  We study the scattering of the Dirac electrons by a point-like nonmagnetic
impurity on the surface of a topological insulator, driven by a time-periodic
gate voltage. It is found that, due to the doublet degenerate crossing points
of different Floquet sidebands, resonant backscattering can happen for the
surface electrons, even without breaking the time-reversal (TR) symmetry of the
topological surface states (TSSs). The energy spectrum is reshuffled in a way
quite different from that for the circularly polarized light, so that new
features are exhibited in the Friedel oscillations of the local charge and spin
density of states. Although the electron scattering is dramatically modified by
the driving voltage, the $1/\rho$ scale law of the spin precession persists for
the TSSs. The TR invariant backscattering provides a possible way to engineer
the Dirac electronic spectrum of the TSSs, without destroying the unique
property of spin-momentum interlocking of the TSSs.
