  In the practice, time variable cannot be negative. The space $L^2(\Bbb R_+)$
of square integrable functions defined on the right half real line $\Bbb R_+$
models causal signal space. This paper focuses on a class of
dilation-and-modulation systems in $L^2(\Bbb R_+)$. The density theorem for
Gabor systems in $L^2(\Bbb R)$ states a necessary and sufficient condition for
the existence of complete Gabor systems or Gabor frames in $L^2(\Bbb R)$ in
terms of the index set alone-independently of window functions. The space
$L^2(\Bbb R_+)$ admits no nontrivial Gabor system since $\Bbb R_+$ is not a
group according to the usual addition. In this paper, we introduce a class of
dilation-and-modulation systems in $L^2(\Bbb R_+)$ and the notion of
$\Theta$-transform matrix. Using $\Theta$-transform matrix method we obtain the
density theorem of the dilation-and-modulation systems in $L^2(\Bbb R_+)$ under
the condition that $\log_ba$ 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
$L^2(\Bbb R_+)$ is that $\log_ba \leq 1$. Simultaneously, we obtain a
$\Theta$-transform matrix-based expression of all complete
dilation-and-modulation systems and all dilation-and-modulation system frames
in $L^2(\Bbb R_+)$.
