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\documentclass{article}
\usepackage{amsmath}
\begin{document}
Here is what resampling we need to do. Content video is at $C_V$ fps, audio at $C_A$.
\section{Easy case 1}
$C_V$ and $C_A$ are both DCI rates, e.g.\ if $C_V = 24$, $C_A = 48\times{}10^3$.
\medskip
\textbf{Nothing to do.}
\section{Easy case 2}
$C_V$ is a DCI rate, $C_A$ is not. e.g.\ if $C_V = 24$, $C_A = 44.1\times{}10^3$.
\medskip
\textbf{Resample $C_A$ to the DCI rate.}
\section{Hard case 1}
\label{sec:hard1}
$C_V$ is not a DCI rate, $C_A$ is, e.g.\ if $C_V = 25$, $C_A =
48\times{}10^3$. We will run the video at a nearby DCI rate $F_V$,
meaning that it will run faster or slower than it should. We resample
the audio to $C_V C_A / F_V$ and mark it as $C_A$ so that it, too,
runs faster or slower by the corresponding factor.
e.g.\ if $C_V = 25$, $F_V = 24$ and $C_A = 48\times{}10^3$, we
resample audio to $25 * 48\times{}10^3 / 24 = 50\times{}10^3$.
\medskip
\textbf{Resample $C_A$ to $C_V C_A / F_V$}
\section{Hard case 2}
Neither $C_V$ nor $C_A$ is not a DCI rate, e.g.\ if $C_V = 25$, $C_A =
44.1\times{}10^3$. We will run the video at a nearby DCI rate $F_V$,
meaning that it will run faster or slower than it should. We first
resample the audio to a DCI rate $F_A$, then perform as with
Section~\ref{sec:hard1} above.
\medskip
\textbf{Resample $C_A$ to $C_V F_A / F_V$}
\section{The general case}
Given a DCP running at $F_V$ and $F_A$ and a piece of content at $C_V$
and $C_A$, resample the audio to $R_A$ where
\begin{align*}
R_A &= \frac{C_V F_A}{F_V}
\end{align*}
Once this is done, consider 1 second's worth of content samples ($C_A$
samples). We have turned them into $R_A$ samples which should still
last 1 second. These samples are then played back at $F_A$ samples
per second, so they last $R_A / F_A$ seconds. Hence there is a
scaling between some content time and some DCP time of $R_A / F_A$
i.e. $C_V / F_V$.
\end{document}
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