  A general class of wireless relay networks with a single source-destination
pair is considered. Intermediate nodes in the network employ an
amplify-and-forward scheme to relay their input signals. In this case the
overall input-output channel from the source via the relays to the destination
effectively behaves as an intersymbol interference channel with colored noise.
Unlike previous work we formulate the problem of the maximum achievable rate in
this setting as an optimization problem with no assumption on the network size,
topology, and received signal-to-noise ratio. Previous work considered only
scenarios wherein relays use all their power to amplify their received signals.
We demonstrate that this may not always maximize the maximal achievable rate in
amplify-and-forward relay networks. The proposed formulation allows us to not
only recover known results on the performance of the amplify-and-forward
schemes for some simple relay networks but also characterize the performance of
more complex amplify-and-forward relay networks which cannot be addressed in a
straightforward manner using existing approaches.
  Using cut-set arguments, we derive simple upper bounds on the capacity of
general wireless relay networks. Through various examples, we show that a large
class of amplify-and-forward relay networks can achieve rates within a constant
factor of these upper bounds asymptotically in network parameters.
