Error Exponent Region for Gaussian Multiple Access Channels and Gaussian Broadcast Channels

Error exponents for single-user additive white Gaussian noise (AWGN) channels provide bounds on the rate of exponential decay of the average probability of error as a function of the block length of the random codebooks. In this paper, we only consider Gallager-type lower bounds of error exponents [1, 2]. For simplicity, we use the term “error exponent” to mean the maximum of random coding exponent and expurgated exponent throughout this paper, though actually should be called a lower bound of error exponents. The concept of error exponents was extended to Gaussian MAC channels [3, 4], where random coding exponents were derived. Recently, Zheng et al. considered error exponents, in high signal-to-noise ratio (SNR), for single-user wireless multi-input-multi-output (MIMO) channels [5], and for wireless MIMO multiple access channels [6]1. Conceptually, the error exponent is a function of the channel capacity C and the transmission rate R in a single user channel (see Fig. 1(a)). However, error exponents for a multi-user network is quite different from the error exponent for a singleuser channel. Each user in a multi-user network is associated with his own error exponent. Therefore, there are multiple error exponents for a given multi-user channel. In contrast to all previous works, however, we make the following observations. Consider the capacity region of a two-user multiple access channel as shown in Fig. 1(b). As expected, the error exponents for the two users are functions of both the transmission rate point A and the channel capacity. However, unlike the case in a single user channel where channel capacity boundary is a single point, in a multi-user channel we have multiple points on the capacity boundary (e.g. A1,A2 in Fig. 1(b)). Thus it is expected that one can get different error exponents depending on which particular point on the capacity boundary is considered. Furthermore, it might be possible to trade off error exponents between users by considering different points on the capacity boundary. For instance, consider a rate point A with respect to a boundary point A1 in Fig. 1(b). It is intuitive to expect that the error exponent for user 1 is smaller than that of user 2. On the other hand, if we consider point A with respect to point A2, we then expect error exponent for user 1 is larger than that of user 2. Therefore, tradeoff of error exponent between users might be possible by considering different points on the capacity boundary. Similar behavior can be observed in other multi-user channels, e.g. the broadcast channels in Fig. 1(c). It is our intention in this paper to formalize these ideas and show that such tradeoff indeed exists and suggest a constructive strategy to achieve it.