Good progress with cleaning up "IntInv vs SNR". Gonna merge with "IntInv vs IntInv" tomorrow.

main.tex

@@ -1719,18 +1719,7 @@ degree of temporal integration~(Section\,\ref{sec:constant_feat}).
 
 \subsection{Intensity invariance versus SNR along the model pathway}
 
-% % Establishing the principle trade-off (should maybe come later?):
-% The output of a transformation is considered to be intensity-invariant if its
-% intensity measure saturates for sufficiently large scales $\sca$, which in turn
-% caps the output SNR to a constant value across these $\sca$. Otherwise, the
-% output SNR will increase monotonically with $\sca$. The trade-off between
-% intensity invariance and SNR refers to the principle that a transformation can
-% either improve intensity invariance or maintain SNR --- it cannot do both at
-% the same time. This principle is most likely not specific to the two mechanisms
-% along the model pathway but rather a general property of transformations that
-% equalize between different input intensities.
-
-% Building a sufficient SNR "buffer":
+% Building a sufficiently large SNR "buffer":
 A stridulating grasshopper generates a song with a specific initial intensity,
 which is steadily attenuated as the song propagates through the
 environment~(\bcite{michelsen1978sound}). A listening grasshopper receives a
@@ -1742,12 +1731,12 @@ filtering of $\raw(t)$ into $\filt(t)$ likely improves the SNR by attenuating
 frequencies outside the relevant range of grasshopper songs. The SNR is further
 improved by the rectification and lowpass filtering of $\filt(t)$ into
 $\env(t)$. The lower the cutoff frequency $\fc$ of the lowpass filter, the
-higher the SNR of $\env(t)$ at a given $\sca$, although $\fc$ must also be
-sufficiently high to preserve the amplitude dynamics of the song pattern.
-Overall, the first processing steps along the pathway are not designed to
-achieve intensity invariance but rather to improve the SNR of the song
-representation beyond the initial SNR of $\raw(t)$.
+higher the SNR of $\env(t)$ for a given $\sca$, although $\fc$ must also be
+sufficiently high to preserve the amplitude dynamics of the song pattern. The
+first processing steps along the pathway are hence designed to improve the SNR
+of the song representation beyond the initial SNR of $\raw(t)$.
 
+% Dependence of log-HP intensity invariance on sufficient SNR (+implications):
 The first mechanism of intensity invariance consists of logarithmic compression
 and adaptation of $\env(t)$ into $\adapt(t)$. In the absence of $\noc(t)$,
 $\adapt(t)$ is a perfectly intensity-invariant representation of $\soc(t)$. In
@@ -1757,28 +1746,36 @@ $\raw(t)$ to $\env(t)$ thus serve to improve the intensity invariance of
 $\adapt(t)$ by shifting the saturation point towards lower $\sca$. However,
 this effect is limited --- if the SNR of $\raw(t)$ at the receiver's position
 does not allow for a sufficiently high SNR of $\env(t)$, $\adapt(t)$ will not
-be intensity-invariant. The initial song intensity that the sender can achieve
-therefore determines the distance at which $\adapt(t)$ is intensity-invariant
-to the receiver.
+be intensity-invariant. In this case, the receiver is presumably less likely to
+recognize $\raw(t)$ as a conspecific song. The limitation of the intensity
+invariance of $\adapt(t)$ by the SNR of $\raw(t)$ might hence at least in parts
+be responsible for the limited maximum distance at which song recognition is
+possible~(\bcite{lang2000acoustic}) and the selection towards song patterns
+that are robust to noise masking~(\bcite{einhaupl2011attractiveness}).
 
-Assuming that intensity invariance of $\adapt(t)$ is required for reliable song
-recognition, 
-
-
-This might be a reason why robustness to noise masking is an
-attractive property of male calling songs~(\bcite{einhaupl2011attractiveness}).
-
-The saturation level of $\adapt$,
-unlike its saturation point, is independent of the SNR of $\env(t)$ because the
-influence of $\noc(t)$ is negligible for sufficiently large $\sca$. The output
-SNR of $\adapt(t)$ saturates at a comparably low value of around 10. This might
-in parts be a consequence of the logarithm, which compresses different higher
-intensities but also amplifies lower intensities, including the noise floor.
-Both the saturation level and the saturation point of $\adapt(t)$ vary between
-different species and individual songs. These differences are likely rooted in
-the way in which logarithmic compression acts on the specific distribution of
-$\env(t)$, which is determined by $\fc$ as well as the temporal structure and
-frequency spectrum of the rectified $\filt(t)$.
+% Trading SNR for log-HP intensity invariance (+variability, +general principle):
+The SNR of each song representation prior to $\adapt(t)$ increases
+monotonically with $\sca$~(excluding $0<\sca\ll1$, noise regime). These
+representations maintain and improve the initial SNR of $\raw(t)$ and hence
+never achieve intensity invariance. In contrast, the SNR of the
+intensity-invariant $\adapt(t)$ never exceeds its saturation level even for
+arbitrarily high $\sca$. The saturation level of $\adapt(t)$ varies across
+species and songs. This variability is likely rooted in the way in which
+logarithmic compression acts on the specific distribution of $\env(t)$, which
+depends on the $\fc$ of the lowpass filter as well as the temporal structure
+and frequency spectrum of the rectified $\filt(t)$. Overall, $\adapt(t)$ has
+never been observed to exceed a SNR of around~10 across all songs. The low SNR
+of $\adapt(t)$ partially results from the amplification of smaller values of
+$\env(t)$ by the logarithm, which raises the noise floor of $\adapt(t)$. Still,
+the reduction in SNR is substantial --- considering that the SNR of preceeding
+song representations has been orders of magnitude higher --- but is likely a
+necessary price to pay for the intensity invariance of $\adapt(t)$. After all,
+a transformation cannot compress a range of different input intensities into a
+constant output intensity without sacrificing some of the corresponding input
+SNR. Accordingly, the trade-off between intensity invariance and SNR is not
+expected to be specific to the particular mechanisms along the pathway but
+presumably applies to any transformation that achieves or improves intensity
+invariance.
 
 Thresholding and temporal averaging renders feature $f_i(t)$
 intensity-invariant for sufficiently large $\sca$. The trade-off between