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diff --git a/source/know/concept/impulse-response/index.md b/source/know/concept/impulse-response/index.md index 661ed3f..8210f5c 100644 --- a/source/know/concept/impulse-response/index.md +++ b/source/know/concept/impulse-response/index.md @@ -8,68 +8,75 @@ categories: layout: "concept" --- -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 +Given a system whose behaviour is described by a linear operator $$\hat{L}$$, +its **impulse response** $$u_\delta(t)$$ is defined as the system's response when forced by the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(t)$$: $$\begin{aligned} \boxed{ - \hat{L} \{ u_p(t) \} = \delta(t) + \hat{L} \{ u_\delta(t) \} + = \delta(t) } \end{aligned}$$ -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 $$u_p(t)$$: +This can be used to find the response $$u(t)$$ of $$\hat{L}$$ +to *any* forcing function $$f(t)$$, +by simply taking the convolution with $$u_\delta(t)$$: $$\begin{aligned} - \hat{L} \{ u(t) \} = f(t) + \hat{L} \{ u(t) \} + = f(t) \quad \implies \quad \boxed{ - u(t) = (f * u_p)(t) + u(t) + = (f * u_\delta)(t) } \end{aligned}$$ {% include proof/start.html id="proof-theorem" -%} -Starting from the definition of $$u_p(t)$$, +Starting from the definition of $$u_\delta(t)$$, we shift the argument by some constant $$\tau$$, -and multiply both sides by the constant $$f(\tau)$$: +and multiply both sides by $$f(\tau)$$: $$\begin{aligned} - \hat{L} \{ u_p(t - \tau) \} &= \delta(t - \tau) + \hat{L} \{ u_\delta(t - \tau) \} + &= \delta(t - \tau) \\ - \hat{L} \{ f(\tau) \: u_p(t - \tau) \} &= f(\tau) \: \delta(t - \tau) + \hat{L} \{ f(\tau) \: u_\delta(t - \tau) \} + &= f(\tau) \: \delta(t - \tau) \end{aligned}$$ -Where $$f(\tau)$$ can be moved inside using the -linearity of $$\hat{L}$$. Integrating over $$\tau$$ then gives us: +Where $$f(\tau)$$ was moved inside thanks to the linearity of $$\hat{L}$$. +Integrating over $$\tau$$ gives us: $$\begin{aligned} - \int_0^\infty \hat{L} \{ f(\tau) \: u_p(t - \tau) \} \dd{\tau} + \int_0^\infty \hat{L} \{ f(\tau) \: u_\delta(t - \tau) \} \dd{\tau} &= \int_0^\infty f(\tau) \: \delta(t - \tau) \dd{\tau} = f(t) \end{aligned}$$ -The integral and $$\hat{L}$$ are operators of different variables, so we reorder them: +The integral and $$\hat{L}$$ are operators of different variables, so we reorder them, +and recognize that the resulting integral is a convolution: $$\begin{aligned} - \hat{L} \int_0^\infty f(\tau) \: u_p(t - \tau) \dd{\tau} - &= (f * u_p)(t) = \hat{L}\{ u(t) \} = f(t) + f(t) + &= \hat{L} \int_0^\infty f(\tau) \: u_\delta(t - \tau) \dd{\tau} + = \hat{L} \Big\{ (f * u_\delta)(t) \Big\} \end{aligned}$$ + +Because $$\hat{L} \{ u(t) \} = f(t)$$ by definition, +we then see that $$(f * u_\delta)(t) = u(t)$$. {% include proof/end.html id="proof-theorem" %} 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](/know/concept/fundamental-solution/) -is preferable. +because any initial condition can be satisfied thanks to linearity, +by choosing the initial values of the homogeneous solution $$\hat{L}\{ u_0(t) \} = 0$$ +such that the total solution $$(f * u_\delta)(t) + u_0(t)$$ has the desired values. + +For boundary value problems, there is the related concept of +a [fundamental solution](/know/concept/fundamental-solution/). |