Categories: Mathematics, Physics.

Impulse response

Given a system whose behaviour is described by a linear operator L^\hat{L}, its impulse response uδ(t)u_\delta(t) is defined as the system’s response when forced by the Dirac delta function δ(t)\delta(t):

L^{uδ(t)}=δ(t)\begin{aligned} \boxed{ \hat{L} \{ u_\delta(t) \} = \delta(t) } \end{aligned}

This can be used to find the response u(t)u(t) of L^\hat{L} to any forcing function f(t)f(t), by simply taking the convolution with uδ(t)u_\delta(t):

L^{u(t)}=f(t)  ⟹  u(t)=(f∗uδ)(t)\begin{aligned} \hat{L} \{ u(t) \} = f(t) \quad \implies \quad \boxed{ u(t) = (f * u_\delta)(t) } \end{aligned}

Starting from the definition of uδ(t)u_\delta(t), we shift the argument by some constant τ\tau, and multiply both sides by f(τ)f(\tau):

L^{uδ(t−τ)}=δ(t−τ)L^{f(τ) uδ(t−τ)}=f(τ) δ(t−τ)\begin{aligned} \hat{L} \{ u_\delta(t - \tau) \} &= \delta(t - \tau) \\ \hat{L} \{ f(\tau) \: u_\delta(t - \tau) \} &= f(\tau) \: \delta(t - \tau) \end{aligned}

Where f(τ)f(\tau) was moved inside thanks to the linearity of L^\hat{L}. Integrating over τ\tau gives us:

∫0∞L^{f(τ) uδ(t−τ)}dτ=∫0∞f(τ) δ(t−τ)dτ=f(t)\begin{aligned} \int_0^\infty \hat{L} \{ f(\tau) \: u_\delta(t - \tau) \} \dd{\tau} &= \int_0^\infty f(\tau) \: \delta(t - \tau) \dd{\tau} = f(t) \end{aligned}

The integral and L^\hat{L} are operators of different variables, so we reorder them, and recognize that the resulting integral is a convolution:

f(t)=L^∫0∞f(τ) uδ(t−τ)dτ=L^{(f∗uδ)(t)}\begin{aligned} f(t) &= \hat{L} \int_0^\infty f(\tau) \: u_\delta(t - \tau) \dd{\tau} = \hat{L} \Big\{ (f * u_\delta)(t) \Big\} \end{aligned}

Because L^{u(t)}=f(t)\hat{L} \{ u(t) \} = f(t) by definition, we then see that (f∗uδ)(t)=u(t)(f * u_\delta)(t) = u(t).

This is useful for solving initial value problems, because any initial condition can be satisfied thanks to linearity, by choosing the initial values of the homogeneous solution L^{u0(t)}=0\hat{L}\{ u_0(t) \} = 0 such that the total solution (f∗uδ)(t)+u0(t)(f * u_\delta)(t) + u_0(t) has the desired values.

For boundary value problems, there is the related concept of a fundamental solution.

References

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 O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.