Suppose that I want to create a techno music soundtrack in which several sounds can be produced; every sound is determined by the frequency spectrum:
$f_k (\omega), k \in \{1,\dots,N\}$. The number of sounds that can be played is $N$. Now one can set which sounds can be played on several times. The input might be
$1;2(5),4(3)$ meaning that at time step 1 the sounds with $k=2$ and $k=4$ will be played, where sound with $k=2$ will be played for 5 time steps and sound with $k=4$ will be played for 3 time steps. Now I am interested which mathematical operations can transform this input into the complete electronic soundtrack. The conversion of the input to Boolean numbers is straightforward. But is it possible to transform the Boolean numbers to the complete soundtrack?
I know that the (analogous) output signal must be of the form:
$F(t) = \frac{1}{2 \pi} \sum_{k}^{(programmed)} \int_{0}^\infty f_k(\omega) e^{i \omega t} d \omega $