  In this paper a unified approach for power allocation (PA) problem in
multi-hop orthogonal frequency division multiplexing (OFDM) amplify-and-forward
(AF) relaying systems has been developed. In the proposed approach, we consider
short and long-term individual and total power constraints at the source and
relays, and devise decentralized low complexity PA algorithms when wireless
links are subject to channel path-loss and small-scale Rayleigh fading. In
particular, aiming at improving the instantaneous rate of multi-hop
transmission systems with AF relaying, we develop (i) a near-optimal iterative
PA algorithm based on the exact analysis of the received SNR at the
destination; (ii) a low complexity-suboptimal iterative PA algorithm based on
an approximate expression at high-SNR regime; and (iii) a low complexity-non
iterative PA scheme with limited performance loss. Since the PA problem in
multi-hop systems is too complex to solve with known optimization solvers, in
the proposed formulations, we adopted a two-stage approach, including a power
distribution phase among distinct subcarriers, and a power allocation phase
among different relays. The individual PA phases are then appropriately linked
through an iterative method which tries to compensate the performance loss
caused by the distinct two-stage approach. Simulation results show the superior
performance of the proposed power allocation algorithms.
