  In spectrum-sharing cognitive radio systems, the transmit power of secondary
users has to be very low due to the restrictions on the tolerable interference
power dictated by primary users. In order to extend the coverage area of
secondary transmission and reduce the corresponding interference region,
multi-hop amplify-and-forward (AF) relaying can be implemented for the
communication between secondary transmitters and receivers. This paper
addresses the fundamental limits of this promising technique. Specifically, the
effect of major system parameters on the performance of spectrum-sharing
multi-hop AF relaying is investigated. To this end, the optimal transmit power
allocation at each node along the multi-hop link is firstly addressed. Then,
the extreme value theory is exploited to study the limiting distribution
functions of the lower and upper bounds on the end-to-end signal-to-noise ratio
of the relaying path. Our results disclose that the diversity gain of the
multi-hop link is always unity, regardless of the number of relaying hops. On
the other hand, the coding gain is proportional to the water level of the
optimal water-filling power allocation at secondary transmitter and to the
large-scale path-loss ratio of the desired link to the interference link at
each hop, yet is inversely proportional to the accumulated noise, i.e. the
product of the number of relays and the noise variance, at the destination.
These important findings do not only shed light on the performance of the
secondary transmissions but also benefit system designers improving the
efficiency of future spectrum-sharing cooperative systems.
