Relationship of average transmitted and received energies in adaptive transmission

This paper studies the analytical relationship between the average transmitted and received energies under several adaptive transmitter power control methods, including water filling, truncated power inversion, and downlink beamforming. The study is applicable to many fading channel scenarios, including frequency-nonselective, frequency-selective, and multiple-input-multiple-output (MIMO) channels. Both the average transmitted and received energies are commonly used in performance comparisons, and the selection depends on what one wants to investigate. The transmitted energy is known to be the basic system resource. In the case of adaptive transmission, the average transmitted energy should, in general, be used instead of the average received energy. The use of transmitted energy leads to the normalization problem of the channel. The ratio of received energy to transmitted energy is the energy gain of the channel. All physical systems follow an energy-conservation law, which implies that the energy gain of the channel is less than or equal to 1. The major approaches for normalization include the setting of either the average energy gain or the peak energy gain to unity. In the normalization, the average energy gain is defined for a signal whose energy is uniformly distributed across the frequency and spatial dimensions. The peak energy gain of many mathematical fading models is not bounded, and those models cannot be normalized by the peak energy gain. We show that the proper normalization of the mathematical model and the selection of the correct performance measure are of critical importance in comparative performance analysis of adaptive transmission systems.