Calibration requirements for Epoch of Reionization 21-cm signal
  observations -- II. Analytical estimation of the bias and variance with
  time-correlated residual gains
  Observation of redshifted 21-cm signals from neutral hydrogen holds the key
to understanding the structure formation and its evolution during the
reionization and post-reionization era. Apart from the presence of orders of
magnitude larger foregrounds in the observed frequency range, the instrumental
effects of the interferometers combined with the ionospheric effects present a
considerable challenge in the extraction of 21-cm signals from strong
foregrounds. The systematic effects of time and frequency correlated residual
gain errors originating from the measurement process introduce a bias and
enhance the variance of the power spectrum measurements. In this work, we study
the effect of time-correlated residual gain errors in the presence of strong
foreground. We present a method to produce analytic estimates of the bias and
vari ance in the power spectrum. We use simulated observations to confirm the
efficacy of this method and then use it to understand various effects of the
gain errors. We find that as the standard deviation in the residual gain errors
increases, the bias in the estimation supersedes the variance. It is observed
that an optimal choice of the time over which the gain solutions are estimated
minimizes the risk. We also find that the interferometers with higher baseline
densities are preferred instruments for these studies.
