  The 21-cm signal of neutral hydrogen - emitted during the Epoch of
Reionization - promises to be an important source of information for the study
of the infant universe. However, its detection is impossible without sufficient
mitigation of other strong signals in the data, which requires an accurate
knowledge of the instrument. Using the result of instrument calibration, a
large part of the contaminating signals are removed and the resulting residual
data is further analyzed in order to detect the 21-cm signal. Direction
dependent calibration (DDC) can strongly affect the 21-cm signal, however, its
effect has not been precisely quantified.
  In the analysis presented here we show how to exactly calculate what part of
the 21-cm signal is removed as a result of the DDC. We also show how a-priori
information about the frequency behavior of the instrument can be used to
reduce signal suppression. The theoretical results are tested using a realistic
simulation based on the LOFAR setup. Our results show that low-order smooth
gain functions (e.g. polynomials) over a bandwidth of ~10\,MHz - over which the
signal is expected to be stationary - is sufficient to allow for calibration
with limited, quantifiable, signal suppression in its power spectrum. We also
show mathematically and in simulations that more incomplete sky models lead to
larger 21-cm signal suppression, even if the gain models are enforced to be
fully smooth. This result has immediate consequences for current and future
radio telescopes with non-identical station beams, where DDC might be necessary
(e.g. SKA-low).
