In two-way networks, nodes act as both sources and destinations of messages. This allows for "adaptation" at or "interaction" between the nodes - a node's channel inputs may be functions of its message(s) and previously received signals. How to best adapt is key to two-way communication, rendering it challenging. However, examples exist of point-to-point channels where adaptation is not beneficial from a capacity perspective. We ask whether analogous examples exist for multi-user two-way networks. We first consider deterministic two-way channel models: the binary modulo-2 addition channel and a generalization thereof, and the linear deterministic channel. For these deterministic models we obtain the capacity region for the two-way multiple access broadcast channel, the two-way Z channel and the two-way interference channel (IC). In all cases we permit all nodes to adapt channel inputs to past outputs (except for portions of the linear deterministic two-way IC where we only permit 2 of the 4 nodes to fully adapt). However, we show that this adaptation is useless from a capacity region perspective and capacity is achieved by strategies where the channel inputs at each use do not adapt to previous inputs. Finally, we consider the Gaussian two-way IC, and show that partial adaptation is useless when the interference is very strong. In the strong and weak interference regimes, we show that the non-adaptive Han and Kobayashi scheme utilized in parallel in both directions achieves to within a constant gap for the symmetric rate of the fully (some regimes) or partially (remaining regimes) adaptive models. The central technical contribution is the derivation of new, computable outer bounds which allow for adaptation. Inner bounds follow from non-adaptive achievability schemes of the corresponding one-way channel models.
The second channel model we consider is the two-way Z channel, with 6 messages. The one-way Z channel (with 3 messages, rather than the Z Interference channel with 2 messages) was first studied in @cite_12 , in which a general outer bound, and a matching inner bound for a special class of degraded Z channels are obtained. The capacity region of the one-way deterministic Z channel with invertibility constraints similar in flavor to those in @cite_40 , is found in @cite_31 , which will be of use here.
@cite_40 @cite_31 @cite_12
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We study the discrete memoryless Z-interference channel where the transmitter of the pair that suffers from interference is cognitive. We first provide an outer bound on the capacity region of this channel. We then show that, when the channel of the transmitter-receiver pair that does not experience interference is deterministic and invertible, our proposed outer bound matches the best known inner bound. The obtained results imply that in the considered channel, superposition encoding at the noncognitive transmitter as well as Gel'fand-Pinsker encoding at the cognitive transmitter is needed in order to minimize the impact of interference. As a byproduct of the obtained capacity region, we obtain the capacity under the generalized Gel'fand-Pinsker setting where a transmitter-receiver pair communicates in the presence of interference noncausally known at the encoder. A two transmitter two receiver channel where independent data is sent on each communication link of the system is considered. We consider a three-link system, termed the "Z" channel, in which one transmitter is connected to both receivers while the other transmitter is only connected to one of the receivers. Thus, the "Z" channel has a three dimensional capacity region. We characterize the capacity region of a special class of degraded "Z" channels and establish an achievable region for the Gaussian "Z" channels. Finally, we use genie-aided techniques previously used for the interference and broadcast channels to obtain an outer bound for general "Z" channels. We generalize the Gel'fand-Pinsker model to encompass the setup of a memoryless multiple-access channel (MAC). According to this setup, only one of the encoders knows the state of the channel (noncausally), which is also unknown to the receiver. Two independent messages are transmitted: a common message and a message transmitted by the informed encoder. We find explicit characterizations of the capacity region with both noncausal and causal state information. Further, we study the noise-free binary case, and we also apply the general formula to the Gaussian case with noncausal channel state information, under an individual power constraint as well as a sum power constraint. In this case, the capacity region is achievable by a generalized writing-on-dirty-paper scheme. We consider the generalized dirty-paper channel Y=X+S+N,E X sup 2 spl les P sub X , where N is not necessarily Gaussian, and the interference S is known causally or noncausally to the transmitter. We derive worst case capacity formulas and strategies for "strong" or arbitrarily varying interference. In the causal side information (SI) case, we develop a capacity formula based on minimum noise entropy strategies. We then show that strategies associated with entropy-constrained quantizers provide lower and upper bounds on the capacity. At high signal-to-noise ratio (SNR) conditions, i.e., if N is weak relative to the power constraint P sub X , these bounds coincide, the optimum strategies take the form of scalar lattice quantizers, and the capacity loss due to not having S at the receiver is shown to be exactly the "shaping gain" 1 2log(2 spl pi e 12) spl ap 0.254 bit. We extend the schemes to obtain achievable rates at any SNR and to noncausal SI, by incorporating minimum mean-squared error (MMSE) scaling, and by using k-dimensional lattices. For Gaussian N, the capacity loss of this scheme is upper-bounded by 1 2log2 spl pi eG( spl Lambda ), where G( spl Lambda ) is the normalized second moment of the lattice. With a proper choice of lattice, the loss goes to zero as the dimension k goes to infinity, in agreement with the results of Costa. These results provide an information-theoretic framework for the study of common communication problems such as precoding for intersymbol interference (ISI) channels and broadcast channels.