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 per-antenna 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 closed-form 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. Last, the theoretical results are illustrated by numerical simulations.

This paper studies optimal transmit strategies for multiple-input multiple-output (MIMO) Gaussian channels with joint sum and per-antenna power constraints. It is shown that if an unconstraint optimal allocation for an antenna exceeds a per-antenna power constraint, then the maximal power for this antenna is used in the constraint optimal transmit strategy. This observation is then used in an iterative algorithm to compute the optimal transmit strategy in closed-form. Finally, a numerical example is provided to illustrate the theoretical results.

The energy efficiency (EE) of a multi-user multi-relay 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 constraint. Relay scheduling (RS) and power allocation (PA) are applied to schedule the relay states (transmitting, sleeping, and so on) 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. Furthermore, the EE is converted into a subtractive form based on the fractional programming. Then, a convex mixed-integer nonlinear problem is eventually obtained. With a generalized outer approximation 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.

We consider multiple-input single-output (MISO) Gaussian channels with joint sum and per-antenna power constraints. A closed-form solution of the optimal beamforming vector is derived which achieves the maximal transmission rate. The result shows that if the sum power constraint only optimal power allocation violates a per-antenna power constraint then the joint power constraint optimal power allocation is at the intersection of the sum power constraint and the per-antenna power constraints.