A new capacity upper bound for "relay-with-delay" channel
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The paper investigates the effect of link delays on the capacity of relay networks. The relay-with-delay is defined as a relay channel with relay encoding delay d isin Z of units, or equivalently, a delay of units on the link from the sender to the relay, zero delay on the links from the transmitter to the receiver and from the relay to the receiver, and zero relay encoding delay. Two special cases are studied. The first is the relay-with-unlimited look-ahead, where each relay transmission can depend on its entire received sequence, and the second is the relay-without-delay, where the relay transmission can depend only on current and past received symbols, i.e., d=0. Upper and lower bounds on capacity for these two channels that are tight in some cases are presented. It is shown that the cut-set bound for the classical relay channel, corresponding to the case where d=1, does not hold for the relay-without-delay. Further, it is shown that instantaneous relaying can be optimal and can achieve higher rates than the classical cut-set bound. Capacity for the classes of degraded and semi-deterministic relay-with-unlimited-look-ahead and relay-without-delay are established. These results are then extended to the additive white Gaussian noise (AWGN) relay-with-delay case, where it is shown that for any dles0, capacity is achieved using amplify-and-forward when the channel from the sender to the relay is sufficiently weaker than the other two channels. In addition, it is shown that a superposition of amplify-and-forward and decode-and-forward can achieve higher rates than the classical cut-set bound. The relay-with-delay model is then extended to feedforward relay networks. It is shown that capacity is determined only by the relative delays of paths from the sender to the receiver and not by their absolute delays. A new cut-set upper bound that generalizes both the classical cut-set bound for the classical relay and the upper bound for the relay-without-delay on capacity is - established.