  In this paper, we study the performance of an amplify-and-forward (AF) based
analog network coding (ANC) relay scheme in a multi-hop wireless network under
individual power constraints. In the first part, a unicast scenario is
considered. The problem of finding the maximum achievable rate is formulated as
an optimization problem. Rather than solving this non-concave maximization
problem, we derive upper and lower bounds for the optimal rate. A cut-set like
upper bound is obtained in a closed form for a layered relay network. A
pseudo-optimal AF scheme is developed for a two-hop parallel network, which is
different from the conventional scheme with all amplification gains chosen as
the maximum possible values. The conditions under which either the novel scheme
or the conventional one achieves a rate within half a bit of the upper bound
are found. Then we provide an AF-based multi-hop ANC scheme with the two
schemes for a layered relay network. It is demonstrated that the lower bound of
the optimal rate can asymptotically achieve the upper bound when the network is
in the generalized high-SNR regime. In the second part, the optimal rate region
for a two-hop multiple access channel (MAC) via AF relays is investigated. In a
similar manner, we first derive an outer bound for it and then focus on
designing low complexity AF-based ANC schemes for different scenarios. Several
examples are given and the numerical results indicate that the achievable rate
region of the ANC schemes can perform close to the outer bound.
