Due　to　its　good　potential　for　digital　signal　processing,　discrete　Gabor　analysis　has　interested　some　mathematicians.　This　paper　addresses　Gabor　systems　on　discrete　periodic　sets,　which　can　model　signals　to　appear　periodically　but　intermittently.　Complete　Gabor　systems　and　Gabor　frames　on　discrete　periodic　sets　are　characterized;　a　sufficient　and　necessary　condition　on　what　periodic　sets　admit　complete　Gabor　systems　is　obtained;　this　condition　is　also　proved　to　be　sufficient　and　necessary　for　the　existence　of　sets　E　such　that　the　Gabor　systems　generated　by　chi　(E)　are　tight　frames　on　these　periodic　sets;　our　proof　is　constructive,　and　all　tight　frames　of　the　above　form　with　a　special　frame　bound　can　be　obtained　by　our　method;　periodic　sets　admitting　Gabor　Riesz　bases　are　characterized;　some　examples　are　also　provided　to　illustrate　the　general　theory.