  This paper studies optimum power control and sum-rate scaling laws for the
distributed cognitive uplink. It is first shown that the optimum distributed
power control policy is in the form of a threshold based water-filling power
control. Each secondary user executes the derived power control policy in a
distributed fashion by using local knowledge of its direct and interference
channel gains such that the resulting aggregate (average) interference does not
disrupt primary's communication. Then, the tight sum-rate scaling laws are
derived as a function of the number of secondary users $N$ under the optimum
distributed power control policy. The fading models considered to derive
sum-rate scaling laws are general enough to include Rayleigh, Rician and
Nakagami fading models as special cases. When transmissions of secondary users
are limited by both transmission and interference power constraints, it is
shown that the secondary network sum-rate scales according to
$\frac{1}{\e{}n_h}\log\logp{N}$, where $n_h$ is a parameter obtained from the
distribution of direct channel power gains. For the case of transmissions
limited only by interference constraints, on the other hand, the secondary
network sum-rate scales according to $\frac{1}{\e{}\gamma_g}\logp{N}$, where
$\gamma_g$ is a parameter obtained from the distribution of interference
channel power gains. These results indicate that the distributed cognitive
uplink is able to achieve throughput scaling behavior similar to that of the
centralized cognitive uplink up to a pre-log multiplier $\frac{1}{\e{}}$,
whilst primary's quality-of-service requirements are met. The factor
$\frac{1}{\e{}}$ can be interpreted as the cost of distributed implementation
of the cognitive uplink.
