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-rw-r--r--DCS.bib23
-rw-r--r--Makefile8
-rw-r--r--chapter04/content_ch04.tex225
-rw-r--r--chapter04/win_blackman.svg1592
-rw-r--r--chapter04/win_gauss.svg1604
-rw-r--r--chapter04/win_hamming.svg1595
-rw-r--r--chapter04/win_hann.svg1596
-rw-r--r--chapter04/win_rect.svg1594
-rw-r--r--chapter04/win_tri.svg1599
-rw-r--r--common/settings.tex9
10 files changed, 9801 insertions, 44 deletions
diff --git a/DCS.bib b/DCS.bib
index 5799af3..9d08198 100644
--- a/DCS.bib
+++ b/DCS.bib
@@ -7,7 +7,7 @@
howpublished = {Wikimedia},
year = {2007},
note = {License: \href{https://creativecommons.org/licenses/by-sa/3.0/deed.en}{CC-BY-SA
- 3.0}},
+ 3.0}, Accessed: 2020-05-26},
owner = {ple},
timestamp = {2020.05.05},
url = {https://en.wikipedia.org/wiki/File:Rainbow_above_Kaviskis_Lake,_Lithuania.jpg}
@@ -20,7 +20,7 @@
month = {11},
year = {2016},
note = {License: \href{https://creativecommons.org/licenses/by-sa/3.0/deed.en}{CC-BY-SA
- 3.0}},
+ 3.0}, Accessed: 2020-05-26},
owner = {ple},
timestamp = {2020.05.17},
url = {https://commons.wikimedia.org/wiki/File:Comparison_convolution_correlation.svg}
@@ -31,7 +31,7 @@
title = {Netowrk topologies},
howpublished = {Wikimedia},
year = {2011},
- note = {License: Public Domain},
+ note = {License: Public Domain, Accessed: 2020-05-26},
owner = {ple},
timestamp = {2020.05.04},
url = {https://en.wikipedia.org/wiki/File:NetworkTopologies.svg}
@@ -42,7 +42,7 @@
title = {Electromagnetic spectrum},
year = {2012},
note = {License: \href{https://creativecommons.org/licenses/by-sa/3.0/deed.en}{CC-BY-SA
- 3.0}},
+ 3.0}, Accessed: 2020-05-26},
owner = {ple},
timestamp = {2020.05.03},
url = {https://en.wikipedia.org/wiki/File:Electromagnetic-Spectrum.svg}
@@ -78,7 +78,7 @@
howpublished = {Wikimedia},
year = {2007},
note = {License: \href{https://creativecommons.org/licenses/by-sa/3.0/deed.en}{CC-BY-SA
- 3.0}},
+ 3.0}, Accessed: 2020-05-26},
owner = {ple},
timestamp = {2020.05.03},
url = {https://en.wikipedia.org/wiki/File:EM_Spectrum_Properties_edit.svg}
@@ -96,6 +96,19 @@
url = {https://en.wikisource.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field}
}
+@MISC{Niemitalo2013,
+ author = {Olli Niemitalo},
+ title = {Window function and frequency response},
+ howpublished = {Wikipedia},
+ month = {02},
+ year = {2013},
+ note = {License: \href{https://creativecommons.org/publicdomain/zero/1.0/deed.en}{CC0
+ 1.0}, Accessed: 2020-05-26},
+ owner = {ple},
+ timestamp = {2020.05.26},
+ url = {https://en.wikipedia.org/wiki/Window_function}
+}
+
@BOOK{Shannon1949,
title = {The Mathematical Theory of Communication},
publisher = {University of Illinois Press},
diff --git a/Makefile b/Makefile
index 9977656..c13a8b4 100644
--- a/Makefile
+++ b/Makefile
@@ -6,7 +6,7 @@ LATEXMK_PVC = $(LATEXMK) -pvc
ALL_CHAPTERS = $(BUILD_DIR)/chapter00.pdf $(BUILD_DIR)/chapter01.pdf $(BUILD_DIR)/chapter02.pdf $(BUILD_DIR)/chapter03.pdf $(BUILD_DIR)/chapter04.pdf
ALL_EXERCISES = $(BUILD_DIR)/exercise00.pdf $(BUILD_DIR)/exercise01.pdf $(BUILD_DIR)/exercise02.pdf $(BUILD_DIR)/exercise03.pdf
-ALL_SVGS = $(BUILD_DIR)/svg/ch01_EM_Spectrum_Properties.pdf $(BUILD_DIR)/svg/ch01_Electromagnetic-Spectrum.pdf $(BUILD_DIR)/svg/ch01_NetworkTopologies.pdf $(BUILD_DIR)/svg/ch03_Conv_Corr_Auto.pdf
+ALL_SVGS = $(BUILD_DIR)/svg/ch01_EM_Spectrum_Properties.pdf $(BUILD_DIR)/svg/ch01_Electromagnetic-Spectrum.pdf $(BUILD_DIR)/svg/ch01_NetworkTopologies.pdf $(BUILD_DIR)/svg/ch03_Conv_Corr_Auto.pdf $(BUILD_DIR)/svg/ch04_win_blackman.pdf $(BUILD_DIR)/svg/ch04_win_hamming.pdf $(BUILD_DIR)/svg/ch04_win_hann.pdf $(BUILD_DIR)/svg/ch04_win_rect.pdf $(BUILD_DIR)/svg/ch04_win_tri.pdf $(BUILD_DIR)/svg/ch04_win_gauss.pdf
COMMON_DEPS = common/settings.tex common/titlepage.tex common/acronym.tex common/imprint.tex DCS.bib
all: chapters exercises complete
@@ -44,6 +44,12 @@ $(BUILD_DIR)/svg/ch01_EM_Spectrum_Properties.pdf: chapter01/EM_Spectrum_Properti
$(BUILD_DIR)/svg/ch01_Electromagnetic-Spectrum.pdf: chapter01/Electromagnetic-Spectrum.svg
$(BUILD_DIR)/svg/ch01_NetworkTopologies.pdf: chapter01/NetworkTopologies.svg
$(BUILD_DIR)/svg/ch03_Conv_Corr_Auto.pdf: chapter03/Comparison_convolution_correlation.svg
+$(BUILD_DIR)/svg/ch04_win_blackman.pdf: chapter04/win_blackman.svg
+$(BUILD_DIR)/svg/ch04_win_hamming.pdf: chapter04/win_hamming.svg
+$(BUILD_DIR)/svg/ch04_win_hann.pdf: chapter04/win_hann.svg
+$(BUILD_DIR)/svg/ch04_win_rect.pdf: chapter04/win_rect.svg
+$(BUILD_DIR)/svg/ch04_win_tri.pdf: chapter04/win_tri.svg
+$(BUILD_DIR)/svg/ch04_win_gauss.pdf: chapter04/win_gauss.svg
%-watch: main/%.tex
mkdir -p $(BUILD_DIR)
diff --git a/chapter04/content_ch04.tex b/chapter04/content_ch04.tex
index e3a90ac..e93f287 100644
--- a/chapter04/content_ch04.tex
+++ b/chapter04/content_ch04.tex
@@ -1061,7 +1061,7 @@ The \ac{DFT} \eqref{eq:ch04:dft} can be expressed as a linear system of equation
\cmplxvect{X} = \underline{\mat{F}} \cdot \cmplxvect{x}
\end{equation}
-The $N \times N$ transformation matrix $\underline{\mat{F}}$ is the \index{DFTmatrx} \textbf{ac{DFT} matrix} with the elements:
+The $N \times N$ transformation matrix $\underline{\mat{F}}$ is the \index{DFT matrx} \textbf{\ac{DFT} matrix} with the elements:
\begin{equation}
\underline{F}_{pq} = \underline{w}^{p \cdot q}
\end{equation}
@@ -1135,7 +1135,8 @@ A sequence of length $N$ is taken out of $\underline{x}[n]$:
\end{equation}
The act of extracting $N$ subsequent samples out of $\underline{x}[n]$ is called \index{windowing} \textbf{windowing}. To illustrate this, imagine that you watch a sequence moving to the left through a window. The movement to the left is the time advance. Because of the window, you see a certain extract of $\underline{x}[n]$ only at one time.
-\todo{Window illustration}
+%TODO
+%\todo{Window illustration}
This sequence is repeated indefinitely (\emph{periodic continuation}):
\begin{equation}
@@ -1169,22 +1170,31 @@ Windowing and periodic continuation is, in fact, a sampling of the \ac{DTFT} (se
The widowing of $\underline{x}[n]$ to obtain $\underline{x}_N[n]$ as discussed before did not change the values of $\underline{x}[n]$. This can be expressed as a multiplication of the original sequence with the rectangular function:
\begin{equation}
- \underline{x}_W[n] = \underline{x}_N[n] \cdot \underbrace{w_{rect,N}[n]}_{\text{Rectangular window}}
+ \underline{x}_W[n] = \underline{x}_N[n] \cdot \underbrace{w_{rect,M}[n]}_{\text{Rectangular window}}
\end{equation}
-where
+where $w_{rect,M}[n]$ is the rectangular function of the length $M$:
\begin{equation}
- w_{rect,N}[n] = \begin{cases}
- 1 &\quad \text{if } \; 0 \geq n < N, \\
+ w_{rect,M}[n] = \begin{cases}
+ 1 &\quad \text{if } \; -\frac{M-1}{2} \leq n \leq \frac{M-1}{2}, \\
0 &\quad \text{else}.
\end{cases}
\end{equation}
+$M$ shall be an odd number.
-$w[n]$ is called \index{window function} \textbf{window function}.
+$w_M[n]$ is called \index{window function} \textbf{window function}.
\begin{definition}{Window function}
- A \index{window function} \textbf{window function} $w[n]$ is applied to the periodically continued sequence $\underline{x}_N[n]$ by multiplication:
+ A \index{window function} \textbf{window function} $w_M[n]$ is applied to the periodically continued sequence $\underline{x}_N[n]$ by multiplication:
\begin{equation}
- \underline{x}_N[n] \equiv \underline{x}[n] \cdot w[n]
+ \underline{x}_N[n] \equiv \underline{x}[n] \cdot w_M[n]
\end{equation}
+
+ Window functions are $w_M[n]$:
+ \begin{itemize}
+ \item limited to a length of $M$
+ \item where $M$ is an odd number,
+ \item always symmetric (even functions), and
+ \item real-valued.
+ \end{itemize}
\end{definition}
\subsubsection{Spectral Leakage}
@@ -1207,6 +1217,16 @@ This has some implications:
\end{itemize}
This effect is called \index{spectral leakage} \textbf{spectral leakage}.
+\begin{fact}
+ The window changes only the amplitude spectrum, but not the phase of the sampled signal.
+\end{fact}
+\begin{itemize}
+ \item The window function is an even, real-valued function.
+ \item Its Fourier transform is therefore always real-valued, too.
+ \item The purely real-valued ($\arg{\cdot} = 0 \text{ or } \pm \pi$) Fourier transform cannot change the phase of the signal.
+\end{itemize}
+
+
\begin{figure}[H]
\centering
@@ -1242,6 +1262,11 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}.
%ytick={0},
]
\addplot[red, thick, smooth, domain=0:4, samples=50] plot(\x, {cos(deg(pi*\x))});
+
+ \pgfplotsinvokeforeach{0,0.25,...,4}{
+ \addplot[blue] coordinates {(#1,0) (#1,{cos(deg(pi*#1))})};
+ \addplot[blue, only marks, mark=o] coordinates {(#1,{cos(deg(pi*#1))})};
+ }
\end{axis}
\end{tikzpicture}
}
@@ -1254,7 +1279,7 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}.
width=0.35\linewidth,
scale only axis,
xlabel={$n$},
- ylabel={$w[n]$},
+ ylabel={$w_9[n]$},
%grid style={line width=.6pt, color=lightgray},
%grid=both,
grid=none,
@@ -1269,7 +1294,7 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}.
at={(ticklabel* cs:1.05)},
anchor=east,
},
- xmin=-0.5,
+ xmin=-8.5,
xmax=8.5,
ymin=0,
ymax=1.2,
@@ -1277,12 +1302,23 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}.
%xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$},
%ytick={0},
]
- \addplot[blue, thick] coordinates {(-0.5,0) (0,0)};
- \addplot[blue, thick] coordinates {(0,1) (8,1)};
- \addplot[blue, thick] coordinates {(8,0) (8.5,0)};
-
- \addplot[blue, thick, dashed] coordinates {(0,0) (0,1)};
- \addplot[blue, thick, dashed] coordinates {(8,1) (8,0)};
+ \pgfplotsinvokeforeach{-4,-3,...,4}{
+ \addplot[blue] coordinates {(#1,0) (#1,1)};
+ \addplot[blue, only marks, mark=o] coordinates {(#1,1)};
+ }
+ \pgfplotsinvokeforeach{-8,-7,...,-5}{
+ \addplot[blue, only marks, mark=o] coordinates {(#1,0)};
+ }
+ \pgfplotsinvokeforeach{5,6,...,8}{
+ \addplot[blue, only marks, mark=o] coordinates {(#1,0)};
+ }
+
+% \addplot[blue, thick] coordinates {(-0.5,0) (0,0)};
+% \addplot[blue, thick] coordinates {(0,1) (8,1)};
+% \addplot[blue, thick] coordinates {(8,0) (8.5,0)};
+%
+% \addplot[blue, thick, dashed] coordinates {(0,0) (0,1)};
+% \addplot[blue, thick, dashed] coordinates {(8,1) (8,0)};
\end{axis}
\end{tikzpicture}
}
@@ -1310,15 +1346,26 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}.
at={(ticklabel* cs:1.05)},
anchor=east,
},
- xmin=0,
- xmax=4.5,
- ymin=-1.2,
+ xmin=-8.5,
+ xmax=8.5,
+ ymin=0,
ymax=1.2,
%xtick={-1, -0.5, 0, 0.5, 1},
%xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$},
%ytick={0},
]
- \addplot[red, thick, smooth, domain=0:4, samples=50] plot(\x, {cos(deg(pi*\x))});
+ \addplot[red] coordinates {(2,0) (2,1)};
+ \addplot[red] coordinates {(-2,0) (-2,1)};
+ \addplot[red, only marks, mark=o] coordinates {(-2,1) (2,1)};
+ \pgfplotsinvokeforeach{-8,-7,...,-3}{
+ \addplot[red, only marks, mark=o] coordinates {(#1,0)};
+ }
+ \pgfplotsinvokeforeach{-1,0,1}{
+ \addplot[red, only marks, mark=o] coordinates {(#1,0)};
+ }
+ \pgfplotsinvokeforeach{3,4,...,8}{
+ \addplot[red, only marks, mark=o] coordinates {(#1,0)};
+ }
\end{axis}
\end{tikzpicture}
}
@@ -1346,20 +1393,65 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}.
at={(ticklabel* cs:1.05)},
anchor=east,
},
- xmin=-4.5,
- xmax=4.5,
+ xmin=-8.5,
+ xmax=8.5,
ymin=0,
- ymax=1.2,
+ ymax=9.5,
%xtick={-1, -0.5, 0, 0.5, 1},
%xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$},
%ytick={0},
]
- \addplot[blue, thick] coordinates {(-0.5,0) (0,0)};
- \addplot[blue, thick] coordinates {(0,1) (8,1)};
- \addplot[blue, thick] coordinates {(8,0) (8.5,0)};
-
- \addplot[blue, thick, dashed] coordinates {(0,0) (0,1)};
- \addplot[blue, thick, dashed] coordinates {(8,1) (8,0)};
+ \addplot[blue, dashed, smooth, domain=-8:8, samples=50] plot(\x, {9*abs(asinc((pi*\x/8), 9))});
+ \pgfplotsinvokeforeach{-8,-7,...,8}{
+ \addplot[red] coordinates {(#1,0) (#1,{9*abs(asinc((pi*#1/8), 9))})};
+ \addplot[red, only marks, mark=o] coordinates {(#1,{9*abs(asinc((pi*#1/8), 9))})};
+ }
+
+ \draw[blue, dashed] (axis cs:1.5,7) -- (axis cs:2.5,7) node[right,align=left,color=blue]{$\underline{W}\left(e^{j\phi}\right)$\\ $\phi=2\pi\frac{k}{N}$};
+ \draw (axis cs:-1,7) node[left,align=right,color=blue]{Main lobe};
+ \draw (axis cs:-2,3) node[left,align=right,color=blue]{Side lobes};
+ \end{axis}
+ \end{tikzpicture}
+ }
+
+
+ \subfloat[Amplitude spectrum of the windowed signal, showing spectral leakage]{
+ \centering
+ \begin{tikzpicture}
+ \begin{axis}[
+ height={0.15\textheight},
+ width=0.7\linewidth,
+ scale only axis,
+ xlabel={$k$},
+ ylabel={$\underline{X}_W[k]$},
+ %grid style={line width=.6pt, color=lightgray},
+ %grid=both,
+ grid=none,
+ legend pos=north east,
+ axis y line=middle,
+ axis x line=middle,
+ every axis x label/.style={
+ at={(ticklabel* cs:1.05)},
+ anchor=north,
+ },
+ every axis y label/.style={
+ at={(ticklabel* cs:1.05)},
+ anchor=east,
+ },
+ xmin=-8.5,
+ xmax=8.5,
+ ymin=0,
+ ymax=12.5,
+ %xtick={-1, -0.5, 0, 0.5, 1},
+ %xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$},
+ %ytick={0},
+ ]
+ \addplot[blue, dashed, smooth, domain=-8:8, samples=50] plot(\x, {9*abs(asinc((pi*(\x+2)/8), 9)) + 9*abs(asinc((pi*(\x-2)/8), 9))});
+ \pgfplotsinvokeforeach{-8,-7,...,8}{
+ \addplot[red] coordinates {(#1,0) (#1,{9*abs(asinc((pi*(#1+2)/8), 9)) + 9*abs(asinc((pi*(#1-2)/8), 9))})};
+ \addplot[red, only marks, mark=o] coordinates {(#1,{9*abs(asinc((pi*(#1+2)/8), 9)) + 9*abs(asinc((pi*(#1-2)/8), 9))})};
+ }
+ \draw[blue, dashed] (axis cs:3,9) -- (axis cs:3.7,9) node[right,align=left,color=blue]{$\left.\underline{X}_W\left(e^{j\phi}\right)\right|_{\phi=2\pi\frac{k}{N}}$};
\end{axis}
\end{tikzpicture}
}
@@ -1367,8 +1459,54 @@ This effect is called \index{spectral leakage} \textbf{spectral leakage}.
\caption{Spectral leakage of a mono-chromatic signal}
\end{figure}
+In the above example:
+\begin{itemize}
+ \item The monochromatic signal has two ideal pulses in its amplitude spectrum.
+ \item The rectangular window has a sinc-like amplitude spectrum.
+ \item Due to the convolution, the ideal pulses of the monochromatic signal are ``leak'' across the frequency axis and take a sinc-like pattern, too.
+\end{itemize}
+
+The selection of the window function has implications on the spectral leakage:
+\begin{itemize}
+ \item The width of the \emph{main lobe}, defines how much the sampled signal is spread across the frequency axis.
+ \item The \emph{main lobe} should be narrow, in order to retain the original shape of the signal.
+ \item The amplitudes of the \emph{side lobes} define the noise, which emerges.
+ \item The \emph{side lobes} should be as weak as possible regarding their amplitudes, in order to reduce the noise and improve the \ac{SNR}.
+\end{itemize}
+
+\begin{fact}
+ Generally, window functions with narrow main lobes have strong side lobes, and vice versa. So, there is always a trade-off.
+\end{fact}
+
\begin{landscape}
\subsubsection{Some Window Functions}
+
+ \begin{longtable}[H]{|l|l|l|l|l|}
+ \hline
+ Name & Function $w[n]$ & Peak side lobe & Main lobe width & Time domain and frequency response \\
+ \hline
+ \hline
+ Rectangular & $w[n] = \begin{cases}1, &\quad 0 \leq n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \SI{-13}{dB} & $\frac{4\pi}{M+1}$ & \includegraphics[width=5cm]{svg/ch04_win_rect.pdf} \\
+ \hline
+ Hamming & $w[n] = \begin{cases}0.54 - 0.46 \cos\left(\frac{2 \pi n}{M}\right), &\quad 0 \leq n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \SI{-41}{dB} & $\frac{8\pi}{M}$ & \includegraphics[width=5cm]{svg/ch04_win_hamming.pdf} \\
+ \hline
+ Hann & $w[n] = \begin{cases}0.5 - 0.5 \cos\left(\frac{2 \pi n}{M}\right), &\quad 0 \leq n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \SI{-31}{dB} & $\frac{8\pi}{M}$ & \includegraphics[width=5cm]{svg/ch04_win_hann.pdf} \\
+ \hline
+ Blackman & $w[n] = \begin{cases}0.42 - 0.5 \cos\left(\frac{2 \pi n}{M}\right) & \\ \quad + 0.08 \cos\left(\frac{4 \pi n}{M}\right), &\quad 0 \leq n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \SI{-57}{dB} & $\frac{12\pi}{M}$ & \includegraphics[width=5cm]{svg/ch04_win_blackman.pdf} \\
+ \hline
+ Bartlett & $w[n] = \begin{cases}2n/M, &\quad 0 \leq n \leq M/2,\\ 2 - 2n/M, &\quad M/2 < n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \SI{-25}{dB} & $\frac{8\pi}{M}$ & \includegraphics[width=5cm]{svg/ch04_win_tri.pdf} \\
+ \hline
+ Gauss & $w[n] = \begin{cases}\exp\left(- \frac{1}{2} \left(\frac{n - M/2}{\sigma M / 2}\right)^2\right), &\quad 0 \leq n \leq M,\\ 0, &\quad \text{otherwise}.\end{cases}$ & \multicolumn{2}{l|}{Adjustable by $\sigma$} & \includegraphics[width=5cm]{svg/ch04_win_gauss.pdf} \\
+ \hline
+ \end{longtable}
+
+ \textit{Picture credits:} \licensequote{\cite{Niemitalo2013}}{Olli Niemitalo}{\href{https://creativecommons.org/publicdomain/zero/1.0/deed.en}{CC0 1.0}}
+
+ \begin{itemize}
+ \item Now, you see the trade-off which must be made between narrow main lobe and weak side lobes.
+ \item The rectangular window has a narrow main lobe but strong side lobes.
+ \item The Blackman window has weak side lobes but a wide main lobe.
+ \end{itemize}
\end{landscape}
@@ -1597,26 +1735,36 @@ All considerations apply for ergodic or \ac{WSS} processes only:
\end{equation}
\end{itemize}
-\subsection{Energy Spectral Density}
-
-%TODO
+\subsubsection{Energy Spectral Density}
Parseval's theorem for discrete systems:
\begin{equation}
\sum\limits_{n=0}^{N-1} \left|\underline{x}[n]\right|^2 = \frac{1}{N} \sum\limits_{k=0}^{N-1} \left|\underline{X}[k]\right|^2
\end{equation}
+The signal energy is defined to:
\begin{equation}
- E = \sum\limits_{n=0}^{N-1} \left|\underline{x}[n]\right|^2
+ E = T_S^2 \sum\limits_{n=0}^{N-1} \left|\underline{x}[n]\right|^2 \stackrel{!}{=} \sum\limits_{n=0}^{N-1} \underline{\mathrm{S}}_{E,xx}[k]
\end{equation}
+The sampling period $T_S$ (spacing between the samples) must be kept, because the physical energy depends on the duration of the samples.
+The energy spectral density is:
\begin{equation}
- \underline{\mathrm{S}}_{E,xx}[k] = \left|\underline{X}[k]\right|^2
+ \underline{\mathrm{S}}_{E,xx}[k] = \frac{T_S^2}{N} \left|\underline{X}[k]\right|^2
\end{equation}
-\todo{PSD}
+\subsubsection{Power Spectral Density}
-%\subsection{Noise}
+The signal power is:
+\begin{equation}
+ P = \frac{T_S^2}{T} \sum\limits_{n=0}^{N-1} \left|\underline{x}[n]\right|^2 \stackrel{!}{=} \sum\limits_{n=0}^{N-1} \underline{\mathrm{S}}_{xx}[k]
+\end{equation}
+Where $T = N T_S$ is the duration of the signal.
+
+Using the Parseval's theorem:
+\begin{equation}
+ \underline{\mathrm{S}}_{xx}[k] = \underline{\mathrm{S}}_{P,xx}[k] = \frac{T_S}{N} \sum\limits_{n=0}^{N-1} \left|\underline{X}[k]\right|^2
+\end{equation}
\section{Digital Signals and Systems}
@@ -1928,7 +2076,8 @@ For example if the \ac{PCM} value is $\left(10010100\right)_2$, its real value i
The \index{least significant bit} \textbf{\ac{LSB}} is the digit with the exponent $2^0$.
\end{excursus}
-\todo{Plot error}
+%TODO
+%\todo{Plot error}
The quantization error superimposes the original signal and is therefore noise -- the \index{quantization noise} \textbf{quantization noise}.
diff --git a/chapter04/win_blackman.svg b/chapter04/win_blackman.svg
new file mode 100644
index 0000000..3432d57
--- /dev/null
+++ b/chapter04/win_blackman.svg
@@ -0,0 +1,1592 @@
+<?xml version="1.0" encoding="windows-1252" standalone="no"?>
+<svg
+ viewBox="0 0 1200 600"
+ xmlns="http://www.w3.org/2000/svg"
+ xmlns:xlink="http://www.w3.org/1999/xlink"
+>
+
+<title>Window function and its Fourier transform &#8211; Blackman</title>
+<desc>Produced by GNUPLOT 5.2 patchlevel 6 </desc>
+
+<g id="gnuplot_canvas">
+
+<rect x="0" y="0" width="1200" height="600" fill="#ffffff"/>
+<defs>
+
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-20</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-10</tspan></text>
+ </g>
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+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
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+</g>
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+ <path stroke='rgb( 0, 0, 0)' opacity='0.25' class="gridline" d='M714.0,103.2 L1175.9,103.2 '/></g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
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+ <path stroke='rgb( 0, 0, 0)' d='M714.0,103.2 L729.6,103.2 M1175.9,103.2 L1160.3,103.2 '/> <g transform="translate(703.5,109.7)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="end">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >0</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-40</tspan></text>
+ </g>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-20</tspan></text>
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+</g>
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+</g>
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+ <path stroke='rgb( 0, 0, 0)' opacity='0.25' class="gridline" d='M945.0,516.0 L945.0,84.1 '/></g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <path stroke='rgb( 0, 0, 0)' d='M945.0,516.0 L945.0,500.4 M945.0,84.1 L945.0,99.7 '/> <g transform="translate(945.0,545.0)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >0</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="3.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="3.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <path stroke='rgb( 0, 0, 0)' opacity='0.25' class="gridline" d='M1060.4,516.0 L1060.4,84.1 '/></g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <path stroke='rgb( 0, 0, 0)' d='M1060.4,516.0 L1060.4,500.4 M1060.4,84.1 L1060.4,99.7 '/> <g transform="translate(1060.4,545.0)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >20</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="3.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
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+ <path stroke='rgb( 0, 0, 0)' opacity='0.25' class="gridline" d='M1175.9,516.0 L1175.9,84.1 '/></g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
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+ <path stroke='rgb( 0, 0, 0)' d='M1175.9,516.0 L1175.9,500.4 M1175.9,84.1 L1175.9,99.7 '/> <g transform="translate(1175.9,545.0)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >40</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <g transform="translate(648.1,300.1) rotate(-90)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >decibels</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <g transform="translate(944.9,581.3)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >bins</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <g transform="translate(944.9,39.9)" stroke="none" fill="rgb(0,0,0)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >Fourier transform</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+ <g id="gnuplot_plot_1b" ><title>gnuplot_plot_1b</title>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >0</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-20</tspan></text>
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+ <path stroke='rgb( 0, 0, 0)' d='M945.0,516.0 L945.0,500.4 M945.0,84.1 L945.0,99.7 '/> <g transform="translate(945.0,545.0)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >0</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >20</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >40</tspan></text>
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+</g>
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+ <g transform="translate(648.1,300.1) rotate(-90)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >decibels</tspan></text>
+ </g>
+</g>
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+ <g transform="translate(944.9,581.3)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >bins</tspan></text>
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+ <g transform="translate(944.9,39.9)" stroke="none" fill="rgb(0,0,0)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >Fourier transform</tspan></text>
+ </g>
+</g>
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diff --git a/chapter04/win_hamming.svg b/chapter04/win_hamming.svg
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >20</tspan></text>
+ </g>
+</g>
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+</g>
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+</g>
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+ <path stroke='rgb( 0, 0, 0)' opacity='0.25' class="gridline" d='M1175.9,516.0 L1175.9,84.1 '/></g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <path stroke='rgb( 0, 0, 0)' d='M1175.9,516.0 L1175.9,500.4 M1175.9,84.1 L1175.9,99.7 '/> <g transform="translate(1175.9,545.0)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >40</tspan></text>
+ </g>
+</g>
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+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <g transform="translate(648.1,300.1) rotate(-90)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >decibels</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <g transform="translate(944.9,581.3)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >bins</tspan></text>
+ </g>
+</g>
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+ <g transform="translate(944.9,39.9)" stroke="none" fill="rgb(0,0,0)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >Fourier transform</tspan></text>
+ </g>
+</g>
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+</g>
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+ <path stroke='rgb( 0, 0, 0)' d='M714.0,516.0 L1175.9,516.0 '/></g>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-10</tspan></text>
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+</g>
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+</g>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >0</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-40</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-20</tspan></text>
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+</g>
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+</g>
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+ <path stroke='rgb( 0, 0, 0)' opacity='0.25' class="gridline" d='M945.0,516.0 L945.0,84.1 '/></g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
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+ <path stroke='rgb( 0, 0, 0)' d='M945.0,516.0 L945.0,500.4 M945.0,84.1 L945.0,99.7 '/> <g transform="translate(945.0,545.0)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >0</tspan></text>
+ </g>
+</g>
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+</g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="3.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="3.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <path stroke='rgb( 0, 0, 0)' opacity='0.25' class="gridline" d='M1060.4,516.0 L1060.4,84.1 '/></g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
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+ <path stroke='rgb( 0, 0, 0)' d='M1060.4,516.0 L1060.4,500.4 M1060.4,84.1 L1060.4,99.7 '/> <g transform="translate(1060.4,545.0)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >20</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="3.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="3.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <path stroke='rgb( 0, 0, 0)' opacity='0.25' class="gridline" d='M1175.9,516.0 L1175.9,84.1 '/></g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <path stroke='rgb( 0, 0, 0)' d='M1175.9,516.0 L1175.9,500.4 M1175.9,84.1 L1175.9,99.7 '/> <g transform="translate(1175.9,545.0)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >40</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <g transform="translate(648.1,300.1) rotate(-90)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >decibels</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <g transform="translate(944.9,581.3)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >bins</tspan></text>
+ </g>
+</g>
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+ <g transform="translate(944.9,39.9)" stroke="none" fill="rgb(0,0,0)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >Fourier transform</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
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diff --git a/chapter04/win_rect.svg b/chapter04/win_rect.svg
new file mode 100644
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >0</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-40</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-20</tspan></text>
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+</g>
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+ <path stroke='rgb( 0, 0, 0)' d='M945.0,516.0 L945.0,500.4 M945.0,84.1 L945.0,99.7 '/> <g transform="translate(945.0,545.0)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >0</tspan></text>
+ </g>
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+</g>
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+</g>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >20</tspan></text>
+ </g>
+</g>
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+</g>
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+</g>
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+</g>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >40</tspan></text>
+ </g>
+</g>
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+</g>
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+ <g transform="translate(648.1,300.1) rotate(-90)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >decibels</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <g transform="translate(944.9,581.3)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >bins</tspan></text>
+ </g>
+</g>
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+ <g transform="translate(944.9,39.9)" stroke="none" fill="rgb(0,0,0)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >Fourier transform</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+ <g id="gnuplot_plot_1b" ><title>gnuplot_plot_1b</title>
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diff --git a/chapter04/win_tri.svg b/chapter04/win_tri.svg
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >0</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-40</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >-20</tspan></text>
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+ <path stroke='rgb( 0, 0, 0)' d='M945.0,516.0 L945.0,500.4 M945.0,84.1 L945.0,99.7 '/> <g transform="translate(945.0,545.0)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="20.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >0</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >20</tspan></text>
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+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >40</tspan></text>
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+ <g transform="translate(648.1,300.1) rotate(-90)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >decibels</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <g transform="translate(944.9,581.3)" stroke="none" fill="rgb(38,38,38)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >bins</tspan></text>
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+</g>
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+ <g transform="translate(944.9,39.9)" stroke="none" fill="rgb(0,0,0)" font-family="Microsoft Sans Serif:Bold" font-size="28.00" text-anchor="middle">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >Fourier transform</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+ <g id="gnuplot_plot_1b" ><title>gnuplot_plot_1b</title>
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+ L1175.7,497.8 M1175.9,516.0 '/></g>
+ </g>
+ <g id="gnuplot_plot_2b" ><title>gnuplot_plot_2b</title>
+<g fill="none" color="black" stroke="currentColor" stroke-width="0.50" stroke-linecap="butt" stroke-linejoin="miter">
+ <path stroke='rgb( 0, 0, 0)' d='M714.0,516.0 L1175.9,516.0 '/></g>
+ </g>
+<g fill="none" color="#FFFFFF" stroke="rgb( 0, 0, 0)" stroke-width="2.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="2.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <g transform="translate(916.1,506.6)" stroke="none" fill="rgb(0,0,0)" font-family="Microsoft Sans Serif:Bold" font-size="10.00" text-anchor="start">
+ <text><tspan font-family="Microsoft Sans Serif" font-weight="bold" >B = 1.333</tspan></text>
+ </g>
+</g>
+<g fill="none" color="black" stroke="rgb( 0, 0, 0)" stroke-width="3.00" stroke-linecap="butt" stroke-linejoin="miter">
+ <path stroke='rgb( 38, 38, 38)' d='M714.0,516.0 L1175.9,516.0 M714.0,84.1 L1175.9,84.1 M714.0,516.0 L714.0,84.1 M1175.9,516.0 L1175.9,84.1
+ '/></g>
+<g fill="none" color="black" stroke="rgb( 38, 38, 38)" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+<g fill="none" color="black" stroke="currentColor" stroke-width="1.00" stroke-linecap="butt" stroke-linejoin="miter">
+</g>
+</g>
+</svg>
+
diff --git a/common/settings.tex b/common/settings.tex
index ba83d93..0e8e28c 100644
--- a/common/settings.tex
+++ b/common/settings.tex
@@ -107,6 +107,15 @@
};
};
}
+\tikzmath{
+ function asinc(\x, \y) {
+ if abs(\x) < .001 then { % (|x| < .001) ~ (x = 0)
+ return 1;
+ } else {
+ return sin(\y * deg(\x) / 2)/(\y * sin(deg(\x)/2));
+ };
+ };
+}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Symbols
generated by cgit v1.1 at 2025-12-06 03:03:16 +0000