  The recent tentative detection by the EDGES Low-Band antenna of the hydrogen
21-cm line from cosmic dawn, if confirmed, is the first ever signature observed
from the epoch of primordial star formation. However, the magnitude and the
shape of this signal are incompatible with standard astrophysical predictions,
requiring either colder than expected gas, or an excess radio background above
the Cosmic Microwave Background (CMB) radiation. In this paper we explore the
latter option, investigating possible 21-cm signals in models with a
phenomenological excess radio background together with the standard
astrophysical modeling. Assuming uniform radiation with a synchrotron-like
spectrum existing throughout cosmic history, we show that such a radio
background, in addition to explaining the EDGES Low-Band signal, enhances the
power spectrum. The signal during cosmic dawn and reionization strongly depends
on both the intensity of the radio background and the astrophysical parameters.
We verify the broad agreement of our models with the detected feature,
including additional constraints from the EDGES High-Band data, high-redshift
quasars, and observational limits on the extragalactic radio background. The
data imply a lower limit on the star formation efficiency of 2.8\%, an upper
limit on the minimum mass of star-forming halos of M$_{\rm h}\sim 10^9$
M$_\odot$ at $z = 17$, and a lower limit on the excess radio background of 1.9
times the CMB at the 78 MHz cosmic dawn frequency (corresponding to 0.1\% of
the CMB at 1.42 GHz). The properties of X-ray sources remain unconstrained. We
also show that during the dark ages, such a radio background saturates the
21-cm signal, imprinting a unique signature in both the global signal and the
power spectrum.
