Deep brain stimulation (DBS) therapies have shown clinical success in the treatment of a number of neurological illnesses, including obsessive-compulsive disorder, epilepsy, and Parkinson's disease. An emerging strategy for increasing the efficacy of DBS therapies is to develop closed-loop, adaptive DBS systems that can sense biomarkers associated with particular symptoms and in response, adjust DBS parameters in real-time. The development of such systems requires extensive analysis of the underlying neural signals while DBS is on, so that candidate biomarkers can be identified and the effects of varying the DBS parameters can be better understood. However, DBS creates high amplitude, high frequency stimulation artifacts that prevent the underlying neural signals and thus the biological mechanisms underlying DBS from being analyzed. Additionally, DBS devices often require low sampling rates, which alias the artifact frequency, and rely on wireless data transmission methods that can create signal recordings with missing data of unknown length. Thus, traditional artifact removal methods cannot be applied to this setting. We present a novel periodic artifact removal algorithm for DBS applications that can accurately remove stimulation artifacts in the presence of missing data and in some cases where the stimulation frequency exceeds the Nyquist frequency. The numerical examples suggest that, if implemented on dedicated hardware, this algorithm has the potential to be used in embedded closed-loop DBS therapies to remove DBS stimulation artifacts and hence, to aid in the discovery of candidate biomarkers in real-time. Code for our proposed algorithm is publicly available on Github.