Categories: Mathematics, Physics.

Impulse response

The impulse response \(u_p(t)\) of a system whose behaviour is described by a linear operator \(\hat{L}\), is defined as the reponse of the system when forced by the Dirac delta function \(\delta(t)\):

\[\begin{aligned} \boxed{ \hat{L} \{ u_p(t) \} = \delta(t) } \end{aligned}\]

This can be used to find the response \(u(t)\) of \(\hat{L}\) to any forcing function \(f(t)\), i.e. not only \(\delta(t)\), by simply taking the convolution with \(u_p(t)\):

\[\begin{aligned} \hat{L} \{ u(t) \} = f(t) \quad \implies \quad \boxed{ u(t) = (f * u_p)(t) } \end{aligned}\]

This is useful for solving initial value problems, because any initial condition can be satisfied due to the linearity of \(\hat{L}\), by choosing the initial values of the homogeneous solution \(\hat{L}\{ u_h(t) \} = 0\) such that the total solution \((f * u_p)(t) + u_h(t)\) has the desired values.

Meanwhile, for boundary value problems, the related fundamental solution is preferable.


  1. O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.