Before revision: Even though this is not what is physically done, an identical output can be generated by applying the hypothetical sequence of Dirac impulses, xs(t), to a linear , time-invariant system, otherwise known as a linear filter with such characteristics (which, for an LTI system, are fully described by the impulse response) so that each input impulse results in the correct piecewise linear function in the output.
After revision: Even though this is not what is physically done an identical output can be generated by applying the hypothetical sequence of Dirac impulses, xs(t), to a linear , time-invariant system, otherwise known as a linear filter with such characteristics (which, for an LTI system, are fully described by the impulse response) so that each input impulse results in the correct piecewise linear function in the output.