[Abridged] Context. The high-energy radiation from black-hole binaries may be due to the reprocessing of a lamp located on the black hole axis, emitting X-rays. The observed spectrum is made of 3 components: the direct spectrum; the thermal bump; and the reflected spectrum made of the Compton hump and the iron-line complex. Aims. We aim at computing accurately the complete reprocessed spectrum (thermal bump + reflected) of black-hole binaries over the entire X-ray band. We also determine the strength of the direct component. Our choice of parameters is adapted to a source showing an important thermal component. Methods. We compute in full GR the illumination of a thin disk by a lamp along the rotation axis. We use the ATM21 radiative transfer code to compute the spectrum emitted along the disk. We ray trace this local spectrum to determine the reprocessed spectrum as observed at infinity. We discuss the dependence of the local and ray-traced spectra on the emission angle and spin. Results. We show the importance of the angle dependence of the total disk specific intensity spectrum emitted by the illuminated atmosphere when the thermal disk emission if fully taken into account. High spin implies high temperature in the inner regions, so the emitted thermal disk spectrum covers the iron-line complex. As a result we locally observe absorption lines produced in the hot disk atmosphere. Absorption lines are narrow and disappear after ray tracing the local spectrum. Conclusions. Our results mainly highlight the importance of considering the angle dependence of the local spectrum when computing reprocessed spectra, as was already found in a recent study. The main new result of our work is to show the importance of computing the thermal bump of the spectrum, as this feature can change considerably the observed iron-line complex.
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