  The problem of characterizing the optimal rate achievable with analog network
coding (ANC) for a unicast communication over general wireless relay networks
is computationally hard. A relay node performing ANC scales and forwards its
input signals. The source-destination channel in such communication scenarios
is, in general, an intersymbol interference (ISI) channel which leads to the
single-letter characterization of the optimal rate in terms of an optimization
problem with nonconvex, non closed-form objective function and non-convex
constraints. For a special class of such networks, called layered networks, a
few key results and insights are however available.
  To gain insights into the nature of the optimal solution and to construct
low-complexity schemes to characterize the optimal rate for general wireless
relay networks, we need (1) network topologies that are regular enough to be
amenable for analysis, yet general enough to capture essential characteristics
of general wireless relay networks, and (2) schemes to approximate the
objective function in closed-form without significantly compromising the
performance. Towards these two goals, this work proposes (1) nonlinear chain
networks, and (2) two approximation schemes. We show that their combination
allows us to tightly characterize the optimal ANC rate with low computational
complexity for a much larger class of general wireless relay networks than
possible with existing schemes.
