Finished a good part of analysis and figure for Thresh-LP invariance (WIP).

main.tex

@@ -624,18 +624,24 @@ the signal for reliable song recognition.
     \centering
     \includegraphics[width=\textwidth]{figures/fig_invariance_log_hp.pdf}
     \caption{\textbf{Intensity invariance by logarithmic compression and
-                     adaptation.}
-                     \textbf{a}:~Synthetic envelopes resulting from the
-                     mixture of song component $\soc(t)$ with scale $\sca$ and
-                     noise component $\noc(t)$ with fixed scale.
-                     \textbf{b}:~Corresponding logarithmically scaled envelopes.
-                     \textbf{c}:~Corresponding intensity-adapted envelopes.
-                     \textbf{d}:~Absolute SNRs of each representation relative
-                     to the representation without song component ($\sca=0$).
-                     \textbf{e}:~Same SNRs as in \textbf{d} but normalized to
-                     the maximum SNR for each representation.
-                     \textbf{f}:~Relative amplification of normalized SNRs
-                     relative to the normalized SNRs of the initial envelope.
+                     adaptation is restricted by the noise floor.}
+                     Synthetic envelope $\env(t)$ is transformed into
+                     logarihmically compressed envelope $\db(t)$ and further
+                     into intensity-adapted envelope $\adapt(t)$. Indicated
+                     time scale is $5\,$s for both \textbf{a} and \textbf{b}
+                     (black bars).
+                     \textbf{a}:~Ideally, if $\env(t)$ consists only of song
+                     component $\soc(t)$ rescaled by $\sca$, then $\adapt(t)$
+                     is fully intensity-invariant across all $\sca$.
+                     \textbf{b}:~In practice, $\env(t)$ also contains
+                     fixed-scale noise component $\noc(t)$, which limits the
+                     effective intensity invariance of $\adapt(t)$ to
+                     sufficiently large $\sca$.
+                     \textbf{c}:~Ratios of the SD of each representation in
+                     \textbf{b} at a given $\sca$ relative to the SD of the
+                     representation for $\sca=0$ (solid lines). The same ratios
+                     for the ideal $\adapt(t)$ in \textbf{a} are shown for
+                     comparison (dashed line).
                      }
     \label{fig:inv_log-hp}
 \end{figure}