The traditional information theoretic approach to studying feedback is to
consider ideal instantaneous high-rate feedback of the channel outputs to the
encoder. This was acceptable in classical work because the results were
negative: Shannon pointed out that even perfect feedback often does not improve
capacity and in the context of symmetric DMCs, Dobrushin showed that it does
not improve the fixed block-coding error exponents in the interesting high rate
regime. However, it has recently been shown that perfect feedback does allow
great improvements in the asymptotic tradeoff between end-to-end delay and
probability of error, even for symmetric channels at high rate. Since gains are
claimed with ideal instantaneous feedback, it is natural to wonder whether
these improvements remain if the feedback is unreliable or otherwise limited.
Here, packet-erasure channels are considered on both the forward and feedback
links. First, the feedback channel is considered as a given and a strategy is
given to balance forward and feedback error correction in the suitable
information-theoretic limit of long end-to-end delays. At high enough rates,
perfect-feedback performance is asymptotically attainable despite having only
unreliable feedback! Second, the results are interpreted in the zero- sum case
of "half-duplex" nodes where the allocation of bandwidth or time to the
feedback channel comes at the direct expense of the forward channel. It turns
out that even here, feedback is worthwhile since dramatically lower asymptotic
delays are possible by appropriately balancing forward and feedback error
correction.
The results easily generalize to channels with strictly positive
zero-undeclared-error capacities.Comment: 20 pages, 6 pages, submitted to IEEE Transactions on Information
Theory, an earlier version was presented at ITA '07 in UCS
