  The problem of maximum rate achievable with analog network coding for a
unicast communication over a layered wireless relay network with directed links
is considered. A relay node performing analog network coding scales and
forwards the signals received at its input. Recently this problem has been
considered under two assumptions: (A) each relay node scales its received
signal to the upper bound of its transmit power constraint, (B) the relay nodes
in specific subsets of the network operate in the high-SNR regime. We establish
that assumption (A), in general, leads to suboptimal end-to-end rate. We also
characterize the performance of analog network coding in class of symmetric
layered networks without assumption (B).
  The key contribution of this work is a lemma that states that a globally
optimal set of scaling factors for the nodes in a layered relay network that
maximizes the end-to-end rate can be computed layer-by-layer. Specifically, a
rate-optimal set of scaling factors for the nodes in a layer is the one that
maximizes the sum-rate of the nodes in the next layer. This critical insight
allows us to characterize analog network coding performance in network
scenarios beyond those that can be analyzed using the existing approaches. We
illustrate this by computing the maximum rate achievable with analog network
coding in one particular layered network, in various communication scenarios.
