  This paper aims to jointly determine linear precoding (LP) vectors, beam
hopping (BH), and discrete DVB-S2X transmission rates for the GEO satellite
communication systems to minimize the payload power consumption and satisfy
ground users' demands within a time window. Regarding constraint on the maximum
number of illuminated beams per time slot, the technical requirement is
formulated as a sparse optimization problem in which the hardware-related beam
illumination energy is modeled in a sparsity form of the LP vectors. To cope
with this problem, the compressed sensing method is employed to transform the
sparsity parts into the quadratic form of precoders. Then, an iterative
window-based algorithm is developed to update the LP vectors sequentially to an
efficient solution. Additionally, two other two-phase frameworks are also
proposed for comparison purposes. In the first phase, these methods aim to
determine the MODCOD transmission schemes for users to meet their demands by
using a heuristic approach or DNN tool. In the second phase, the LP vectors of
each time slot will be optimized separately based on the determined MODCOD
schemes.
