Abstract-In this paper, we study an optimal transmit strategy for multiple-input single-output (MISO) Gaussian channels with joint sum and per-antenna power constraints. We study in detail the interesting case where the sum of the per-antenna power constraints is larger than sum power constraint. A closed-form characterization of an optimal beamforming strategy is derived. It is shown that we can always find an optimal beamforming transmit strategy that allocates the maximal sum power with phases matched to the complex channel coefficients. The main result is a simple recursive algorithm to compute the optimal power allocation. Whenever the optimal power allocation of the corresponding problem with sum power constraint only exceeds per-antenna power constraints, it is optimal to allocate maximal per-antenna power to those antennas to satisfy the per-antenna power constraints. The remaining power is divided amongst the other antennas whose optimal allocation follows from a reduced joint sum and per-antenna power constraints problem of smaller channel coefficient dimension and reduced sum power constraint. Finally, the theoretical results are illustrated by numerical examples.

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This paper studies the optimal tradeoff between secrecy and non-secrecy rates of the MISO wiretap channels for different power constraint settings: sum power constraint only, per-antenna power constraints only and joint sum and perantenna power constraints. The problem is motivated by the fact that channel capacity and secrecy capacity are generally achieved by different transmit strategies. First, a necessary and sufficient condition to ensure a positive secrecy capacity is shown. The optimal tradeoff between secrecy rate and transmission rate is characterized by a weighted rate sum maximization problem. Since this problem is not necessarily convex, equivalent problem formulations are introduced to derive the optimal transmit strategies. Under sum power constraint only, a closedform solution is provided. Under per-antenna power constraints, necessary conditions to find the optimal power allocation are derived. Sufficient conditions are provided for the special case of two transmit antennas. For the special case of aligned channels, the optimal transmit strategies can deduced from an equivalent point-to-point channel problem. Lastly, the theoretical results are illustrated by numerical simulations.Index Terms-Wiretap channels, per-antenna power constraints, secrecy rate, optimal transmit strategy.

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Abstract-The energy efficiency (EE) of a multi-user multirelay system with the maximum diversity network coding (MDNC) is studied. We explicitly find the connection among the outage probability, energy consumption and EE and formulate the maximizing EE problem under the outage probability constraints. Relay scheduling (RS) and power allocation (PA) are applied to schedule the relay states (transmitting, sleeping, etc) and optimize the transmitting power under the practical channel and power consumption models. Since the optimization problem is NP-hard, to reduce computational complexity, the outage probability is first tightly approximated to a log-convex form. Further, the EE is converted into a subtractive form based on the fractional programming. Then a convex mixedinteger nonlinear problem (MINLP) is eventually obtained. With a generalized outer approximation (GOA) algorithm, RS and PA are solved in an iterative manner. The Pareto-optimal curves between the EE and the target outage probability show the EE gains from PA and RS. Moreover, by comparing with the no network coding (NoNC) scenario, we conclude that with the same number of relays, MDNC can lead to EE gains. However, if RS is implemented, NoNC can outperform MDNC in terms of the EE when more relays are needed in the MDNC scheme.Index Terms-nergy efficiency-outage probability tradeoff, network coding, relay scheduling, power allocation, generalized outer approximation (GOA).nergy efficiency-outage probability tradeoff, network coding, relay scheduling, power allocation, generalized outer approximation (GOA).E