diff --git a/ASCEE_lasp.lyx b/ASCEE_lasp.lyx
index b34ed14..5641ab0 100644
--- a/ASCEE_lasp.lyx
+++ b/ASCEE_lasp.lyx
@@ -4,7 +4,7 @@
 \begin_header
 \save_transient_properties true
 \origin unavailable
-\textclass article
+\textclass memoir
 \begin_preamble
 \input{tex/preamble_article.tex}
 \end_preamble
@@ -15,8 +15,8 @@
 \language_package babel
 \inputencoding utf8
 \fontencoding global
-\font_roman "libertine" "Linux Libertine O"
-\font_sans "default" "Courier New"
+\font_roman "libertine" "Libertinus Sans"
+\font_sans "default" "default"
 \font_typewriter "default" "default"
 \font_math "libertine-ntxm" "auto"
 \font_default_family default
@@ -31,7 +31,7 @@
 \default_output_format default
 \output_sync 1
 \output_sync_macro "\synctex=1"
-\bibtex_command biber
+\bibtex_command default
 \index_command default
 \paperfontsize 10
 \spacing single
@@ -551,6 +551,10 @@ confidential
 
 \end_layout
 
+\begin_layout Chapter
+Power spectra estimation
+\end_layout
+
 \begin_layout Section
 Fourier transform vs discrete Fourier transform
 \end_layout
@@ -718,7 +722,7 @@ Sample frequency:
 \end_layout
 
 \begin_layout Standard
-Frequency resolution 
+Frequency resolution:
 \begin_inset Formula 
 \begin{equation}
 \Delta f=\frac{f_{s}}{N_{\mathrm{DFT}}},
@@ -735,7 +739,7 @@ i.e.
 \end_layout
 
 \begin_layout Standard
-Definition of the DFT:
+Our definition of the DFT:
 \end_layout
 
 \begin_layout Standard
@@ -1089,11 +1093,11 @@ w_{N}=0.01
 
 \end_layout
 
-\begin_layout Section
+\begin_layout Chapter
 Impedance tube
 \end_layout
 
-\begin_layout Subsection
+\begin_layout Section
 Design aspects
 \end_layout
 
@@ -1742,7 +1746,7 @@ where
 ).
 \end_layout
 
-\begin_layout Subsection
+\begin_layout Section
 Method for computing the sample impedance, absorption and reflection coefficient
 \end_layout
 
@@ -2017,12 +2021,65 @@ Nonlinearities
 Noise
 \end_layout
 
+\begin_layout Chapter
+Digital signal processing
+\end_layout
+
 \begin_layout Section
 Filter bank design
 \end_layout
 
 \begin_layout Subsection
-Overlap-save method
+FIR Filter
+\end_layout
+
+\begin_layout Standard
+An FIR filter performs the operation:
+\begin_inset Formula 
+\begin{equation}
+y[n]=\boldsymbol{h}*\boldsymbol{x}[n]\equiv\sum_{m=0}^{N-1}h[m]x[n-m].
+\end{equation}
+
+\end_inset
+
+This operation can be implemented in time domain, or in frequency domain.
+ Notably the 
+\begin_inset Quotes eld
+\end_inset
+
+fast convolution form
+\begin_inset Quotes erd
+\end_inset
+
+ is of interest, as it requires fewer operations for high 
+\begin_inset Formula $N$
+\end_inset
+
+ (smaller complexity).
+\end_layout
+
+\begin_layout Subsubsection
+Fast convolution: overlap-save method
+\end_layout
+
+\begin_layout Standard
+Definitions:
+\end_layout
+
+\begin_layout Itemize
+FIR Filter length: 
+\begin_inset Formula $N$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Block length of inputs: 
+\begin_inset Formula $L$
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
@@ -2047,7 +2104,7 @@ status open
 \align center
 \begin_inset Graphics
 	filename img/overlap_save.pdf
-	width 90text%
+	width 70text%
 
 \end_inset
 
@@ -2055,12 +2112,19 @@ status open
 \begin_inset Caption Standard
 
 \begin_layout Plain Layout
-
+Overlap-save algorithm
 \end_layout
 
 \end_inset
 
 
+\begin_inset CommandInset label
+LatexCommand label
+name "fig:schematic_overlapsave"
+
+\end_inset
+
+
 \end_layout
 
 \end_inset
@@ -2069,7 +2133,7 @@ status open
 \end_layout
 
 \begin_layout Standard
-The yellow blocks are saved for the next 
+The yellow blocks are saved for the next block.
 \end_layout
 
 \begin_layout Section
@@ -2246,7 +2310,7 @@ H\left(\omega\right)=H[z=e^{i\omega T}],
 For the DSP equation:
 \begin_inset Formula 
 \begin{equation}
-H(\omega)=\frac{\sum\limits _{p=0}^{P}a_{p}e^{-ip\Omega}}{1-\sum\limits _{m=1}^{M}b_{m}e^{-im\Omega}}
+H(\omega)=\frac{\sum\limits _{p=0}^{P-1}a_{p}e^{-ip\Omega}}{1-\sum\limits _{m=1}^{M-1}b_{m}e^{-im\Omega}}
 \end{equation}
 
 \end_inset
@@ -2595,6 +2659,146 @@ Example: 1000 Hz octave band: pass-band is from 750 Hz to 1500 Hz.
  Use a Hamming window to reduce the pass and stop-band ripple.
 \end_layout
 
+\begin_layout Subsection
+First order digital high pass
+\end_layout
+
+\begin_layout Standard
+A simple digital high-pass filter can be implemented using:
+\begin_inset Formula 
+\begin{equation}
+G(s)=\frac{\tau s}{1+\tau s},
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\tau$
+\end_inset
+
+ is the 
+\begin_inset Formula $-3$
+\end_inset
+
+
+\begin_inset space ~
+\end_inset
+
+dB time constant, as when 
+\begin_inset Formula $\omega=\tau^{-1}$
+\end_inset
+
+,
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+ 
+\begin_inset Formula $|G|=\frac{1}{\sqrt{2}}$
+\end_inset
+
+.
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+\begin_inset Formula $2\left(\omega\tau\right)^{2}=1+\left(\omega\tau\right)^{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $\left(\omega\tau\right)^{2}=1$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+ Such that the cut-on frequency is:
+\begin_inset Formula 
+\begin{equation}
+\frac{1}{2\pi f_{c}}=\tau.
+\end{equation}
+
+\end_inset
+
+Filling in, we find:
+\begin_inset Formula 
+\begin{equation}
+G(s)=\frac{\frac{s}{2\pi f_{c}}}{1+\frac{s}{2\pi f_{c}}},
+\end{equation}
+
+\end_inset
+
+ Applying the bilinear transform, we find:
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $G(s)=\frac{2\pi f_{c}s}{1+2\pi f_{c}s},$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+Bilinear: 
+\begin_inset Formula $2f_{s}\frac{z-1}{z+1},$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{2f_{s}\tau\left(z-1\right)}{2f_{s}\tau\left(z-1\right)+z+1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+Readjust:
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{2f_{s}\tau\left(1-z^{-1}\right)}{2f_{s}\tau\left(1-z^{-1}\right)+1+z^{-1}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+Make denominator a0 1:
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{\frac{2f_{s}\tau}{\left(1+2f_{s}\tau\right)}\left(1-z^{-1}\right)}{1+\frac{\left(1-2f_{s}\tau\right)}{\left(1+2f_{s}\tau\right)}z^{-1}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{equation}
+G[z]=\frac{\frac{2f_{s}\tau}{\left(1+2f_{s}\tau\right)}\left(1-z^{-1}\right)}{1+\frac{\left(1-2f_{s}\tau\right)}{\left(1+2f_{s}\tau\right)}z^{-1}}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Section
 Sound level meter implementation
 \end_layout
@@ -2611,7 +2815,7 @@ Fast time-weighted sound level,of the A-weighted pressure signal
 :
 \begin_inset Formula 
 \begin{equation}
-L_{AF}=10\log_{10}\left(\frac{1}{\tau_{F}}\int\limits _{-\infty}^{t}p_{A}^{2}\left(\xi\right)e^{-\left(t-\xi\right)/\tau_{F}}\right)-10\log_{10}\left(p_{\mathrm{ref}}^{2}\right),
+L_{AF}=10\log_{10}\left(\frac{1}{\tau_{F}}\int\limits _{-\infty}^{t}p_{A}^{2}\left(\xi\right)e^{-\left(t-\xi\right)/\tau_{F}}\mathrm{d}\xi\right)-10\log_{10}\left(p_{\mathrm{ref}}^{2}\right),
 \end{equation}
 
 \end_inset
@@ -2662,6 +2866,26 @@ ms
 Implementation of single pole low pass filter
 \end_layout
 
+\begin_layout Standard
+The time weighting is specified as:
+\begin_inset Formula 
+\begin{equation}
+L_{\mathrm{AF}}(t)=10\log_{10}\left[\frac{1}{p_{\mathrm{ref}}^{2}}\frac{1}{\tau_{F}}\int_{-\infty}^{t}p_{\mathrm{A}}^{2}(t)\exp\left(-\frac{\left(t-\xi\right)}{\tau_{\mathrm{F}}}\right)\mathrm{d}\xi\right].
+\end{equation}
+
+\end_inset
+
+The Laplace transform of the integral is:
+\begin_inset Formula 
+\begin{equation}
+\mathcal{L}\left[\frac{1}{\tau_{F}}\int_{-\infty}^{t}p_{\mathrm{A}}^{2}\exp\left(-\frac{\left(t-\xi\right)}{\tau_{\mathrm{F}}}\right)\mathrm{d}\xi\right]=p_{\mathrm{A}}^{2}(s)\frac{1}{1+\tau_{F}s}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout Standard
 A single pole low pass filter has a frequency response of
 \begin_inset Formula 
@@ -2671,6 +2895,14 @@ G_{\mathrm{splp}}=\frac{1}{1+\tau s},
 
 \end_inset
 
+
+\end_layout
+
+\begin_layout Subsubsection
+Implementation using Bilinear transform
+\end_layout
+
+\begin_layout Standard
 we create a digital filter from this one using the bilinear transform:
 \begin_inset Formula 
 \begin{equation}
@@ -2691,22 +2923,32 @@ status collapsed
 \end_layout
 
 \begin_layout Plain Layout
-\begin_inset Formula $G_{\mathrm{splp},d}=\frac{z+1}{\left(1+\tau2f_{s}\right)z+1-\tau2f_{s}}$
+Filling in 
+\begin_inset Formula $z=\exp(i\omega/f_{s})$
+\end_inset
+
+ and setting 
+\begin_inset Formula $\omega=0$
 \end_inset
 
 
 \end_layout
 
 \begin_layout Plain Layout
-Divide by 
-\begin_inset Formula $z$
+
+\series bold
+\begin_inset Formula $G_{\mathrm{splp},d}(\omega=0)=1$
 \end_inset
 
-, both numerator and denominator:
+
 \end_layout
 
 \begin_layout Plain Layout
-\begin_inset Formula $G_{\mathrm{splp},d}=\frac{1+z^{-1}}{\left(1+\tau2f_{s}\right)+\left(1-\tau2f_{s}\right)z^{-1}}$
+==============
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $G_{\mathrm{splp},d}=\frac{1+z^{-1}}{\left(1+2f_{s}\tau\right)+\left(1-2f_{s}\tau\right)z^{-1}}$
 \end_inset
 
 
@@ -2741,7 +2983,31 @@ Normalizing such that
 \end_layout
 
 \begin_layout Plain Layout
-–
+– Check: filling in 
+\begin_inset Formula $z=0$
+\end_inset
+
+ should give a unit gain:
+\end_layout
+
+\begin_layout Plain Layout
+
+\family roman
+\series medium
+\shape up
+\size normal
+\emph off
+\bar no
+\strikeout off
+\xout off
+\uuline off
+\uwave off
+\noun off
+\color none
+\begin_inset Formula $G_{\mathrm{splp},d}(z=0)=\frac{\left(1+2\tau f_{s}\right)^{-1}}{\frac{\left(1-\tau2f_{s}\right)}{\left(1+2\tau f_{s}\right)}}$
+\end_inset
+
+
 \end_layout
 
 \end_inset
@@ -2749,7 +3015,7 @@ Normalizing such that
 
 \begin_inset Formula 
 \begin{equation}
-G_{\mathrm{splp},d}=\frac{1+z^{-1}}{\left(\frac{1}{\tau}+2f_{s}\right)+\left(\frac{1}{\tau}-2f_{s}\right)z^{-1}},
+G_{\mathrm{splp},d}=\frac{\left(1+2\tau f_{s}\right)^{-1}\left(1+z^{-1}\right)}{1+\frac{\left(1-\tau2f_{s}\right)}{\left(1+2\tau f_{s}\right)}z^{-1}},
 \end{equation}
 
 \end_inset
@@ -2775,9 +3041,10 @@ No correction for frequency warping has been done, as for all cases
 
 .
  The output frequency of the sound level meter will be decimated to a sampling
- frequency, where the single pole low pass filter has a -20 dB point:
+ frequency, where the single pole low pass filter has a -20 dB point, such
+ that aliasing is at max ~ 0.1 times any oscillation:
 \begin_inset Note Note
-status open
+status collapsed
 
 \begin_layout Plain Layout
 \begin_inset Formula $\frac{1-0.01}{0.01}=\left|\tau s\right|\Rightarrow\left|s\right|=\frac{1}{\tau}\frac{1-0.01}{0.01}\Rightarrow f_{s,\mathrm{slm}}=$
@@ -2861,6 +3128,156 @@ id>N-o
 \end_inset
 
 
+\end_layout
+
+\begin_layout Subsubsection
+Implementation using the matched Z-transform method
+\begin_inset Formula 
+\begin{equation}
+G_{\mathrm{splp},d}=\frac{1-\exp\left(-\left(f_{s}\tau\right)^{-1}\right)}{1-\exp\left(-\left(f_{s}\tau\right)^{-1}\right)z^{-1}}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+By setting unity gain for 
+\begin_inset Formula $\omega=0$
+\end_inset
+
+, this results in the following simple difference equation:
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+\begin_inset Formula $Y[z]=\frac{1-\exp\left(\left(f_{s}\tau_{d}\right)^{-1}\right)}{1-\exp\left(\left(f_{s}\tau_{d}\right)^{-1}\right)z^{-1}}X[z]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $y[n]=\exp\left(\left(f_{s}\tau_{d}\right)^{-1}\right)y[n-1]+\left[1-\exp\left(\left(f_{s}\tau_{d}\right)^{-1}\right)\right]x[n]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align}
+y[n] & =\alpha y[n-1]+\left(1-\alpha\right)x[n],\\
+\alpha & =\exp\left(-\frac{1}{f_{s}\tau}\right)
+\end{align}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+This way: the number of dB's / s, that the level is decaying, after signal
+ stop is:
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+y[n]=\alpha y[n-1]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+For 
+\begin_inset Formula $\alpha>0$
+\end_inset
+
+:
+\begin_inset Formula 
+\[
+|y[n]|=\alpha|y[n-1]|
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+y[n]=y[0]\alpha^{n}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+L\left(y[n]\right)=20\log\left(|y[n]\right)=20\log\left(|y(0)|\right)+20\log\left(\alpha^{n}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\begin{align*}
+L(y(t)) & =20\log\left(|y[n=\frac{t}{\Delta t}]\right)=20\log\left(|y(0)|\right)+\frac{t}{\Delta t}20\log\left(\alpha\right)\\
+ & =20\log\left(|y(0)|\right)+t\frac{1}{\Delta t}\underbrace{20\log\left(\alpha\right)}
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{equation}
+\mathrm{Decay\,rate\,dB\,/\,s}\approx\frac{20}{\Delta t}\log\left(\alpha\right)
+\end{equation}
+
+\end_inset
+
+
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+And the other way around:
+\begin_inset Formula $20\log\left(\alpha\right)=d\Delta t$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $\alpha=10^{\frac{d\Delta t}{20}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Subsection
@@ -3864,6 +4281,533 @@ h=s^{-1}*\left(y[n]-e[n]\right)
 \end_inset
 
 
+\end_layout
+
+\begin_layout Section
+Transfer function estimation using least squares estimation and a FIR filter
+\end_layout
+
+\begin_layout Standard
+This is a time domain method to estimate the transfer function by minimizing
+ the error.
+ For a FIR system we can write
+\begin_inset Formula 
+\begin{equation}
+y[n]=\sum_{l=0}^{L-1}w[n-l]x[n-l]+d[n]=\boldsymbol{w}\cdot\boldsymbol{x}[n]+d[n],
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $\boldsymbol{w}$
+\end_inset
+
+ is the vector of FIR coefficients 
+\begin_inset Formula $d[n]$
+\end_inset
+
+ is an unknown disturbance (noise), and 
+\begin_inset Formula $\boldsymbol{x}[n]$
+\end_inset
+
+ the time history at discrete time point 
+\begin_inset Formula $n$
+\end_inset
+
+ containing 
+\begin_inset Formula $L$
+\end_inset
+
+ points.
+ If we write
+\begin_inset Formula 
+\begin{equation}
+e[n]=y[n]-\boldsymbol{w}\cdot\boldsymbol{x}[n],
+\end{equation}
+
+\end_inset
+
+we can generate the cost function
+\begin_inset Formula 
+\begin{equation}
+J=\sum_{n=0}^{N-1}e^{2}[n]
+\end{equation}
+
+\end_inset
+
+to be minimized, we can estimate the optimal filter 
+\begin_inset Formula $\boldsymbol{w}$
+\end_inset
+
+ by taking the derivative of 
+\begin_inset Formula $J$
+\end_inset
+
+ w.r.t.
+ to 
+\begin_inset Formula $\boldsymbol{w}$
+\end_inset
+
+ and setting that to 0.
+ This results in:
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{\partial J}{\partial\boldsymbol{w}}=\frac{\partial}{\partial\boldsymbol{w}}\sum_{n=0}^{N-1}\left[e[n]^{2}\right]=\sum_{n=0}^{N-1}\left[2e[n]\frac{\partial e[n]}{\partial\boldsymbol{w}}\right]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+Working out:
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $\frac{\partial J}{\partial\boldsymbol{w}}=\sum_{n=0}^{N-1}\left[2e[n]\boldsymbol{x}[n]\right]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+Setting this to 
+\begin_inset Formula $\boldsymbol{0}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula 
+\[
+\sum_{n=0}^{N-1}\left[2\left(y[n]-\boldsymbol{w}\cdot\boldsymbol{x}[n]\right)\boldsymbol{x}[n]\right]=\boldsymbol{0}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+Defining the matrix 
+\begin_inset Formula $X_{nl}$
+\end_inset
+
+ containing the time history at time at time instance 
+\begin_inset Formula $n$
+\end_inset
+
+ and history point 
+\begin_inset Formula $l$
+\end_inset
+
+, and 
+\begin_inset Formula $Y_{nl}$
+\end_inset
+
+ containing the output signal at time instance 
+\begin_inset Formula $n$
+\end_inset
+
+, we can write 
+\begin_inset Formula $\hat{\boldsymbol{y}}=\boldsymbol{X}\cdot\boldsymbol{w}$
+\end_inset
+
+, such that:
+\begin_inset Formula 
+\[
+\sum_{n=0}^{N-1}\left[2\left(y[n]-\boldsymbol{w}\cdot\boldsymbol{x}[n]\right)\boldsymbol{x}[n]\right]=\boldsymbol{0}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+For all 
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Vector fitting
+\end_layout
+
+\begin_layout Itemize
+Method to estimate poles and zeros (rational transfer function estimation)
+ of an arbitrary transfer function.
+ Guaranteed stable poles.
+\end_layout
+
+\begin_layout Standard
+Suppose we want to fit the transfer function 
+\begin_inset Formula $f(s)$
+\end_inset
+
+ using a rational function approximation as: 
+\begin_inset Formula 
+\begin{equation}
+f(s)\approx\sum_{n=1}^{N}\frac{c_{n}}{s-a_{n}}+d+sh,
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $a_{n}$
+\end_inset
+
+ are the poles.
+ We introduce an 
+\emph on
+unknown
+\emph default
+ function 
+\begin_inset Formula $\sigma(s)$
+\end_inset
+
+ which shares the same poles as 
+\begin_inset Formula $f(s)$
+\end_inset
+
+, but with different zeros:
+\begin_inset Formula 
+\begin{equation}
+\sigma(s)=\sum_{n=1}^{N}\frac{\tilde{c}_{n}}{s-a_{n}}+1
+\end{equation}
+
+\end_inset
+
+We create the augmented problem:
+\begin_inset Formula 
+\begin{equation}
+\left[\begin{array}{c}
+\sigma(s)f(s)\\
+\sigma(s)
+\end{array}\right]\approx\left[\begin{array}{c}
+\sum\limits _{n=1}^{N}\frac{c_{n}}{s-a_{n}}+d+sh\\
+\sum_{n=1}^{N}\frac{\tilde{c}_{n}}{s-a_{n}}+k(s)
+\end{array}\right].
+\end{equation}
+
+\end_inset
+
+So we assume that we can fit 
+\begin_inset Formula $\sigma(s)f(s)$
+\end_inset
+
+ with the top equation, and 
+\begin_inset Formula $\sigma(s)$
+\end_inset
+
+ with the bottom equation.
+ By multiplying the bottom equation with 
+\begin_inset Formula $f(s)$
+\end_inset
+
+, we find:
+\begin_inset Formula 
+\begin{equation}
+\underbrace{\sum\limits _{n=1}^{N}\frac{c_{n}}{s-a_{n}}+d+sh}_{\left(\sigma f\right)_{\mathrm{fit}}}=\underbrace{\left(\sum_{n=1}^{N}\frac{\tilde{c}_{n}}{s-a_{n}}+1\right)f(s)}_{\sigma_{\mathrm{fit}}(s)f(s)},
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+For a certain set of starting poles 
+\begin_inset Formula $a_{n}$
+\end_inset
+
+, we are able to fit the zeros.
+\end_layout
+
+\begin_layout Section
+Soft gain transitioning
+\end_layout
+
+\begin_layout Standard
+Equation governing gain evolution, running at each sample:
+\begin_inset Formula 
+\begin{equation}
+\Delta g=\alpha\left(g_{\mathrm{required}}-g_{\mathrm{old}}\right),
+\end{equation}
+
+\end_inset
+
+which can be written as:
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+\begin_inset Formula $G[z]\left(1-z^{-1}\right)=\alpha\left(G_{\mathrm{required}}[z]-z^{-1}G[z]\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $G[z]\left(1-z^{-1}\right)+\alpha z^{-1}G[z]=\alpha G_{\mathrm{required}}[z]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $G[z]=\frac{\alpha}{1+\left(\alpha-1\right)z^{-1}}G_{\mathrm{required}}[z]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{equation}
+G[z]=\frac{\alpha}{1+\left(\alpha-1\right)z^{-1}}G_{\mathrm{required}}(z),
+\end{equation}
+
+\end_inset
+
+which is an approximate first order digital low-pass filter:
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Filling in for 
+\begin_inset Formula $H(s)=H[z=e^{sT}],$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $G\left(s\right)=\frac{\alpha}{1+\left(\alpha-1\right)\exp\left(-s/f_{s}\right)}G_{\mathrm{required}}(s)$
+\end_inset
+
+,for 
+\begin_inset Formula $s\ll f_{s}$
+\end_inset
+
+ can be written as:
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $G\left(s\right)\approx\frac{\alpha}{\alpha+\left(1-\alpha\right)s/f_{s}}G_{\mathrm{required}}(s)$
+\end_inset
+
+ which has a pole at approximately: 
+\begin_inset Formula $s_{p}=f_{s}\left[1-\frac{1}{1-\alpha}\right]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+Fill in:
+\begin_inset Formula $\alpha=0.1\Rightarrow s_{p}=f_{s}\left[1-\frac{1}{0.9}\right]$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula $s=f_{s}\left[-9\right]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{equation}
+G\left(s\right)\approx\frac{\alpha}{\alpha+s/f_{s}}G_{\mathrm{required}}(s)\qquad s\ll f_{s},0<\alpha\ll1
+\end{equation}
+
+\end_inset
+
+Hence for the transition time constant, 
+\begin_inset Formula $s_{p}=-\frac{2\pi}{\tau_{p}}$
+\end_inset
+
+, we find:
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\alpha=1-\frac{1}{1+\frac{2\pi}{\tau_{p}f_{s}}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{equation}
+\alpha=1-\frac{1}{1+\frac{2\pi}{\tau_{p}f_{s}}}
+\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Programming aspects
+\end_layout
+
+\begin_layout Section
+Array ordering
+\end_layout
+
+\begin_layout Itemize
+A 2D array of is called in 
+\emph on
+row-major order
+\emph default
+ if each of the rows is subsequentially put in memory.
+ This means, we can find element 
+\begin_inset Formula $A_{\mathrm{row},\mathrm{col}}=A_{ij}$
+\end_inset
+
+ in memory 
+\begin_inset Formula $A[i+j\mathrm{n_{rows}}]$
+\end_inset
+
+.
+ This is also called a C-contiguous array.
+\end_layout
+
+\begin_layout Itemize
+A 2D array of is called in 
+\emph on
+column-major order
+\emph default
+ if each of the columns is subsequentially put in memory.
+ This means, we can find element 
+\begin_inset Formula $A_{\mathrm{row},\mathrm{col}}=A_{ij}$
+\end_inset
+
+ in memory 
+\begin_inset Formula $A[in_{\mathrm{cols}}+j]$
+\end_inset
+
+.
+ This is also called a Fortran-contiguous array.
+ For the 
+\family typewriter
+dmat
+\family default
+ structure in 
+\family typewriter
+LASP
+\family default
+, we have chosen to use the Fortran-contiguous convention.
+\end_layout
+
+\begin_layout Standard
+Numpy works under the hood with 
+\family typewriter
+strides
+\family default
+, strides are the values with which each array index needs to be multiplied
+ to obtain the value of each index in an 
+\begin_inset Formula $N$
+\end_inset
+
+-dimensional array.
+ 
+\family typewriter
+Strides
+\family default
+ is a tuple containing 
+\begin_inset Formula $(a,b,c,\dots$
+\end_inset
+
+).
+ Such that 
+\begin_inset Formula $A[i,j,k,\dots]=A[ai+bj+ck+\dots]$
+\end_inset
+
+.
+ Strides are a method to have an arbitrary memory layout, and also provide
+ a very easy way to transpose an array over one dimension (just swap the
+ stride values in the tuple).
+\end_layout
+
+\begin_layout Chapter
+Beamforming
+\end_layout
+
+\begin_layout Section
+Near-field cardioid beamformer
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{equation}
+\boldsymbol{x}(\omega)=\boldsymbol{a}(\omega)s,
+\end{equation}
+
+\end_inset
+
+where 
+\begin_inset Formula $s$
+\end_inset
+
+ is the source signal, and 
+\begin_inset Formula $\boldsymbol{a}(\omega)$
+\end_inset
+
+ the transfer from the source signal to the measured signal 
+\begin_inset Formula $\boldsymbol{x}$
+\end_inset
+
+.
+ Now, we would like to generate a signal 
+\begin_inset Formula $y=\boldsymbol{w}^{H}\boldsymbol{x}$
+\end_inset
+
+, that is sensitive to signals at position 
+\begin_inset Formula $\boldsymbol{r}_{s}$
+\end_inset
+
+ and not sensitive to signals at other positions.
+ Therefore, the optimal beamformer is such that:
+\begin_inset Formula 
+\begin{align}
+\boldsymbol{w}^{H}\boldsymbol{a}s_{\boldsymbol{r}=\boldsymbol{r}_{0}} & =1,\\
+\boldsymbol{w}^{H}\boldsymbol{a}s_{\boldsymbol{r}\neq\boldsymbol{r}_{0}} & =0.
+\end{align}
+
+\end_inset
+
+ Now, the sensitivity to a source is:
+\begin_inset Formula 
+\begin{equation}
+\boldsymbol{a}=\left\{ \begin{array}{c}
+\exp\left(-i\omega R_{1}/c_{0}\right)/R_{1}\\
+\exp\left(-i\omega R_{2}/c_{0}\right)/R_{2}\\
+\vdots\\
+\\
+\end{array}\right\} 
+\end{equation}
+
+\end_inset
+
+
 \end_layout
 
 \begin_layout Standard
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