  Redshifted H{\sc\,i} 21 cm emission from unresolved low-redshift large scale
structure is a promising window for ground-based Baryon Acoustic Oscillations
(BAO) observations. A major challenge for this method is separating the cosmic
signal from the foregrounds of Galactic and extra-Galactic origins that are
stronger by many orders of magnitude than the former. The smooth frequency
spectrum expected for the foregrounds would nominally contaminate only very
small $k_\parallel$ modes; however the chromatic response of the telescope
antenna pattern at this wavelength to the foreground introduces non-smooth
structure, pervasively contaminating the cosmic signal over the physical scales
of our interest. Such contamination defines a wedged volume in Fourier space
around the transverse modes that is inaccessible for the cosmic signal. In this
paper, we test the effect of this contaminated wedge on the future 21 cm BAO
surveys using Fisher information matrix calculation. We include the signal
improvement due to the BAO reconstruction technique that has been used for
galaxy surveys and test the effect of this wedge on the BAO reconstruction as a
function of signal to noises and incorporate the results in the Fisher matrix
calculation. We find that the wedge effect expected at $z=1-2$ is very
detrimental to the angular diameter distances: the errors on angular diameter
distances increased by 3-4.4 times, while the errors on H(z) increased by a
factor of 1.5-1.6. We conclude that calibration techniques that clean out the
foreground "wedge" would be extremely valuable for constraining angular
diameter distances from intensity-mapping 21 cm surveys.
