Finished (:D) fig_invariance_log_hp.pdf.
Added movable label string to time_bar().
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42
main.tex
42
main.tex
@@ -47,6 +47,12 @@
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% \newcommand{\eref}[1]{\mbox{\cref{#1}}}
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% \newcommand{\eref}[1]{\mbox{Eq.\,\ref{#1}}}
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% Subplot lettering:
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\newcommand{\figa}{\textbf{a}}
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\newcommand{\figb}{\textbf{b}}
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\newcommand{\figc}{\textbf{c}}
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\newcommand{\figd}{\textbf{d}}
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% Math shorthands - Standard symbols:
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\newcommand{\dec}{\log_{10}} % Logarithm base 10
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\newcommand{\infint}{\int_{-\infty}^{+\infty}} % Indefinite integral
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@@ -625,23 +631,25 @@ the signal for reliable song recognition.
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\includegraphics[width=\textwidth]{figures/fig_invariance_log_hp.pdf}
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\caption{\textbf{Intensity invariance by logarithmic compression and
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adaptation is restricted by the noise floor.}
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Envelope $\env(t)$ is transformed into logarihmically
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compressed envelope $\db(t)$ and further into
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intensity-adapted envelope $\adapt(t)$. Indicated time
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scale is $5\,$s for both \textbf{a} and \textbf{b} (black
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bars).
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\textbf{a}:~Ideally, if $\env(t)$ consists only of song
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component $\soc(t)$ rescaled by $\sca$, then $\adapt(t)$
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is fully intensity-invariant across all $\sca$.
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\textbf{b}:~In practice, $\env(t)$ also contains
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fixed-scale noise component $\noc(t)$, which limits the
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effective intensity invariance of $\adapt(t)$ to
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sufficiently large $\sca$.
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\textbf{c}:~Ratios of the SD of each representation in
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\textbf{b} at a given $\sca$ relative to the SD of the
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representation for $\sca=0$ (solid lines). The same ratios
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for the ideal $\adapt(t)$ in \textbf{a} are shown for
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comparison (dashed line).
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Synthetic input $\filt(t)$ consists of song component
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$\soc(t)$ scaled by $\sca$ with (\figc{} and \figd) or
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without (\figa{} and \figb) additive noise component
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$\noc(t)$. Input $\filt(t)$ is transformed into envelope
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$\env(t)$, logarithmically compressed envelope $\db(t)$,
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and intensity-adapted envelope $\adapt(t)$.
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\textbf{Left}:~$\env(t)$, $\db(t)$, and $\adapt(t)$ for
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different scales $\sca$.
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\textbf{Right}:~Ratios of the standard deviation of
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$\env(t)$, $\db(t)$, and $\adapt(t)$ relative to the
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respective reference standard deviation for input
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$\filt(t)=\noc(t)$.
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\figa{} and \figb:~Ideally, if $\filt(t)=\sca\cdot\soc(t)$, then
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$\adapt(t)$ is intensity-invariant across all $\sca$.
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\figc{} and \figd:~In practice, if
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$\filt(t)=\sca\cdot\soc(t)+\noc(t)$, the intensity
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invariance of $\adapt(t)$ is limited to sufficiently large
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$\sca$. Shaded area indicates saturation of $\adapt(t)$ at
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$95\,\%$ curve span.
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}
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\label{fig:inv_log-hp}
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\end{figure}
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