Vector-　valued　frames　were　first　introduced　under　the　name　of　superframes　by　Balan　in　the　context　of　signal　multiplexing　and　by　Han　and　Larson　from　the　mathematical　aspect.　Since　then,　the　wavelet　and　Gabor　frames　in　L2(R,　CL)　have　interested　many　mathematicians.　The　space　L2(R+,　CL)　models　vector-　valued　causal　signal　spaces　because　of　the　time　variable　being　nonnegative.　But　it　admits　no　nontrivial　shift-　invariant　system　and　thus　no　wavelet　or　Gabor　frame　since　R+　is　not　a　group　by　addition　(not　as　R).　Observing　that　R+　is　a　group　by　multiplication,　we,　in　this　paper,　introduce　a　class　of　multiplication-　based　dilation-　and-　modulation　(..)　systems,　and　investigate　the　theory　of..　frames　in　L2(R+,　CL).　Since　L2(R+,　CL)　is　not　closed　under　the　Fourier　transform,　the　Fourier　transform　does　not　fit　L2(R+,　CL).　We　introduce　the　notion　of　Ta　transform　in　L2(R+,　CL),　and　using　Ta-　transform　matrix　method,　we　characterize..　frames,　Riesz　bases,　and　dual　frames　in　L2(R+,　CL)　and　obtain　an　explicit　expression　of..　duals　for　an　arbitrary　given..　frame.　An　example　theorem　is　also　presented.