We consider a cognitive radio system, consisting of a primary transmitter(PUtx), a primary receiver (PUrx), a secondary transmitter (SUtx), and asecondary receiver (SUrx). The secondary users (SUs) are equipped withsteerable directional antennas. We assume the SUs and primary users (PUs)coexist and the SUtx knows the geometry of network. We find the ergodiccapacity of the channel between SUtx and SUrx , and study how spectrum sensingerrors affect the capacity. In our system, the SUtx first senses the spectrumand then transmits data at two power levels, according to the result ofsensing. The optimal SUtx transmit power levels and the optimal directions ofSUtx transmit antenna and SUrx receive antenna are obtained by maximizing theergodic capacity, subject to average transmit power and average interferencepower constraints. To study the effect of fading channel, we considered threescenarios: 1) when SUtx knows fading channels between SUtx and PUrx, PUtx andSUrx, SUtx and SUrx, 2) when SUtx knows only the channel between SUtx and SUrx,and statistics of the other two channels, and, 3) when SUtx only knows thestatistics of these three fading channels. For each scenario, we explore theoptimal SUtx transmit power levels and the optimal directions of SUtx and SUrxantennas, such that the ergodic capacity is maximized, while the powerconstraints are satisfied.
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