  We study the localization transition in periodically driven one-dimensional
non-Hermitian lattices where the piece-wise two-step drive is constituted by
uniform coherent tunneling and incommensurate onsite gain and loss. We find
that the system can be in localized, delocalized, or mixed-phase depending on
the driving frequency and the phase shift of complex potential. Two critical
driving frequencies of the system are identified, the first one corresponds to
the largest phase shift of the complex potential so that the quasi-energy
spectrum is still real and all the states are extended, the second one
corresponds to the disappear of full real spectrum, and very weak complex
potential leads to the emergence of localized states when the driving frequency
is lower than this critical frequency. In the high frequency limit, we find the
critical phase shift that separates the two regions with respectively real and
complex spectrum tends to a constant value that can be captured by an effective
non-Hermitian Hamiltonian.
