The AWGN Red Alert Problem

Consider the following unequal error protection scenario. One special message, dubbed the "red alert" message, is required to have an extremely small probability of missed detection. The remainder of the messages must keep their average probability of error and probability of false alarm below a certain threshold. The goal then is to design a codebook that maximizes the error exponent of the red alert message while ensuring that the average probability of error and probability of false alarm go to zero as the blocklength goes to infinity. This red alert exponent has previously been characterized for discrete memoryless channels. This paper completely characterizes the optimal red alert exponent for additive white Gaussian noise channels with block power constraints.Comment: 13 pages, 10 figures, To appear in IEEE Transactions on Information Theor

Balancing forward and feedback error correction for erasure channels with unreliable feedback

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

Fundamental limitations on communication channels with noisy feedback: information flow, capacity and bounds

Since the success of obtaining the capacity (i.e. the maximal achievable transmission rate under which the message can be recovered with arbitrarily small probability of error) for non-feedback point-to-point communication channels by C. Shannon (in 1948), Information Theory has been proved to be a powerful tool to derive fundamental limitations in communication systems. During the last decade, motivated by the emerging of networked systems, information theorists have turned lots of their attention to communication channels with feedback (through another channel from receiver to transmitter). Under the assumption that the feedback channel is noiseless, a large body of notable results have been derived, although much work still needs to be done. However, when this ideal assumption is removed, i.e., the feedback channel is noisy, only few valuable results can be found in the literature and many challenging problems are still open. This thesis aims to address some of these long-standing noisy feedback problems, with concentration on the channel capacity. First of all, we analyze the fundamental information flow in noisy feedback channels. We introduce a new notion, the residual directed information, in order to characterize the noisy feedback channel capacity for which the standard directed information can not be used. As an illustration, finite-alphabet noisy feedback channels have been studied in details. Next, we provide an information flow decomposition equality which serves as a foundation of other novel results in this thesis. With the result of information flow decomposition in hand, we next investigate time-varying Gaussian channels with additive Gaussian noise feedback. Following the notable Cover-Pombra results in 1989, we define the n-block noisy feedback capacity and derive a pair of n-block upper and lower bounds on the n-block noisy feedback capacity. These bounds can be obtained by efficiently solving convex optimization problems. Under the assumption of stationarity on the additive Gaussian noises, we show that the limits of these n-block bounds can be characterized in a power spectral optimization form. In addition, two computable lower bounds are derived for the Shannon capacity. Next, we consider a class of channels where feedback could not increase the capacity and thus the noisy feedback capacity equals to the non-feedback capacity. We derive a necessary condition (characterized by the directed information) for the capacity-achieving channel codes. The condition implies that using noisy feedback is detrimental to achievable rate, i.e, the capacity can not be achieved by using noisy feedback. Finally, we introduce a new framework of communication channels with noisy feedback where the feedback information received by the transmitter is also available to the decoder with some finite delays. We investigate the capacity and linear coding schemes for this extended noisy feedback channels. To summarize, this thesis firstly provides a foundation (i.e. information flow analysis) for analyzing communications channels with noisy feedback. In light of this analysis, we next present a sequence of novel results, e.g. channel coding theorem, capacity bounds, etc., which result in a significant step forward to address the long-standing noisy feedback problem