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 \newlabel{eq:toy_highpass_pure}{{13}{13}{}{}{}}
 \newlabel{eq:toy_env_noise}{{14}{13}{}{}{}}
 \newlabel{eq:toy_log_noise}{{15}{13}{}{}{}}
-\newlabel{eq:toy_highpass_noise}{{16}{13}{}{}{}}
+\newlabel{eq:toy_highpass_noise}{{16}{14}{}{}{}}
 \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces \textbf  {Intensity invariance through logarithmic compression and adaptation is restricted by the noise floor and decreases SNR.} Input $x_{\text  {filt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is successively transformed into envelope $x_{\text  {env}}(t)$, logarithmically compressed envelope $x_{\text  {log}}(t)$, and intensity-adapted envelope $x_{\text  {adapt}}(t)$. \textbf  {Top}:~Example representations of $x_{\text  {env}}(t)$, $x_{\text  {log}}(t)$, and $x_{\text  {adapt}}(t)$ for different $\alpha $. \textbf  {a}:~Noiseless case. \textbf  {b}:~Noisy case. \textbf  {Bottom}:~Intensity metrics over a range of $\alpha $. \textbf  {c}:~Noiseless case: Standard deviations $\sigma _x$ of $x_{\text  {env}}(t)$, $x_{\text  {log}}(t)$, and $x_{\text  {adapt}}(t)$. \textbf  {d}:~Noisy case: Ratios of $\sigma _x$ of $x_{\text  {env}}(t)$, $x_{\text  {log}}(t)$, and $x_{\text  {adapt}}(t)$ to the respective reference standard deviation $\sigma _{\eta }$ for input $x_{\text  {filt}}(t)=\eta (t)$. Shaded areas indicate $5\,\%$ (dark grey) and $95\,\%$ (light grey) curve span for $x_{\text  {adapt}}(t)$. \textbf  {e}:~Ratios of $\sigma _x$ to $\sigma _{\eta }$ of $x_{\text  {adapt}}(t)$ as in \textbf  {d} for different species (averaged over songs and recordings, see appendix Fig\,\ref {fig:app_log-hp_curves}). Dots indicate $95\,\%$ curve span per species. }}{15}{}\protected@file@percent }
 \newlabel{fig:log-hp}{{5}{15}{}{}{}}
 \@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Thresholding nonlinearity \& temporal averaging}{16}{}\protected@file@percent }
 \newlabel{eq:pdf_split}{{17}{16}{}{}{}}
 \newlabel{eq:feat_avg}{{18}{16}{}{}{}}
 \newlabel{eq:feat_prop}{{19}{16}{}{}{}}
-\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces \textbf  {Intensity invariance through thresholding and temporal averaging is mediated by the interaction of threshold value and noise floor.} Input $x_{\text  {adapt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is transformed into single kernel response $c(t)$, binary response $b(t)$, and feature $f(t)$. Different color shades indicate different threshold values $\Theta $ (multiples of reference standard deviation $\sigma _{\eta }$ of $c(t)$ for input $x_{\text  {adapt}}(t)=\eta (t)$, with darker colors for higher $\Theta $). \textbf  {Left}:~Noisy case: Example representations of $x_{\text  {adapt}}(t)$ as well as $c(t)$, $b(t)$, and $f(t)$ for different $\alpha $. \textbf  {a}:~$x_{\text  {adapt}}(t)$ with kernel $k(t)$ in black. \textbf  {b\,-\,d}: $c(t)$, $b(t)$, and $f(t)$ based on the same $x_{\text  {adapt}}(t)$ from \textbf  {a} but with different $\Theta $. \textbf  {Right}:~Average value $\mu _f$ of $f(t)$ for each $\Theta $ from \textbf  {b\,-\,d}, once for the noisy case (solid lines) and once for the noiseless case (dotted lines). Dots indicate $95\,\%$ curve span (noisy case). \textbf  {e}:~$\mu _f$ over a range of $\alpha $. \textbf  {f}:~$\mu _f$ over the standard deviation of noisy input $x_{\text  {adapt}}$ corresponding to the values of $\alpha $ shown in \textbf  {e}. }}{18}{}\protected@file@percent }
+\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces \textbf  {Intensity invariance through thresholding and temporal averaging is mediated by the interaction of threshold value and noise floor.} Input $x_{\text  {adapt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is transformed into single kernel response $c(t)$, binary response $b(t)$, and feature $f(t)$. Different color shades indicate different threshold values $\Theta $ (multiples of reference standard deviation $\sigma _{\eta }$ of $c(t)$ for input $x_{\text  {adapt}}(t)=\eta (t)$, with darker colors for higher $\Theta $). \textbf  {Left}:~Noisy case: Example representations of $x_{\text  {adapt}}(t)$ as well as $c(t)$, $b(t)$, and $f(t)$ for different $\alpha $. \textbf  {a}:~$x_{\text  {adapt}}(t)$ with kernel $k(t)$ in black. \textbf  {b\,-\,d}: $c(t)$, $b(t)$, and $f(t)$ based on the same $x_{\text  {adapt}}(t)$ from \textbf  {a} but with different $\Theta $. \textbf  {Right}:~Average value $\mu _f$ of $f(t)$ for each $\Theta $ from \textbf  {b\,-\,d}. Dots indicate $95\,\%$ curve span (noisy case). \textbf  {e}:~$\mu _f$ over a range of $\alpha $, once for the noisy case (solid lines) and once for the noiseless case (dotted lines). \textbf  {f}:~Noisy case: $\mu _f$ over the standard deviation of input $x_{\text  {adapt}}$ corresponding to the values of $\alpha $ shown in \textbf  {e}. Shaded area indicates standard deviations that would be capped in the output $x_{\text  {adapt}}(t)$ of the previous transformation pair (see Fig.\,\ref {fig:log-hp}cd). }}{18}{}\protected@file@percent }
 \newlabel{fig:thresh-lp_single}{{6}{18}{}{}{}}
 \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces \textbf  {Feature representation of different species-specific songs saturates at different points in feature space.} Same input and processing as in Fig.\,\ref {fig:thresh-lp_single} but with three different kernels $k_i$, each with a single kernel-specific threshold value $\Theta _i=0.5\cdot \sigma _{\eta _i}$. \textbf  {a}:~Examples of species-specific grasshopper songs. \textbf  {Middle}:~Average value $\mu _{f_i}$ of each feature $f_i(t)$ over $\alpha $ per species (averaged over songs and recordings, see appendix Figs.\,\ref {fig:app_thresh-lp_pure} and \ref {fig:app_thresh-lp_noise}). Different color shades indicate different kernels $k_i$. Dots indicate $95\,\%$ curve span per $k_i$. \textbf  {b}:~Noiseless case. \textbf  {c}:~Noisy case. \textbf  {Bottom}:~2D feature spaces spanned by each pair of $f_i(t)$. Each trajectory corresponds to a species-specific combination of $\mu _{f_i}$ that develops with $\alpha $ (colorbars). Horizontal dashes in the colorbar indicate $5\,\%$ (dark grey) and $95\,\%$ (light grey) curve span of the norm across all three $\mu _{f_i}$ per species. \textbf  {d}:~Noiseless case. \textbf  {e}:~Noisy case. Shaded areas indicate the average minimum $\mu _{f_i}$ across all species-specific trajectories. }}{19}{}\protected@file@percent }
 \newlabel{fig:thresh-lp_species}{{7}{19}{}{}{}}