  This paper studies energy efficiency (EE) and average throughput maximization
for cognitive radio systems in the presence of unslotted primary users. It is
assumed that primary user activity follows an ON-OFF alternating renewal
process. Secondary users first sense the channel possibly with errors in the
form of miss detections and false alarms, and then start the data transmission
only if no primary user activity is detected. The secondary user transmission
is subject to constraints on collision duration ratio, which is defined as the
ratio of average collision duration to transmission duration. In this setting,
the optimal power control policy which maximizes the EE of the secondary users
or maximizes the average throughput while satisfying a minimum required EE
under average/peak transmit power and average interference power constraints
are derived. Subsequently, low-complexity algorithms for jointly determining
the optimal power level and frame duration are proposed. The impact of
probabilities of detection and false alarm, transmit and interference power
constraints on the EE, average throughput of the secondary users, optimal
transmission power, and the collisions with primary user transmissions are
evaluated. In addition, some important properties of the collision duration
ratio are investigated. The tradeoff between the EE and average throughput
under imperfect sensing decisions and different primary user traffic are
further analyzed.
