In　practice,　the　time　variable　cannot　be　negative.　The　space　L-2(R+)　of　square　integrable　functions　defined　on　the　right　half　real　line　R+　models　the　causal　signal　space.　This　paper　focuses　on　a　class　of　dilationand-modulation　systems　in　L-2(R+).　The　density　theorem　for　Gabor　systems　in　L-2　(R)　states　a　necessary　and　sufficient　condition　for　the　existence　of　complete　Gabor　systems　or　Gabor　frames　in　L-2　(R)　in　terms　of　the　index　set　alone-independently　of　window　functions.　The　space　L-2(R+)　admits　no　nontrivial　Gabor　system　since　R+　is　not　a　group　according　to　the　usual　addition.　In　this　paper,　we　introduce　a　class　of　dilation-and-modulation　systems　in　L-2　(R+)　and　the　notion　of　T-transform　matrix.　Using　the　T-transform　matrix　method　we　obtain　the　density　theorem　of　the　dilation-and-modulation　systems　in　L-2　(R+)　under　the　condition　that　logb　a　is　a　positive　rational　number,　where　a　and　b　are　the　dilation　and　modulation　parameters　respectively.　Precisely,　we　prove　that　a　necessary　and　sufficient　condition　for　the　existence　of　such　a　complete　dilation-and-modulation　system　or　dilation-and-modulation　system　frame　in　L-2　(R+)　is　that　log(b)　a　＜=　1.　Simultaneously,　we　obtain　a　T-transform　matrix-based　expression　of　all　complete　dilation-and-modulation　systems　and　all　dilation-and-modulation　system　frames　in　L-2(R+).