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Diffstat (limited to 'source/know/concept/impulse-response')
| -rw-r--r-- | source/know/concept/impulse-response/index.md | 31 |
1 files changed, 16 insertions, 15 deletions
diff --git a/source/know/concept/impulse-response/index.md b/source/know/concept/impulse-response/index.md index 533580d..397ac2d 100644 --- a/source/know/concept/impulse-response/index.md +++ b/source/know/concept/impulse-response/index.md @@ -8,9 +8,9 @@ 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 -when forced by the [Dirac delta function](/know/concept/dirac-delta-function/) $\delta(t)$: +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](/know/concept/dirac-delta-function/) $$\delta(t)$$: $$\begin{aligned} \boxed{ @@ -18,9 +18,9 @@ $$\begin{aligned} } \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)$$, i.e. not only $$\delta(t)$$, +by simply taking the convolution with $$u_p(t)$$: $$\begin{aligned} \hat{L} \{ u(t) \} = f(t) @@ -35,9 +35,9 @@ $$\begin{aligned} <label for="proof-main">Proof</label> <div class="hidden" markdown="1"> <label for="proof-main">Proof.</label> -Starting from the definition of $u_p(t)$, -we shift the argument by some constant $\tau$, -and multiply both sides by the constant $f(\tau)$: +Starting from the definition of $$u_p(t)$$, +we shift the argument by some constant $$\tau$$, +and multiply both sides by the constant $$f(\tau)$$: $$\begin{aligned} \hat{L} \{ u_p(t - \tau) \} &= \delta(t - \tau) @@ -45,8 +45,8 @@ $$\begin{aligned} \hat{L} \{ f(\tau) \: u_p(t - \tau) \} &= f(\tau) \: \delta(t - \tau) \end{aligned}$$ -Where $f(\tau)$ can be moved inside using the -linearity of $\hat{L}$. Integrating over $\tau$ then gives us: +Where $$f(\tau)$$ can be moved inside using the +linearity of $$\hat{L}$$. Integrating over $$\tau$$ then gives us: $$\begin{aligned} \int_0^\infty \hat{L} \{ f(\tau) \: u_p(t - \tau) \} \dd{\tau} @@ -54,20 +54,21 @@ $$\begin{aligned} = f(t) \end{aligned}$$ -The integral and $\hat{L}$ are operators of different variables, so we reorder them: +The integral and $$\hat{L}$$ are operators of different variables, so we reorder them: $$\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) \end{aligned}$$ + </div> </div> This is useful for solving initial value problems, because any initial condition can be satisfied -due to the linearity of $\hat{L}$, -by choosing the initial values of the homogeneous solution $\hat{L}\{ u_h(t) \} = 0$ -such that the total solution $(f * u_p)(t) + u_h(t)$ +due to the linearity of $$\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, |