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authorNao Pross <np@0hm.ch>2021-10-31 00:20:51 +0200
committerNao Pross <np@0hm.ch>2021-10-31 00:29:54 +0200
commitadc00a39baff866a3b68b52ffc55a4aae71e61c4 (patch)
treee4898c12f2199cba12507f7ed6a611c0f1093ddf /doc
parentDoku Implementation Teil angefangen. (diff)
downloadFading-adc00a39baff866a3b68b52ffc55a4aae71e61c4.tar.gz
Fading-adc00a39baff866a3b68b52ffc55a4aae71e61c4.zip
Fix math error in QAM, improve QAM modulator diagram, rename (wrong) QPSK to PSK
Diffstat (limited to '')
-rw-r--r--doc/thesis/Fading.tex1
-rw-r--r--doc/thesis/Makefile2
-rw-r--r--doc/thesis/chapters/theory.tex18
-rw-r--r--doc/thesis/figures/tikz/psk-constellation.tex (renamed from doc/thesis/figures/tikz/qpsk-constellation.tex)0
-rw-r--r--doc/thesis/figures/tikz/qam-modulator.tex58
-rw-r--r--doc/thesis/tex/docmacros.sty1
6 files changed, 65 insertions, 15 deletions
diff --git a/doc/thesis/Fading.tex b/doc/thesis/Fading.tex
index f8a5dfd..7ddd44c 100644
--- a/doc/thesis/Fading.tex
+++ b/doc/thesis/Fading.tex
@@ -35,6 +35,7 @@
\usetikzlibrary{
calc,
+ positioning,
backgrounds,
matrix,
}
diff --git a/doc/thesis/Makefile b/doc/thesis/Makefile
index 7095689..55b7a0d 100644
--- a/doc/thesis/Makefile
+++ b/doc/thesis/Makefile
@@ -15,7 +15,7 @@ SOURCES := \
chapters/conclusions.tex \
\
figures/tikz/overview.tex \
- figures/tikz/qpsk-constellation.tex \
+ figures/tikz/psk-constellation.tex \
figures/tikz/qam-modulator.tex \
figures/tikz/qam-constellation.tex
diff --git a/doc/thesis/chapters/theory.tex b/doc/thesis/chapters/theory.tex
index e3809fd..9477340 100644
--- a/doc/thesis/chapters/theory.tex
+++ b/doc/thesis/chapters/theory.tex
@@ -52,10 +52,10 @@ The two component \(m_i(t)\) and \(m_q(t)\) are mixed with two different periodi
\begin{subequations} \label{eqn:orthonormal-conditions}
\begin{align}
\langle \phi_i | \phi_q \rangle
- &= \int_T \phi_i^* \phi_q \, dt = \int_T \phi_i \phi_q^* \, dt
+ &= \int_T \phi_i \phi_q^* \, dt
= 0, \text{ and } \\
\langle \phi_k | \phi_k \rangle
- &= \int_T \phi_k^* \phi_k \,dt = 1,
+ &= \int_T \phi_k \phi_k^* \,dt = 1,
\text{ where } k \text{ is either } i \text{ or } q.
\end{align}
\end{subequations}
@@ -66,10 +66,10 @@ In practice typically \(\phi_i(t) = \cos(\omega_c t)\) and \(\phi_q(t) = j\sin(\
%% TODO: is this assumption correct?
Notice that assuming \(m_i\) and \(m_q\) are constant\footnote{This is an approximation assuming that the signal changes much slower relative to the carrier.} over the carrier's period \(T\),
\begin{align*}
- \langle s | \phi_i \rangle = \int_T s^* \phi_i \,dt
- &= \int m_i \phi_i^* \phi_i + m_q \phi_q^* \phi_i \,dt \\
- &= m_i \underbrace{\int_T \phi_i^* \phi_i \,dt}_{1}
- + m_q \underbrace{\int_T \phi_q^* \phi_i \,dt}_{0} = m_i,
+ \langle s | \phi_i \rangle = \int_T s \phi_i^* \,dt
+ &= \int m_i \phi_i \phi_i^* + m_q \phi_q \phi_i^* \,dt \\
+ &= m_i \underbrace{\int_T \phi_i \phi_i^* \,dt}_{1}
+ + m_q \underbrace{\int_T \phi_q \phi_i^* \,dt}_{0} = m_i,
\end{align*}
which effectively means that it is possible to isolate a single component \(m_i(t)\) out of \(s(t)\). The same of course works with \(\phi_q\) as well resulting in \(\langle s | \phi_q \rangle = m_q\). Thus (remarkably) it is possible to send two signals on the same frequency, without them interfering with each other. Since each signal can represent one of \(\sqrt{M}\) values, by having two we obtain \(\sqrt{M} \cdot \sqrt{M} = M\) possible combinations.
@@ -83,8 +83,8 @@ A better way to see what QAM does, is to observe a so called \emph{constellation
\end{subfigure}
\hfill
\begin{subfigure}{.4\linewidth}
- \input{figures/tikz/qpsk-constellation}
- \caption{8-PSK\label{fig:qpsk-constellation}}
+ \input{figures/tikz/psk-constellation}
+ \caption{8-PSK\label{fig:psk-constellation}}
\end{subfigure}
\hfill
\caption{
@@ -92,7 +92,7 @@ A better way to see what QAM does, is to observe a so called \emph{constellation
}
\end{figure}
-\section{Quadrature phase shift keying (\(M\)-QPSK)}
+\section{Quadrature phase shift keying (\(M\)-PSK)}
PSK is a popular modulation type for data transmission\cite{Meyer2011}. With a bipolar binary signal, the amplitude remains constant and only the phase will be changed with phase jumps of 180 degrees, which can be seen as a multiplication of the carrier signal with $\pm$ 1. That is alow known as binary phase shift keying.
diff --git a/doc/thesis/figures/tikz/qpsk-constellation.tex b/doc/thesis/figures/tikz/psk-constellation.tex
index d927b19..d927b19 100644
--- a/doc/thesis/figures/tikz/qpsk-constellation.tex
+++ b/doc/thesis/figures/tikz/psk-constellation.tex
diff --git a/doc/thesis/figures/tikz/qam-modulator.tex b/doc/thesis/figures/tikz/qam-modulator.tex
index 59d989f..a55ff29 100644
--- a/doc/thesis/figures/tikz/qam-modulator.tex
+++ b/doc/thesis/figures/tikz/qam-modulator.tex
@@ -57,7 +57,7 @@
\draw (B2Lq.east) -- (Mq.west);
\draw (Mq.east) -| (SUM.south);
- \draw (SUM) -- (S);
+ \draw (SUM.east) -- (S);
\draw (OSC.east) -| (Mi.south);
\draw (OSC.west) -- (H.east);
@@ -77,13 +77,61 @@
\node[above left] at (si) {\(s_i(t)\)};
\node[below left] at (sq) {\(s_q(t)\)};
+ % Draw digital signals
+ \begin{scope}[font = \ttfamily\footnotesize, text = blue!70!white]
+ \node[above = 1mm of M, xshift = 2mm] {\(\ldots 1100101\)};
+ \node[above = 7mm of vmi, xshift = 3mm]
+ {\(\overbracket[.8pt]{\,11\ldots 00\,}^{\sqrt{M} \text{ bits}}\)};
+ \end{scope}
+
+ % Draw analog waveform
+ \begin{scope}[font = \ttfamily\tiny]
+ \coordinate (O) at ($(mi)+(-2mm,10.5mm)$);
+
+ \node[left, red!70!white, anchor = east, text width = 8mm, align = right]
+ at ($(O) + (-2mm,0)$) {\(2^{\sqrt{M}}\) levels};
+
+ \foreach \y in {-3mm,0,3mm} {
+ \draw[gray, densely dotted] (O) ++(-2mm,\y) -- ++(22mm,0);
+ }
+
+ \draw[thick, draw = red!70!white] (O)
+ -- ++(3mm,0) -- ++(0,-3mm) -- ++(3mm,0) -- ++(0,6mm)
+ -- ++(3mm,0) -- ++(0,-3mm) -- ++(3mm,0) -- ++(0,-3mm)
+ -- ++(3mm,0) -- ++(0,3mm) -- ++(3mm,0);
+ \end{scope}
+
+ % Draw constellation diagram
+ \begin{scope}
+ \coordinate (O) at ($(S)+(-7mm,8mm)$);
+ \draw[gray, -latex] (O) ++(-2mm,0) -- ++(12mm,0) node[right] {\tiny \(\phi_i\)};
+ \draw[gray, -latex] (O) ++(0,-2mm) -- ++(0,12mm) node[above] {\tiny \(\phi_q\)};
+
+ \node[
+ circle, thick,
+ minimum size = 3pt,
+ inner sep = 0, outer sep = .8pt,
+ draw = gray, fill = red!50!white
+ ] (P) at ($(O)+(5mm, 4mm)$) {};
+
+ \node[gray, above right] at (P) {\tiny \(s\)};
+
+ \draw[gray, densely dotted]
+ (P) -- (P |- O)
+ (P) -- (P -| O);
+ \end{scope}
+
+
+ % Background elements
\begin{pgfonlayer}{background}
\fill[left color = white, right color = blue!20, draw = white]
- ($(B2Li.north) + (0,1)$) coordinate (D) rectangle ($(B2Lq.south) - (3,1)$);
+ ($(B2Li.north) + (0,1.1)$) coordinate (D) rectangle ($(B2Lq.south) - (3,1)$);
\fill[right color = white, left color = red!20, draw = white]
- ($(B2Li.north) + (0,1)$) coordinate (A) rectangle ($(B2Lq.south) + (9,-1)$);
+ ($(B2Li.north) + (0,1.1)$) coordinate (A) rectangle ($(B2Lq.south) + (9,-1)$);
- \node[blue!50, anchor = south east, font = \ttfamily\bfseries, xshift = -4mm] at (D) {\bfseries\ttfamily Digital bits};
- \node[red!50, anchor = south west, font = \bfseries\ttfamily, xshift = 4mm] at (A) {Analog waveform};
+ \node[blue!50, anchor = south east, font = \ttfamily\bfseries, xshift = -4mm]
+ at (D) {\bfseries\ttfamily Digital bits};
+ \node[red!50, anchor = south west, font = \bfseries\ttfamily, xshift = 4mm]
+ at (A) {Analog waveform};
\end{pgfonlayer}
\end{circuitikz}
diff --git a/doc/thesis/tex/docmacros.sty b/doc/thesis/tex/docmacros.sty
index 433fe19..0005e76 100644
--- a/doc/thesis/tex/docmacros.sty
+++ b/doc/thesis/tex/docmacros.sty
@@ -1,6 +1,7 @@
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{tex/docmacros}[2021/09/27 v0.1 Document Macros]
+\RequirePackage{mathtools}
\RequirePackage{amsmath}
\RequirePackage{bm}