  Spectral measurements of the 21 cm monopole background have the promise of
revealing the bulk energetic properties and ionization state of our universe
from z ~ 6-30. Synchrotron foregrounds are orders of magnitude larger than the
cosmological signal, and are the principal challenge faced by these
experiments. While synchrotron radiation is thought to be spectrally smooth and
described by relatively few degrees of freedom, the instrumental response to
bright foregrounds may be much more complex. To deal with such complexities, we
develop an approach that discovers contaminated spectral modes using spatial
fluctuations of the measured data. This approach exploits the fact that
foregrounds vary across the sky while the signal does not. The discovered modes
are projected out of each line-of-sight of a data cube. An angular weighting
then optimizes the cosmological signal amplitude estimate by giving preference
to lower-noise regions. Using this method, we show that it is essential for the
passband to be stable to at least ~10^{-4}. In contrast, the constraints on the
spectral smoothness of the absolute calibration are mainly aesthetic if one is
able to take advantage of spatial information. To the extent it is understood,
controlling polarization to intensity leakage at the ~10^{-2} level will also
be essential to rejecting Faraday rotation of the polarized synchrotron
emission.

Cosmology and Nongalactic Astrophysics