Cooperative　Adaptive　Cruise　Control　(CACC)　based　vehicle　platooning　can　increase　the　safety　and　efficiency　of　traffic.　This　work　looks　into　the　communication　and　control　problems　of　vehicle　platooning,　and　proposes　a　control　and　communication　co-design　for　CACC.　First,　an　integrated　radar　system　is　presented.　This　system　integrates　sensing　of　relative　position,　speed,　and　communication　between　a　predecessor　and　its　follower.　Second,　a　working　scheme　for　the　integrated　radar　system　is　presented.　This　scheme　allows　the　radar　systems　to　switch　periodically　between　different　working　modes　without　interferences　from　other　modes.　Therefore,　the　relative　position,　speed,　and　communication　can　be　asynchronously　periodically　updated　to　the　controller.　Third,　a　periodic　event-triggered　control　approach　is　presented.　This　approach　allows　asynchronous　periodic　sampling　of　the　output,　and　is　deeply　co-designed　with　the　radar　system　and　its　working　scheme.　Delays　are　also　considered　in　the　control　approach.　The　co-design　CACC　approach　can　guarantee　the　vehicle　platoons　to　be　string　stable.　A　numerical　example　has　shown　the　feasibility　of　the　approach.