diff options
Diffstat (limited to 'content/know/concept/impulse-response/index.pdc')
| -rw-r--r-- | content/know/concept/impulse-response/index.pdc | 19 |
1 files changed, 12 insertions, 7 deletions
diff --git a/content/know/concept/impulse-response/index.pdc b/content/know/concept/impulse-response/index.pdc index b055fe7..fa921fa 100644 --- a/content/know/concept/impulse-response/index.pdc +++ b/content/know/concept/impulse-response/index.pdc @@ -35,9 +35,14 @@ $$\begin{aligned} } \end{aligned}$$ -*__Proof.__ Starting from the definition of $u_p(t)$, +<div class="accordion"> +<input type="checkbox" id="proof-main"/> +<label for="proof-main">Proof</label> +<div class="hidden"> +<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 $f(\tau)$:* +and multiply both sides by the constant $f(\tau)$: $$\begin{aligned} \hat{L} \{ u_p(t - \tau) \} &= \delta(t - \tau) @@ -45,8 +50,8 @@ $$\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} @@ -54,14 +59,14 @@ $$\begin{aligned} = f(t) \end{aligned}$$ -*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: $$\begin{aligned} \hat{L} \int_0^\infty f(\tau) \: u_p(t - \tau) \dd{\tau} &= (f * u_p)(t) = \hat{L}\{ u(t) \} = f(t) \end{aligned}$$ - -*__Q.E.D.__* +</div> +</div> |