  In cognitive radio systems, employing sensing-based spectrum access
strategies, secondary users are required to perform channel sensing in order to
detect the activities of primary users. In realistic scenarios, channel sensing
occurs with possible errors due to miss-detections and false alarms. As another
challenge, time-varying fading conditions in the channel between the secondary
transmitter and the secondary receiver have to be learned via channel
estimation. In this paper, performance of causal channel estimation methods in
correlated cognitive radio channels under imperfect channel sensing results is
analyzed, and achievable rates under both channel and sensing uncertainty are
investigated. Initially, cognitive radio channel model with channel sensing
error and channel estimation is described. Then, using pilot symbols, minimum
mean square error (MMSE) and linear-MMSE (L-MMSE) estimation methods are
employed at the secondary receiver to learn the channel fading coefficients.
Expressions for the channel estimates and mean-squared errors (MSE) are
determined, and their dependencies on channel sensing results, and pilot symbol
period and energy are investigated. Since sensing uncertainty leads to
uncertainty in the variance of the additive disturbance, channel estimation
strategies and performance are interestingly shown to depend on the sensing
reliability. It is further shown that the L-MMSE estimation method, which is in
general suboptimal, performs very close to MMSE estimation. Furthermore,
assuming the channel estimation errors and the interference introduced by the
primary users as zero-mean and Gaussian distributed, achievable rate
expressions of linear modulation schemes and Gaussian signaling are determined.
Subsequently, the training period, and data and pilot symbol energy allocations
are jointly optimized to maximize the achievable rates for both signaling
schemes.
