  This paper investigates the optimum energy efficiency (EE) and the
corresponding spectral efficiency (SE) for a communication link operating over
a flat-fading channel. The EE is evaluated by the total energy consumption for
transmitting per message bit. Three channel cases are considered, namely static
channel with channel state information available at transmitter (CSIT),
fast-varying (FV) channel with channel distribution information available at
transmitter (CDIT), and FV channel with CSIT. A general circuit power model is
considered. For all the three channel cases, the tradeoff between the EE and SE
is studied. It is shown that the EE improves strictly as the SE increases from
0 to the optimum SE, and then strictly degrades as the SE increases beyond the
optimum SE. The impact of {\kappa}, {\rho} and other system parameters on the
optimum EE and corresponding SE is investigated to obtain insight.Some of the
important and interesting results for all the channel cases include: (1) when
{\kappa} increases the SE corresponding to the optimum EE should keep unchanged
if {\phi}(R) = R, but reduced if {\phi}(R) is strictly convex of R; (2) when
the rate-independent circuit power {\rho} increases, the SE corresponding to
the optimum EE has to be increased. A polynomial-complexity algorithm is
developed with the bisection method to find the optimum SE. The insight is
corroborated and the optimum EE for the three cases are compared by simulation
results.
