Achievable Rate and Outage Probability of Cognitive Radio with Finite-Alphabet Inputs under Imperfect Spectrum Sensing

In this paper, we propose an effective method to calculate the average achievable rate and outage probability of a practical cognitive radio (CR) link with finite-alphabet inputs under imperfect spectrum sensing in fast and slow Rayleigh fading, respectively. In the considered CR system, the secondary user (SU) senses and dynamically exploits the spectrum pool via dynamic frequency hopping. Since spectrum sensing is not perfect, miss-detection occurs. Under this event, the interference emerged from collisions due to the simultaneous spectrum access of both primary and cognitive users leads to a non-Gaussian CR link. This makes it very challenging to evaluate the information theoretical limits, especially when finite-alphabet inputs are used. To overcome such challenge, we first introduce a simple method to calculate the instantaneous differential entropy of the channel output for a given fading gain using Laguerre-Gauss quadrature formulas. Using this result, we propose a piece-wise linear curve fitting (PLCF)-based method to calculate the average output entropy and outage probability, respectively. It is then demonstrated that the average achievable rate in fast fading and the outage probability in slow fading of the considered CR channel can be calculated effectively to achieve any predetermined accuracy level for a given finite-alphabet input.