Source coding for multiple descriptions

This paper discusses an idealization of the situation in which it is required to send information over two separate channels, as in a packet communication network, and it is desired to recover as much as possible of the original information should one of the channels break down. Let {X
k
}
^∞
k–1
be a sequence of independent copies of the binary random variable X, where Pr[X = 0} = Pr{X = 1} = 1/2. Assume that this sequence appears at a rate of one symbol per second as the output of a data source. An encoder observes this sequence and emits two binary sequences at rates R
1
, R
2
≤ 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
1
, D
2
, D
0
be the error rates that result when the streams at rate R
1
, rate R
2
, and both streams are used by a decoder, respectively, our problem is to determine (in the usual Shannon sense) the set of achievable quintuples (R
1
, R
2
, D
1
, D
1
, D
2
). Our main result is a “converse” theorem that gives a necessary condition on the achievable quintuples.