  One important source of systematics in galaxy redshift surveys comes from the
estimation of the galaxy window function. Up until now, the impact of the
uncertainty in estimating the galaxy window function on parameter inference has
not been properly studied. In this paper, we show that the uncertainty and the
bias in estimating the galaxy window function will be salient for ongoing and
next-generation galaxy surveys using a simulation-based approach. With a
specific case study of cross-correlating Emission-line galaxies from the DESI
Legacy Imaging Surveys and the Planck CMB lensing map, we show that neural
network-based regression approaches to modelling the window function are
superior in comparison to linear regression-based models. We additionally show
that the definition of the galaxy overdensity estimator can impact the overall
signal-to-noise of observed power spectra. Finally, we show that the additive
biases coming from the window functions can significantly bias the modes of the
inferred parameters and also degrade their precision. Thus, a careful
understanding of the window functions will be essential to conduct cosmological
experiments.
