  In this paper we consider the capacity of the cognitive radio channel in
different fading environments under a low interference regime. First we derive
the probability that the low interference regime holds under shadow fading as
well as Rayleigh and Rician fast fading conditions. We demonstrate that this is
the dominant case, especially in practical cognitive radio deployment
scenarios. The capacity of the cognitive radio channel depends critically on a
power loss parameter, $\alpha$, which governs how much transmit power the
cognitive radio dedicates to relaying the primary message. We derive a simple,
accurate approximation to $\alpha$ in Rayleigh and Rician fading environments
which gives considerable insight into system capacity. We also investigate the
effects of system parameters and propagation environment on $\alpha$ and the
cognitive radio capacity. In all cases, the use of the approximation is shown
to be extremely accurate.
