diff options
Diffstat (limited to 'source/know/concept/convolution-theorem/index.md')
| -rw-r--r-- | source/know/concept/convolution-theorem/index.md | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/source/know/concept/convolution-theorem/index.md b/source/know/concept/convolution-theorem/index.md index 3f9eafb..8462fcc 100644 --- a/source/know/concept/convolution-theorem/index.md +++ b/source/know/concept/convolution-theorem/index.md @@ -24,10 +24,10 @@ and $$A$$ and $$B$$ are the constants from its definition: $$\begin{aligned} \boxed{ \begin{aligned} - A \cdot (f * g)(x) + A \: (f * g)(x) &= \hat{\mathcal{F}}{}^{-1}\Big\{ \tilde{f}(k) \: \tilde{g}(k) \Big\} \\ - B \cdot (\tilde{f} * \tilde{g})(k) + B \: (\tilde{f} * \tilde{g})(k) &= \hat{\mathcal{F}}\Big\{ f(x) \: g(x) \Big\} \end{aligned} } @@ -45,7 +45,7 @@ $$\begin{aligned} \\ &= A \int_{-\infty}^\infty g(x') \: f(x - x') \dd{x'} \\ - &= A \cdot (f * g)(x) + &= A \: (f * g)(x) \end{aligned}$$ Then we do the same again, @@ -59,7 +59,7 @@ $$\begin{aligned} \\ &= B \int_{-\infty}^\infty \tilde{g}(k') \: \tilde{f}(k - k') \dd{k'} \\ - &= B \cdot (\tilde{f} * \tilde{g})(k) + &= B \: (\tilde{f} * \tilde{g})(k) \end{aligned}$$ {% include proof/end.html id="proof-fourier" %} |