Asymptotic Analysis for Spectrum-sharing Systems with TAS/MRC Using Extreme Value Theory: An Overlooked Aspect

We investigate the asymptotic behavior for an overlooked aspect of spectrum-sharing systems when the number of transmit antennas nt at the secondary transmitter (ST) grows to infinity. Considering imperfect channel state information (CSI), we apply the transmit antenna selection and the maximal-ratio combining techniques at the ST and the secondary receiver (SR), respectively. First, we obtain the signalto-noise ratio (SNR) distributions received by the SR under perfect and imperfect CSI conditions. Then we show that the SNR distributions are tail-equivalent in the sense that the right tails of the two distributions decay in the same rate as the number of transmit antennas nt grows to infinity. Based on the extreme value theory, when the transmit power of the ST is solely limited by the interference constraint, we show that the limiting SNR at the SR is Fréchet-distributed and the limiting rate scales as log(nt). When the transmit power of ST is determined by both the maximal transmit power and the interference power constraints, the limiting SNR is Gumbel-distributed and the limiting rate scales as log(log(nt)). We further show that the average rate can be estimated by the corresponding easier-to-obtain outage rate. Numerical results indicate that the derived asymptotic rate expressions represent accurate approximations even when nt is “not-solarge”. Finally, we study the robustness of the secondary transmissions by analyzing the corresponding average symbol error rates (SER) under general modulation and coding schemes. The findings indicate that the SER is Weibull distributed, when the maximal transmit power and interference power constraints are comparable.