  We present the results of a radial velocity survey of 20 white dwarf plus M
dwarf binaries selected as a follow up to a \textit{Hubble Space Telescope}
study that aimed to spatially resolve suspected binaries. Our candidates are
taken from the list of targets that were spatially unresolved with
\textit{Hubble}. We have determined the orbital periods for 16 of these compact
binary candidates. The period distribution ranges from 0.14 to 9.16\,d and
peaks near 0.6\,d. The original sample therefore contains two sets of binaries,
wide orbits ($\approx100-1000$\,au) and close orbits ($\lesssim1-10$\,au), with
no systems found in the $\approx10-100$\,au range. This observational evidence
confirms the bimodal distribution predicted by population models and is also
similar to results obtained in previous studies. We find no binary periods in
the months to years range, supporting the post common envelope evolution
scenario. One of our targets, WD\,1504+546, was discovered to be an eclipsing
binary with a period of 0.93\,d.
