Recently, the energy efficiency (EE) has become an important factor when designing new wireless communication systems. Due to economic and environmental challenges, new trends and efforts are oriented toward “green” communication especially for energy-constrained applications such as wireless sensors network and cognitive radio. To this end, we analyze the power allocation scheme that maximizes the EE defined as rate over the total power including circuit power. We derive an explicit expression of the optimal power with instantaneous channel gain based on EE criterion. We show that the relation between the EE and the spectral efficiency (SE) when the optimal power is adopted is strictly increasing in contrast with the SE-EE trade-off discussed in the literature. We also solve a non-convex problem and compute explicitly the optimal power for ergodic EE under either a peak or an average power constraint. When the instantaneous channel is not available, we provide the optimal power equation and compute simple sub-optimal power. In the numerical results, we show that the sup-optimal solution is very close to the optimal solution. In addition, we show that the absence of the channel state information (CSI) only affects the EE and the SE performances at high power regime compared to the full CSI case.