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.

## References

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 O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.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