From 395cfb98ab7d25bfa7df566206083889a7212d86 Mon Sep 17 00:00:00 2001 From: j-hartling Date: Tue, 16 Jun 2026 18:23:14 +0200 Subject: [PATCH] Good progress with cleaning up "IntInv vs SNR". Gonna merge with "IntInv vs IntInv" tomorrow. --- main.pdf | Bin 33850878 -> 33851638 bytes main.tex | 73 ++++++++++++++++++++++++++----------------------------- 2 files changed, 35 insertions(+), 38 deletions(-) diff --git a/main.pdf b/main.pdf index 81040ed30ef116c5c68628cfcdd3fab40b1dfc99..901627d70323aba8f5f4848dcda84ff515f6c9e3 100644 GIT binary patch delta 14305 zcmb8!bx__=*C_gi($Yc;#obDAcZX7}xVsg1cYSCZpg?f<0>#~-xI2{M?$F}ya{9dY z`_7zm@0~ew{>X1uGD#-0leO30I}f4JZ!V&u-}r+7hyXqX5y5BRa}Wu90loy0K@{*6 z_!>k7(ZDw#I*0*cf>kATdY+l7eI)IrtW&0N;U>AQear(txxe z9Y_x{fQ%p$$PBW8tRNf64sw8;AQ#9D@_@V`AIJ|1fP$b9C=9*_MZgcBC@2PsgA$-5 z_z{!>r9l}`7L)_!K?P6|R05Sj6;Ksa1JywdP!rSwwLu+F7t{mwK?Cp;Xb2jC#-Isk z3YvlDpap0NT7lM}4QLD6f%c#S=mez%lSQI1WyLli(CM4bFhG;2by) zE`W>R61WVmfUDpdxDIZBo8T6>4gLZDf;-?YxCico2jC%i1RjGY;3;?po`V{mAXfl#10Z(* z@&F)D0P+GLZvgTEAYTCT10Wax`2)~r0Qv$zUjYaNpa1|01fU=Q`UXJ302Bg1p#TH{ zP#6G(15gA2MFLP107U~(3;@LfP#gfo15g40B?3?q03`!Z3IL@7P#OTG15gIg^c~>; 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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