Electrocorticogram (ECoG)-based brain computer interfaces (BCI) can potentially control upper extremity prostheses to restore independent function to paralyzed individuals. However, current research is mostly restricted to the offline decoding of finger or 2D arm movement trajectories, and these results are modest. This study seeks to improve the fundamental understanding of the ECoG signal features underlying upper extremity movements to guide better BCI design. Subjects undergoing ECoG electrode implantation performed a series of elementary upper extremity movements in an intermittent flexion and extension manner. It was found that movement velocity, $\dot\theta$, had a high positive (negative) correlation with the instantaneous power of the ECoG high-$\gamma$ band (80-160 Hz) during flexion (extension). Also, the correlation was low during idling epochs. Visual inspection of the ECoG high-$\gamma$ band revealed power bursts during flexion/extension events that have a waveform that strongly resembles the corresponding flexion/extension event as seen on $\dot\theta$. These high-$\gamma$ bursts were present in all elementary movements, and were spatially distributed in a somatotopic fashion. Thus, it can be concluded that the high-$\gamma$ power of ECoG strongly encodes for movement trajectories, and can be used as an input feature in future BCIs.