© 2019 American Institute of Mathematical Sciences. We study the existence and suppression of artifacts for a Doppler-based Synthetic Aperture Radar (DSAR) system. The idealized air-or spaceborne system transmits a continuous wave at a fixed frequency and a co-located receiver measures the resulting scattered waves; a windowed Fourier transform then converts the raw data into a function of two variables: slow time and frequency. Under simplifying assumptions, we analyze the linearized forward scattering map and the feasibility of inverting it via filtered backprojection, using techniques of microlocal analysis which robustly describe how sharp features in the target appear in the data. For DSAR with a straight flight path, there is, as with conventional SAR, a left-right ambiguity artifact in the DSAR image, which can be avoided via beam forming to the left or right. For a circular flight path, the artifact has a more complicated structure, but filtering out echoes coming from straight ahead or behind the transceiver, as well as those outside a critical range, produces an artifact-free image. We show that these results are qualitatively robust; although initially derived under an approximation widely used for range-based SAR, they are either structurally stable or robust with respect to a more accurate model.