Added documentation how to calculate transfer matrix from impedance tube measurements

.gitignore

@@ -1,2 +1,3 @@
 .ipynb_checkpoints
 .spyproject
+.pdf

lrftubes.lyx

@@ -8269,6 +8269,19 @@ in which
  can be 'measured' by averaging it over the port's boundary.
 \end_layout
 
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TO DO: redraw image
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 \begin_inset Float figure
 wide false
@@ -8345,10 +8358,6 @@ Electrical and mechanical model of the speaker
 \end_inset
 
 
-\end_layout
-
-\begin_layout Plain Layout
-
 \end_layout
 
 \end_inset
@@ -10894,7 +10903,7 @@ T_{21} & T_{22}
 \end{array}\right]\left\{ \begin{array}{c}
 p_{o}\\
 Q_{o}
-\end{array}\right\} ,
+\end{array}\right\} ,\label{eq:transfer_matrix_COMSOL}
 \end{equation}
 
 \end_inset
@@ -10946,6 +10955,455 @@ LookupModel
 .
 \end_layout
 
+\begin_layout Chapter
+Measuring the transmission matrix using the four microphone method
+\end_layout
+
+\begin_layout Standard
+Based on Brüel Kjaer - Transmission loss in impedance tube.pdf in /home/anne/next
+cloud/wip_redusone/2021-Steegmuller/measurement_setup
+\end_layout
+
+\begin_layout Standard
+Modifications: volume flow U instead of velocity v; impedance Z instead
+ of characteristic impedance z; transfer functions Hir instead of cross
+ correlations (?).
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TO DO:
+\end_layout
+
+\begin_layout Plain Layout
+draw own image image
+\end_layout
+
+\begin_layout Plain Layout
+fix citation
+\end_layout
+
+\begin_layout Plain Layout
+Transfer matrix according to our own definition instead of the definition
+ of Bruel & Kjaer = definition of COMSOL
+\end_layout
+
+\begin_layout Plain Layout
+Consistently use Q or U for volume flow? Also in text above about COMSOL.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The transfer matrix of a device can be measured using a four microphone
+ setup as shown in figure 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "fig:meas_transmatrix_4mic"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+ The microphones record acoustic pressure and plane waves are assumed.
+ In the following equations, time dependency 
+\begin_inset Formula $\exp(+j*\omega*t)$
+\end_inset
+
+ is not shown.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\align center
+\begin_inset Graphics
+	filename img/Bruel_Kjaer_fig1.png
+	lyxscale 50
+	width 80text%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Experimental setup to measure the transfer matrix, using the four microphone
+ method
+\end_layout
+
+\end_inset
+
+
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:meas_transmatrix_4mic"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The transfer matrix coefficients are calculated based on sound pressure
+ 
+\begin_inset Formula $p$
+\end_inset
+
+ and volume velocity 
+\begin_inset Formula $U$
+\end_inset
+
+, as related by equation 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "eq:transfer_matrix_COMSOL"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+ Note that this definition is different than the definition used in LRFtubes
+ and therefore 
+\begin_inset Formula $T$
+\end_inset
+
+ should be inverted for further use.
+ Subscrips 
+\begin_inset Formula $i$
+\end_inset
+
+ and 
+\begin_inset Formula $d$
+\end_inset
+
+ refer to 
+\begin_inset Formula $x=0$
+\end_inset
+
+ and 
+\begin_inset Formula $x=d$
+\end_inset
+
+ respectively.
+ There are two equations and four unknowns, so two sets of measurements
+ are required.
+ The second set, indicated by superscript 
+\begin_inset Formula $*$
+\end_inset
+
+, must be performed with a different acoustic termination.
+ Together this results in four equations for four unknowns.
+\end_layout
+
+\begin_layout Standard
+\align left
+\begin_inset Formula 
+\begin{equation}
+\left\{ \begin{array}{c}
+p_{i}\\
+Q_{i}
+\end{array}\begin{array}{c}
+p_{i}^{*}\\
+Q_{i}^{*}
+\end{array}\right\} =\left[\begin{array}{cc}
+T_{11} & T_{12}\\
+T_{21} & T_{22}
+\end{array}\right]\left\{ \begin{array}{c}
+p_{o}\\
+Q_{o}
+\end{array}\begin{array}{c}
+p_{o}^{*}\\
+Q_{o}^{*}
+\end{array}\right\} ,\label{eq:transfer_matrix-double}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Solving for 
+\begin_inset Formula $T$
+\end_inset
+
+ yields:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\left[\begin{array}{cc}
+T_{11} & T_{12}\\
+T_{21} & T_{22}
+\end{array}\right]=\frac{1}{p_{d}Q_{d}^{*}-p_{d}^{*}Q_{d}}\left[\begin{array}{cc}
+p_{i}Q_{d}^{*}-p_{i}^{*}Q_{d} & -p_{i}p_{d}^{*}+p_{i}^{*}p_{d}\\
+Q_{i}Q_{d}^{*}-Q_{i}^{*}Q_{d} & -p_{d}^{*}Q_{i}+p_{d}Q_{i}^{*}
+\end{array}\right]
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $p$
+\end_inset
+
+ and 
+\begin_inset Formula $Q$
+\end_inset
+
+ at 
+\begin_inset Formula $x=0$
+\end_inset
+
+ and 
+\begin_inset Formula $x=d$
+\end_inset
+
+ can be calculated from travelling 
+\begin_inset Formula $A$
+\end_inset
+
+, 
+\begin_inset Formula $B$
+\end_inset
+
+, 
+\begin_inset Formula $C$
+\end_inset
+
+ and 
+\begin_inset Formula $D$
+\end_inset
+
+.
+ The calculation of their second measurement counterparts 
+\begin_inset Formula $*$
+\end_inset
+
+ goes analogously and uses 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $B^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $C^{*}$
+\end_inset
+
+ and 
+\begin_inset Formula $D^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+p_{i}=A+B
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+Q_{i}=\frac{A-B}{Z_{0}}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+p_{d}=C\cdot e^{-jkd}+D\cdot e^{jkd}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+Q_{d}=\frac{C\cdot e^{-jkd}-D\cdot e^{jkd}}{Z_{0}}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+in which 
+\begin_inset Formula $Z_{0}=\frac{z_{0}}{S}$
+\end_inset
+
+ is the impedance of an infinite duct, with 
+\begin_inset Formula $z_{0}$
+\end_inset
+
+ the characteristic impedance and 
+\begin_inset Formula $S$
+\end_inset
+
+ the cross-sectional area, 
+\begin_inset Formula $j=\sqrt{-1}$
+\end_inset
+
+, 
+\begin_inset Formula $k$
+\end_inset
+
+ the wavenumber.
+ Travelling waves 
+\begin_inset Formula $A$
+\end_inset
+
+, 
+\begin_inset Formula $B$
+\end_inset
+
+, 
+\begin_inset Formula $C$
+\end_inset
+
+ and 
+\begin_inset Formula $D$
+\end_inset
+
+ can be calculated from transfer functions 
+\begin_inset Formula $H_{ir}$
+\end_inset
+
+ from reference signal 
+\begin_inset Formula $r$
+\end_inset
+
+, as sent to the loudspeaker, to the recorded signal of microphone 
+\begin_inset Formula $i$
+\end_inset
+
+.
+ The calculation of their second measurement counterparts 
+\begin_inset Formula $*$
+\end_inset
+
+ goes analogously and uses 
+\begin_inset Formula $H_{ir}^{*}$
+\end_inset
+
+.
+\begin_inset Formula 
+\begin{equation}
+A=\frac{j\left(H_{1r}\cdot e^{jkx_{2}}-H_{2r}\cdot e^{jkx_{1}}\right)}{2\sin\left(k\left(x_{1}-x_{2}\right)\right)}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+B=\frac{j\left(H_{2r}\cdot e^{-jkx_{1}}-H_{1r}\cdot e^{-jkx_{2}}\right)}{2\sin\left(k\left(x_{1}-x_{2}\right)\right)}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+C=\frac{j\left(H_{3r}\cdot e^{jkx_{4}}-H_{4r}\cdot e^{jkx_{3}}\right)}{2\sin\left(k\left(x_{3}-x_{4}\right)\right)}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+D=\frac{j\left(H_{4r}\cdot e^{-jkx_{3}}-H_{3r}\cdot e^{-jkx_{4}}\right)}{2\sin\left(k\left(x_{3}-x_{4}\right)\right)}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\sqrt{G_{rr}}$
+\end_inset
+
+ has been removed from the equations because Caspers thinks that 
+\begin_inset Formula $H_{ir}$
+\end_inset
+
+ refers to the cross spectrum instead of the transfer function.
+ If the transfer function is used, then 
+\begin_inset Formula $\sqrt{G_{rr}}$
+\end_inset
+
+ shall be left out.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Note: if no reference signal has been recorded, the reference signal can
+ be set to the signal captured by microphone 1.
+ The equations have no way to figure out whether the loudspeaker really
+ was driven by such a signal.
+ Then a requirement is that all microphones are recorded simultaneously
+ and with synchronized ADC clocks.
+\end_layout
+
 \begin_layout Chapter
 IEC Coupler impedances
 \end_layout