  In this paper, we study an energy efficiency maximization problem in uplink
for D2D communications underlaid with cellular networks on multiple bands.
Utilizing stochastic geometry, we derive closed-form expressions for the
average sum rate, successful transmission probability, and energy efficiency of
cellular and D2D users. Then, we formulate an optimization problem to jointly
maximize the energy efficiency of D2D and cellular users and obtain optimum
transmission power of both D2D and cellular users. In the optimization problem,
we guarantee the QoS of users by taking into account the success transmission
probability on each link. To solve the problem, first we convert the problem
into canonical convex form. Afterwards, we solve the problem in two phases,
energy efficiency maximization of devices and energy efficiency maximization of
cellular users. In the first phase, we maximize the energy efficiency of D2D
users and feed the solution to the second phase where we maximize the energy
efficiency of cellular users. Simulation results reveal that significant energy
efficiency can be attained e.g., 10% energy efficiency improvement compared to
fix transmission power in high density scenario.
