arXiv:2611.07767v1 Announce Type: new Abstract: In this paper, we propose a precoding framework for an integrated satellite terrestrial self-backhauled (ISTS) system, where a low Earth orbit (LEO) satellite transmits a backhaul stream to an earth station (ES), and the ES forwards its payload to the base station (BS) serving the users. We formulate a user sum spectral efficiency (SE) maximization problem subject to a self-backhaul constraint, which ensures that the user data forwarded by the BS do not exceed the data delivered to the ES. We then reformulate the resulting non-convex precoding problem into a form amenable to generalized power iteration (GPI). The GPI update is reinterpreted as projected preconditioned gradient ascent, revealing that its fixed step size may cause overshooting and non-monotonic behavior. Building on this interpretation, we first develop an alternating convergent GPI (AC-GPI), in which the fixed step is replaced by an Armijo backtracking rule, and show that a single Armijo-accepted step per block already guarantees sufficient ascent. This leads to the fast alternating convergent GPI (FAC-GPI), which performs one Armijo-accepted satellite update followed by one BS update in every round. Unlike existing backtracking-based GPI methods, whose convergence has not been established, we establish finite backtracking termination, monotone ascent, convergence to stationary points, along with a global stationarity-rate guarantee. Simulation results demonstrate stable and fast convergence as well as performance gains over existing methods.