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Cooperative adaptive cruise control (CACC) is one of the popular connected and automated vehicle (CAV) applications for cooperative driving automation with combined connectivity and automation technologies to improve string stability. This study aimed to derive the string stability conditions of a CACC controller and analyze the impacts of CACC on string stability for both a fleet of homogeneous CAVs and for heterogeneous traffic with human-driven vehicles (HDVs), connected vehicles (CVs) with connectivity technologies only, and autonomous vehicles (AVs) with automation technologies only. We mathematically analyzed the impact of CACC on string stability for both homogeneous and heterogeneous traffic flow. We adopted parameters from literature for HDVs, CVs, and AVs for the heterogeneous traffic case. We found there was a minimum constant time headway required for each parameter design to ensure stability in homogeneous CACC traffic. In addition, the constant time headway and the length of control time interval had positive correlation with stability, but the control parameter had a negative correlation with stability. The numerical analysis also showed that CACC vehicles could maintain string stability better than CVs and AVs under low HDV market penetration rates for the mixed traffic case.