diff options
Diffstat (limited to 'content/know/concept/convolution-theorem')
| -rw-r--r-- | content/know/concept/convolution-theorem/index.pdc | 26 |
1 files changed, 20 insertions, 6 deletions
diff --git a/content/know/concept/convolution-theorem/index.pdc b/content/know/concept/convolution-theorem/index.pdc index 9d1a666..1454cc0 100644 --- a/content/know/concept/convolution-theorem/index.pdc +++ b/content/know/concept/convolution-theorem/index.pdc @@ -32,7 +32,12 @@ $$\begin{aligned} } \end{aligned}$$ -To prove this, we expand the right-hand side of the theorem and +<div class="accordion"> +<input type="checkbox" id="proof-fourier"/> +<label for="proof-fourier">Proof</label> +<div class="hidden"> +<label for="proof-fourier">Proof.</label> +We expand the right-hand side of the theorem and rearrange the integrals: $$\begin{aligned} @@ -45,8 +50,8 @@ $$\begin{aligned} = A \cdot (f * g)(x) \end{aligned}$$ -Then we do the same thing again, this time starting from a product in -the $x$-domain: +Then we do the same again, +this time starting from a product in the $x$-domain: $$\begin{aligned} \hat{\mathcal{F}}\{f(x) \: g(x)\} @@ -57,6 +62,8 @@ $$\begin{aligned} &= B \int_{-\infty}^\infty \tilde{g}(k') \tilde{f}(k - k') \dd{k'} = B \cdot (\tilde{f} * \tilde{g})(k) \end{aligned}$$ +</div> +</div> ## Laplace transform @@ -76,9 +83,14 @@ $$\begin{aligned} \boxed{\hat{\mathcal{L}}\{(f * g)(t)\} = \tilde{f}(s) \: \tilde{g}(s)} \end{aligned}$$ -We prove this by expanding the left-hand side. Note that the lower -integration limit is 0 instead of $-\infty$, because we set both $f(t)$ -and $g(t)$ to zero for $t < 0$: +<div class="accordion"> +<input type="checkbox" id="proof-laplace"/> +<label for="proof-laplace">Proof</label> +<div class="hidden"> +<label for="proof-laplace">Proof.</label> +We expand the left-hand side. +Note that the lower integration limit is 0 instead of $-\infty$, +because we set both $f(t)$ and $g(t)$ to zero for $t < 0$: $$\begin{aligned} \hat{\mathcal{L}}\{(f * g)(t)\} @@ -98,6 +110,8 @@ $$\begin{aligned} &= \int_0^\infty \tilde{f}(s) g(t') \exp(- s t') \dd{t'} = \tilde{f}(s) \: \tilde{g}(s) \end{aligned}$$ +</div> +</div> |