol per second as the output of a data source. (Refer to Fig. 1.) An encoder observes this sequence and emits two binary sequences at rates R, R2 ^ 1. These sequences are such that by observing either one, a decoder can recover a good approximation to the source output, and by observing both sequences, a decoder can obtain a better approximation to the source output. Letting D, D2, and Do be the error rates which result when the streams at rate R i, rate R2, and both streams are used by a decoder, respectively, our problem is to determine (in the usual Shannon sense) the set of achievable quintuples (Rh R2, Do, Di, D2). Our main result is a "converse" theorem which gives a necessary condition on the achiev2281 Fig. 1--Communication system.