Many indoor localization techniques have been proposed as a substitute for GPS, but most of them can not measure the relative direction between the source and the target. This capability is essential for many applications such as unmanned aerial vehicles (UAVs) and robots. In this paper, we propose a UWB-based omnidirectional DOA (Direction of Arrival) measurement method. The proposed method is based on compressive sensing theory, which is more robust to noise and offers higher accuracy. Furthermore, we present a grid refinement algorithm aimed at enhancing the accuracy of DOA estimation while simultaneously reducing computational costs. We implement the proposed method on a UWB antenna array, and experimental results validate its accuracy.