Wavelet　and　Gabor　systems　are　based　on　translation-and-dilation　and　translation-and-modulation　operators,　respectively,　and　have　been　studied　extensively.　However,　dilation-and-modulation　systems　cannot　be　derived　from　wavelet　or　Gabor　systems.　This　study　aims　to　investigate　a　class　of　dilation-and-modulation　systems　in　the　causal　signal　spaceL(2)(Double-struck　capital　R+).L-2(Double-struck　capital　R+)　can　be　identified　as　a　subspace　ofL(2)(Double-struck　capital　R),　which　consists　of　allL(2)(Double-struck　capital　R)-functions　supported　on　Double-struck　capital　R(+)but　not　closed　under　the　Fourier　transform.　Therefore,　the　Fourier　transform　method　does　not　work　inL(2)(Double-struck　capital　R+).　Herein,　we　introduce　the　notion　of　Theta(a)-transform　inL(2)(Double-struck　capital　R+)　and　characterize　the　dilation-and-modulation　frames　and　dual　frames　inL(2)(Double-struck　capital　R+)　using　the　Theta(a)-transform;　and　present　an　explicit　expression　of　all　duals　with　the　same　structure　for　a　general　dilation-and-modulation　frame　forL(2)(Double-struck　capital　R+).　Furthermore,　it　has　been　proven　that　an　arbitrary　frame　of　this　form　is　always　nonredundant　whenever　the　number　of　the　generators　is　1　and　is　always　redundant　whenever　the　number　is　greater　than　1.　Finally,　some　examples　are　provided　to　illustrate　the　generality　of　our　results.