  In wireless relay networks, noise at the relays can be correlated possibly
due to common interference or noise propagation from preceding hops. In this
work we consider a parallel relay network with noise correlation. For the relay
strategy of amplify-and-forward (AF), we determine the optimal rate maximizing
relay gains when correlation knowledge is available at the relays. The effect
of correlation on the performance of the relay networks is analyzed for the
cases where full knowledge of correlation is available at the relays and when
there is no knowledge about the correlation structure. Interestingly we find
that, on the average, noise correlation is beneficial regardless of whether the
relays know the noise covariance matrix or not. However, the knowledge of
correlation can greatly improve the performance. Typically, the performance
improvement from correlation knowledge increases with the relay power and the
number of relays. With perfect correlation knowledge the system is capable of
canceling interference if the number of interferers is less than the number of
relays.
  For a dual-hop multiple access parallel network, we obtain closed form
expressions for the maximum sum-rate and the optimal relay strategy. The relay
optimization for networks with three hops is also considered. For any relay
gains for the first stage relays, this represents a parallel relay network with
correlated noise. Based on the result of two hop networks with noise
correlation, we propose an algorithm for solving the relay optimization problem
for three-hop networks.
