  We analyze the problem of recovering a source term of the form
$h(t)=\sum_{j}h_j\phi(t-t_j)\chi_{[t_j, \infty)}(t)$ from space-time samples of
the solution $u$ of an initial value problem in a Hilbert space of functions.
In the expression of $h$, the terms $h_j$ belong to the Hilbert space, while
$\phi$ is a generic real-valued function with exponential decay at $\infty$.
The design of the sampling strategy takes into account noise in measurements
and the existence of a background source.
