  This paper considers the cooperation between a cognitive system and a primary
system where multiple cognitive base stations (CBSs) relay the primary user's
(PU) signals in exchange for more opportunity to transmit their own signals.
The CBSs use amplify-and-forward (AF) relaying and coordinated beamforming to
relay the primary signals and transmit their own signals. The objective is to
minimize the overall transmit power of the CBSs given the rate requirements of
the PU and the cognitive users (CUs). We show that the relaying matrices have
unit rank and perform two functions: Matched filter receive beamforming and
transmit beamforming. We then develop two efficient algorithms to find the
optimal solution. The first one has linear convergence rate and is suitable for
distributed implementation, while the second one enjoys superlinear convergence
but requires centralized processing. Further, we derive the beamforming vectors
for the linear conventional zero-forcing (CZF) and prior zero-forcing (PZF)
schemes, which provide much simpler solutions. Simulation results demonstrate
the improvement in terms of outage performance due to the cooperation between
the primary and cognitive systems.
