From 8b94ac1a7bab2e7fc7b1379aeb429efe9951e299 Mon Sep 17 00:00:00 2001
From: Jan Benda <jan.benda@uni-tuebingen.de>
Date: Fri, 12 Mar 2021 22:58:55 +0100
Subject: [PATCH] [spectral] updated transfer function for noisy systems

---
 spectral/lecture/spectral.tex | 32 ++++++++++++--------------------
 1 file changed, 12 insertions(+), 20 deletions(-)

diff --git a/spectral/lecture/spectral.tex b/spectral/lecture/spectral.tex
index e00d405..eb745a5 100644
--- a/spectral/lecture/spectral.tex
+++ b/spectral/lecture/spectral.tex
@@ -46,6 +46,7 @@ predict the mean response $\langle R \rangle_n$, averaged over the
 noise, i.e. averaged over responses evoked by several presentations
 of the same, frozen stimulus:
 \begin{equation}
+  \label{transfernoise}
   \langle R(\omega) \rangle_n = H(\omega) S(\omega)
 \end{equation}
 
@@ -74,26 +75,17 @@ averages by $\langle \cdot \rangle$:
 The transfer function of a noisy system is estimated by dividing the
 cross spectrum by the power spectrum of the stimulus.
 
-Computing the squared gain like this
-\begin{equation}
-  |H(\omega)|^2 = \frac{R(\omega)R^*(\omega)}{S(\omega)S^*(\omega)}
-\end{equation}
-is not possible, it again requires to average over the noise
-\begin{equation}
-  |H(\omega)|^2 = \frac{\langle R(\omega)R^*(\omega) \rangle_n}{S(\omega)S^*(\omega)}
-\end{equation}
-Subsequent averaging over stimuli leads to
-\begin{equation}
-  |H(\omega)|^2 = \left\langle\frac{\langle R(\omega)R^*(\omega) \rangle_n}{S(\omega)S^*(\omega)} \right\rangle_s
-\end{equation}
-which is \emph{not} just the power spectrum $\langle R R^* \rangle$ of
-the response devided by the power spectrum $\langle S S^* \rangle$ of
-the stimulus
-\begin{equation}
-  |H(\omega)|^2 \ne \frac{\langle\langle R(\omega)R^*(\omega) \rangle_n\rangle_s}{\langle S(\omega)S^*(\omega)\rangle_s}
-\end{equation}
-The gain can not be computed by simply dividing the response spectrum
-by the stimulus spectrum.
+If we are interested in the gain of the transfer function, i.e. its magnitude, we get starting from Eq.~\eqref{transfernoise}
+\begin{eqnarray}
+  |\langle R(\omega) \rangle_n| & = & |H(\omega)| |S(\omega)| \\
+  \langle R(\omega) \rangle_n \langle R(\omega) \rangle_n^* & = & |H(\omega)|^2 S(\omega)S^*(\omega)\\
+  \langle \langle R(\omega) \rangle_n \langle R(\omega) \rangle_n^* \rangle_s & = & |H(\omega)|^2 \langle S(\omega)S^*(\omega) \rangle_s \\
+  |H(\omega)|^2 & = & \frac{\langle \langle R(\omega) \rangle_n \langle R(\omega) \rangle_n^* \rangle_s}{\langle S(\omega)S^*(\omega) \rangle_s} \\
+  |H(\omega)|^2 & \ne & \frac{\langle \langle R(\omega) R^*(\omega) \rangle_n \rangle_s}{\langle S(\omega)S^*(\omega) \rangle_s}
+\end{eqnarray}
+For noisy systems, dividing the power spectrum of the response by the
+power spectrum of the stimulus is not resulting in the squared gain of
+the transfer function. Only for noise-free systems does this work.
 
 \section{Coherence function}