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Pull request for solution of L06 Q4 - fixed simple display bug in LaTeX display of one equation #11

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Sep 23, 2024
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Pull request for solution of L06 Q4 - fixed simple display bug in LaTeX display of one equation #11

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35 changes: 17 additions & 18 deletions lectures/L04_scipy.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -3043,24 +3043,23 @@
"source": [
"A convolution is defined as: \n",
"\n",
" \\begin{equation} \n",
" (f \\star g)(t) \\equiv \\int_{-\\infty}^{\\infty} f(\\tau) g(t - \\tau) d\\tau \n",
" \\end{equation} \n",
"\n",
" It is easy to compute this with FFTs, via the _convolution theorem_, \n",
" \\begin{equation} \n",
" \\mathcal{F}\\{f \\star g\\} = \\mathcal{F}\\{f\\} \\, \\mathcal{F}\\{g\\} \n",
" \\end{equation} \n",
" That is: the Fourier transform of the convolution of $f$ and $g$ is simply\n",
" the product of the individual transforms of $f$ and $g$. This allows us\n",
" to compute the convolution via multiplication in Fourier space and then take\n",
" the inverse transform, $\\mathcal{F}^{-1}\\{\\}$, to recover the convolution in real space:\n",
" \n",
" \\begin{equation}\n",
" f \\star g = \\mathcal{F}^{-1}\\{ \\mathcal{F}\\{f\\} \\, \\mathcal{F}\\{g\\}\\}\n",
" \\end{equation}\n",
" \n",
"A common use of a convolution is to smooth noisy data, for example by convolving noisy data with a Gaussian. We'll do that here."
"\\begin{equation} \n",
" (f \\star g)(t) \\equiv \\int_{-\\infty}^{\\infty} f(\\tau) g(t - \\tau) d\\tau \\tag{1} \n",
"\\end{equation} \n",
"\n",
"It is easy to compute this with FFTs, via the _convolution theorem_, \n",
"\n",
"\\begin{equation} \n",
" \\mathcal{F}\\{f \\star g\\} = \\mathcal{F}\\{f\\} \\, \\mathcal{F}\\{g\\} \\tag{2} \n",
"\\end{equation} \n",
"\n",
"That is: the Fourier transform of the convolution of $f$ and $g$ is simply the product of the individual transforms of $f$ and $g$. This allows us to compute the convolution via multiplication in Fourier space and then take the inverse transform, $\\mathcal{F}^{-1}\\{\\}$, to recover the convolution in real space:\n",
"\n",
"\\begin{equation} \n",
" f \\star g = \\mathcal{F}^{-1}\\{ \\mathcal{F}\\{f\\} \\, \\mathcal{F}\\{g\\}\\} \\tag{3} \n",
"\\end{equation} \n",
"\n",
"A common use of a convolution is to smooth noisy data, for example by convolving noisy data with a Gaussian. We'll do that here.\n"
]
},
{
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