Fix math error in QAM, improve QAM modulator diagram, rename (wrong) QPSK to PSK

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}