From 569193b000a5df2caa9ef9127a06882bdb0c428a Mon Sep 17 00:00:00 2001
From: Philipp Le <philipp-le-prviat@freenet.de>
Date: Sat, 6 Jun 2020 03:06:39 +0200
Subject: WIP: Chapter 5 - Digital Modulation

---
 chapter05/content_ch05.tex | 828 +++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 803 insertions(+), 25 deletions(-)

(limited to 'chapter05/content_ch05.tex')

diff --git a/chapter05/content_ch05.tex b/chapter05/content_ch05.tex
index 7252131..15d8a58 100644
--- a/chapter05/content_ch05.tex
+++ b/chapter05/content_ch05.tex
@@ -1662,11 +1662,26 @@ All digital modulation techniques take time-discrete and value-discrete data.
 	\item We already know these terms from the chapter about sampling.
 	\begin{itemize}
 		\item The time-discrete data is transferred back to a time-continuous, but still value-discrete domain.
-		\item Each instantaneous value of the time-discrete data points is prolonged to the symbol period $T_{sym}$.
+		\item Each instantaneous value of the time-discrete data points is prolonged to the symbol period $T_{sym}$. In fact, it becomes a rectangle function.
 		\item The result is a series of symbols $x_{sym}(t)$.
 	\end{itemize}
 \end{itemize}
 
+The process of converting time-discrete symbols to time-continuous rectangle functions can be mathematically described by:
+\begin{equation}
+	x_{sym}(t) = \sum\limits_{n} x_{sym}[n] \cdot \mathrm{rect}_{T_{sym}}\left(t - n T_{sym}\right)
+	\label{eq:ch05:sym_rect}
+\end{equation}
+
+\begin{excursus}{The rectangle function}
+	\begin{equation}
+		\mathrm{rect}_{T_{sym}}\left(t\right) = \frac{1}{T_{sym}} \begin{cases}
+			0 &\quad \text{if } |t| > \frac{1}{2} T_{sym}, \\
+			1 &\quad \text{if } |t| \leq \frac{1}{2} T_{sym}
+		\end{cases}
+	\end{equation}
+\end{excursus}
+
 \begin{figure}[H]
 	\centering
 	\begin{tikzpicture}
@@ -1698,7 +1713,7 @@ All digital modulation techniques take time-discrete and value-discrete data.
 			%xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$},
 			%ytick={0},
 			ytick={0, 0.25, 0.5, 0.75, 1},
-			yticklabels={0, $(00)_2 \mapsto 0.25$, $(01)_2 \mapsto 0.5$, $(10)_2 \mapsto 0.75$, $(11)_2 \mapsto 1$},
+			yticklabels={0, $0$, $1$, $2$, $3$},
 		]
 			\draw[blue] (axis cs:0,1) -- (axis cs:2,1) -- (axis cs:2,0.25) -- (axis cs:4,0.25) -- (axis cs:4,0.75) -- (axis cs:6,0.75) -- (axis cs:6,0.5) -- (axis cs:8,0.5) -- (axis cs:8,1) -- (axis cs:9,1);
 			
@@ -1710,7 +1725,7 @@ All digital modulation techniques take time-discrete and value-discrete data.
 %			\draw (7,-0.5) node{01};
 		\end{axis}
 	\end{tikzpicture}
-	\caption{Series of symbols, each encoding 2 bits}
+	\caption[Series of symbols]{Series of symbols of the set $\left\{0, 1, 2, 3\right\}$. The set is exemplary and can be any set of discrete values.}
 \end{figure}
 
 \subsubsection{Symbol Mapping}
@@ -1730,13 +1745,13 @@ However, the modulation is capable of encoding only $K_m$ discrete values. \text
 		Data & Symbol \\
 		\hline
 		\hline
-		$\left[0, 0\right]^{\mathrm{T}}$ & 0.25 \\
+		$\left[0, 0\right]^{\mathrm{T}}$ & 0 \\
 		\hline
-		$\left[1, 0\right]^{\mathrm{T}}$ & 0.5 \\
+		$\left[1, 0\right]^{\mathrm{T}}$ & 1 \\
 		\hline
-		$\left[0, 1\right]^{\mathrm{T}}$ & 0.75 \\
+		$\left[0, 1\right]^{\mathrm{T}}$ & 2 \\
 		\hline
-		$\left[1, 1\right]^{\mathrm{T}}$ & 1 \\
+		$\left[1, 1\right]^{\mathrm{T}}$ & 3 \\
 		\hline
 	\end{tabular}
 \end{table}
@@ -1814,7 +1829,7 @@ Remember that the received signal is subject to noise (thermal noise, quantizati
 	\caption{Abstract receiver signal chain of demodulation and data detection}
 \end{figure}
 
-\subsection{Amplitude-Shift Keying and Symbol Mapping}
+\subsection{Amplitude-Shift Keying}
 
 The application of the \ac{AM} in digital communication system is the \index{amplitude-shift keying} \acf{ASK}.
 \begin{itemize}
@@ -1824,9 +1839,15 @@ The application of the \ac{AM} in digital communication system is the \index{amp
 	\item Drawback: The amplitude can be strongly affected by disturbances. \ac{ASK} signals are not very immune against noise.
 \end{itemize}
 
-The amplitude can take $K_m$ discrete values. The data is mapped to symbols $x_{sym}(t)$ which alter the amplitude of the carrier.
+The amplitude can take $K_m$ discrete values (symbols). The symbols are mapped to amplitude levels between $\hat{A}_L$ and the maximum amplitude $\hat{A}_H$:
+\begin{equation}
+	x_{A}(t) = \hat{A}_L + \frac{\hat{A}_H - \hat{A}_L}{K_m - 1} x_{sym}(t)
+\end{equation}
+where $x_{sym}(t) \in \left\{0, 1, \cdots, K_m - 1\right\}$.
+
+The \ac{ASK} is:
 \begin{equation}
-	x_{ASK}(t) = x_{sym}(t) \underbrace{\cos\left(\omega_{RF} t\right)}_{\text{Carrier}}
+	x_{ASK}(t) = x_{A}(t) \underbrace{\cos\left(\omega_{RF} t\right)}_{\text{Carrier}}
 \end{equation}
 
 \begin{figure}[H]
@@ -1879,7 +1900,7 @@ The amplitude can take $K_m$ discrete values. The data is mapped to symbols $x_{
 			\draw (7,-1.5) node{0};
 		\end{axis}
 	\end{tikzpicture}
-	\caption{\acs{ASK} with $K_m = 2$ discrete states capable of encoding 1 bit}
+	\caption{\acs{ASK} with $K_m = 2$ discrete states (with $\hat{A}_L = 0.2$ and $\hat{A}_H = 1$) capable of encoding 1 bit}
 \end{figure}
 
 \begin{figure}[H]
@@ -1922,13 +1943,13 @@ The amplitude can take $K_m$ discrete values. The data is mapped to symbols $x_{
 		\draw[olive, dashed] (axis cs:0,1) -- (axis cs:2,1) -- (axis cs:2,0.2) -- (axis cs:4,0.2) -- (axis cs:4,0.8) -- (axis cs:6,0.8) -- (axis cs:6,0.5) -- (axis cs:8,0.5) -- (axis cs:8,1) -- (axis cs:9,1);
 		%\addlegendentry{Envelope of $x_B(t)$};
 		
-		\draw (1,-1.5) node{11};
-		\draw (3,-1.5) node{00};
-		\draw (5,-1.5) node{10};
-		\draw (7,-1.5) node{01};
+		\draw (1,-1.5) node{3};
+		\draw (3,-1.5) node{0};
+		\draw (5,-1.5) node{2};
+		\draw (7,-1.5) node{1};
 	\end{axis}
 	\end{tikzpicture}
-	\caption{\acs{ASK} with $K_m = 4$ discrete states capable of encoding 2 bits}
+	\caption{\acs{ASK} with $K_m = 4$ discrete states (with $\hat{A}_L = 0.25$ and $\hat{A}_H = 1$) capable of encoding 2 bits}
 \end{figure}
 
 \subsubsection{Phasor Representation}
@@ -1938,10 +1959,13 @@ The above examples depicted different constellations for $K_m$.
 The \ac{ASK} is a multiplication of the symbol stream $x_{sym}(t)$ which the mono-chromatic carrier. This causes an amplitude change with each new symbol.
 
 \textbf{An alternate representation is known from Chapter 2 -- the phasors.} Each symbol of the $K_m$ states is assigned a phasor.
+\begin{equation}
+	\underline{X}_{ASK}(t) = \hat{A}_L + \frac{\hat{A}_H - \hat{A}_L}{K_m - 1} x_{sym}(t)
+\end{equation}
 
 \begin{table}[H]
 	\centering
-	\caption{Example phasor representation of the symbols}
+	\caption{Example phasor representation of the symbols with $K_m = 4$ discrete states (with $\hat{A}_L = 0.25$ and $\hat{A}_H = 1$)}
 	\begin{tabular}{|l|l|}
 		\hline
 		Symbol & Phasor \\
@@ -1963,28 +1987,782 @@ At each transition to a new symbol, the phasor is switched. The new phasor alter
 \subsection{Phase-Shift Keying}
 
 The application of the \ac{PM} in digital communication system is the \index{phase-shift keying} \acf{PSK}.
+\begin{itemize}
+	\item The phase of the carrier is altered to modulate the data.
+	\item Non-coherent demodulation will not work. Generally, a coherent demodulator is required.
+	\item Advantage: The modulated carrier is more immune to disturbances causing fluctuations of the amplitude.
+	\item Drawback: The hardware is more complex than that for \ac{ASK}.
+\end{itemize}
+
+\begin{fact}
+	\ac{PSK} must be demodulated coherently.
+\end{fact}
+
+The general form of a \ac{PSK} is:
+\begin{equation}
+	x_{PSK}(t) = \sqrt{\frac{2 E_{sym}}{T_{sym}}} \cos\left(\omega_{RF} t + \phi_{sym}(t)\right)
+\end{equation}
+where
+\begin{itemize}
+	\item $E_{sym}$ is the energy per symbol,
+	\item $T_{sym}$ is the symbol period, and
+	\item $\phi_{sym}(t)$ represents the symbol converted to a phase-shift.
+\end{itemize}
+
+Each of the $K_m$ symbol represents a phase-shift:
+\begin{equation}
+	\phi_{sym}(t) = \frac{2\pi}{K_m} x_{sym}(t)
+\end{equation}
+The discrete phase-shifts are equally distributed.
+
+The phase-shifts can be represented by a phasor:
+\begin{equation}
+	\underline{X}_{PSK}(t) = \sqrt{\frac{2 E_{sym}}{T_{sym}}} e^{j \phi_{sym}(t)} = \sqrt{\frac{2 E_{sym}}{T_{sym}}} e^{j \frac{2\pi}{K_m} x_{sym}(t)}
+\end{equation}
+
+\begin{figure}[H]
+	\centering
+	
+	\subfloat[\acs{PSK} with $K_m = 2$ in the time-domain]{
+		\centering
+		\begin{tikzpicture}
+			\begin{axis}[
+				height={0.15\textheight},
+				width=0.35\linewidth,
+				scale only axis,
+				xlabel={$t$},
+				ylabel={$x_{PSK}(t)$},
+				%grid style={line width=.6pt, color=lightgray},
+				%grid=both,
+				grid=none,
+				legend pos=outer 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=-0.5,
+				xmax=9.5,
+				ymin=-1.7,
+				ymax=1.7,
+				%xtick={0,0.125,...,1},
+				%xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$},
+				%ytick={0},
+			]
+				\addplot[red, smooth, domain=0:2, samples=50] plot(\x, {cos(deg(2*pi*1*\x)+180)});
+				\addplot[red, smooth, domain=2:4, samples=50] plot(\x, {cos(deg(2*pi*1*\x))});
+				\addplot[red, smooth, domain=4:6, samples=50] plot(\x, {cos(deg(2*pi*1*\x)+180)});
+				\addplot[red, smooth, domain=6:8, samples=50] plot(\x, {cos(deg(2*pi*1*\x))});
+				\addplot[red, smooth, domain=8:9, samples=50] plot(\x, {cos(deg(2*pi*1*\x))});
+				
+				\draw[dashed] (axis cs:2,-1.6) -- (axis cs:2,1.2);
+				\draw[dashed] (axis cs:4,-1.6) -- (axis cs:4,1.2);
+				\draw[dashed] (axis cs:6,-1.6) -- (axis cs:6,1.2);
+				\draw[dashed] (axis cs:8,-1.6) -- (axis cs:8,1.2);
+				\draw[latex-latex] (axis cs:4,1.1) -- node[midway,above,align=center]{Symbol period $T_{sym}$} (axis cs:6,1.1);
+				
+				\draw (1,-1.5) node{1};
+				\draw (3,-1.5) node{0};
+				\draw (5,-1.5) node{1};
+				\draw (7,-1.5) node{0};
+			\end{axis}
+		\end{tikzpicture}
+	}
+	\hfill
+	\subfloat[Constellation diagram of the \acs{PSK} with $K_m = 2$]{
+		\centering
+		\begin{tikzpicture}
+			\draw[->] (-2.2,0) -- (2.2,0) node[below right, align=left]{$\Re\left\{\underline{X}_{PSK}(t)\right\}$};
+			\draw[->] (0,-2.2) -- (0,2.2) node[left, align=right]{$\Im\left\{\underline{X}_{PSK}(t)\right\}$};
+			
+			\draw[black,thick,fill=gray!60] (0:1) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{0};
+			\draw[black,thick,fill=gray!60] (180:1) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{1};
+		\end{tikzpicture}
+	}
+
+	\caption{\acs{PSK} with $K_m = 2$ discrete states (also known as \ac{BPSK}) capable of encoding 1 bit}
+\end{figure}
+
+\begin{figure}[H]
+	\centering
+	
+	\subfloat[\acs{PSK} with $K_m = 4$ in the time-domain]{
+		\centering
+		\begin{tikzpicture}
+			\begin{axis}[
+				height={0.15\textheight},
+				width=0.35\linewidth,
+				scale only axis,
+				xlabel={$t$},
+				ylabel={$x_{PSK}(t)$},
+				%grid style={line width=.6pt, color=lightgray},
+				%grid=both,
+				grid=none,
+				legend pos=outer 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=-0.5,
+				xmax=9.5,
+				ymin=-1.7,
+				ymax=1.7,
+				%xtick={0,0.125,...,1},
+				%xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$},
+				%ytick={0},
+			]
+				\addplot[red, smooth, domain=0:2, samples=50] plot(\x, {cos(deg(2*pi*1*\x))});
+				\addplot[red, smooth, domain=2:4, samples=50] plot(\x, {cos(deg(2*pi*1*\x)+270)});
+				\addplot[red, smooth, domain=4:6, samples=50] plot(\x, {cos(deg(2*pi*1*\x)+90)});
+				\addplot[red, smooth, domain=6:8, samples=50] plot(\x, {cos(deg(2*pi*1*\x)+180)});
+				\addplot[red, smooth, domain=8:9, samples=50] plot(\x, {cos(deg(2*pi*1*\x))});
+				
+				\draw[dashed] (axis cs:2,-1.6) -- (axis cs:2,1.2);
+				\draw[dashed] (axis cs:4,-1.6) -- (axis cs:4,1.2);
+				\draw[dashed] (axis cs:6,-1.6) -- (axis cs:6,1.2);
+				\draw[dashed] (axis cs:8,-1.6) -- (axis cs:8,1.2);
+				\draw[latex-latex] (axis cs:4,1.1) -- node[midway,above,align=center]{Symbol period $T_{sym}$} (axis cs:6,1.1);
+				
+				\draw (1,-1.5) node{0};
+				\draw (3,-1.5) node{3};
+				\draw (5,-1.5) node{1};
+				\draw (7,-1.5) node{2};
+			\end{axis}
+		\end{tikzpicture}
+	}
+	\hfill
+	\subfloat[Constellation diagram of the \acs{PSK} with $K_m = 4$]{
+		\centering
+		\begin{tikzpicture}
+			\draw[->] (-2.2,0) -- (2.2,0) node[below right, align=left]{$\Re\left\{\underline{X}_{PSK}(t)\right\}$};
+			\draw[->] (0,-2.2) -- (0,2.2) node[left, align=right]{$\Im\left\{\underline{X}_{PSK}(t)\right\}$};
+			
+			\draw[black,thick,fill=gray!60] (0:1) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{0};
+			\draw[black,thick,fill=gray!60] (90:1) ++(-0.2,0) arc(-180:180:0.2) node[left,align=right]{1};
+			\draw[black,thick,fill=gray!60] (180:1) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{2};
+			\draw[black,thick,fill=gray!60] (270:1) ++(-0.2,0) arc(-180:180:0.2) node[left,align=right]{3};
+		\end{tikzpicture}
+	}
+	
+	\caption{\acs{PSK} with $K_m = 4$ discrete states (also known as \ac{QPSK}) capable of encoding 2 bit}
+\end{figure}
+
+The two examples above depicted the phasor constellations in the complex plane besides the time-domain function of the \ac{PSK} signal.
 
-\todo{Constellation Diagrams}
+\begin{definition}{Constellation diagram}
+	The \index{symbol constellation} \textbf{symbol constellation} describes all possible phasor values in the complex plane with a symbol associated to it.
+	
+	The \index{constellation diagram} \textbf{constellation diagram} depicts the symbol constellation graphically.
+\end{definition}
 
 \subsection{Quadrature Amplitude Modulation}
 
-\todo{What is a symbol?}
+Now, \ac{ASK} and \ac{PSK} techniques are combined.
+\begin{itemize}
+	\item Both amplitude and phase of the carrier are altered.
+	\item Using these two dimensions enlarges the set of symbols. Consequently, more data can be transmitted in one symbol.
+	\item The modulation is called \index{quadrature amplitude modulation} \textbf{\acf{QAM}}.
+	\item The abbreviation \acs{QAM} is often prefixed like \emph{$K_m$-\acs{QAM}}, where $K_m$ is the number of symbols.
+	\begin{itemize}
+		\item $2$-\acs{QAM} is equivalent to the \ac{BPSK}. The amplitude is constant.
+		\item $4$-\acs{QAM} is equivalent to the \ac{QPSK}. The amplitude is constant.
+		\item $16$-\acs{QAM} encodes 4 bits in 16 symbols. Both amplitude and phase are used.
+		\item $64$-\acs{QAM} encodes 6 bits in 64 symbols. Both amplitude and phase are used.
+		\item $256$-\acs{QAM} encodes 8 bits (1 byte) in 256 symbols. Both amplitude and phase are used.
+		\item Any other value of can be selected. However, practical implementations only support powers of 2 because binary data (bits) is encoded.
+	\end{itemize}
+	\item The symbols are distributed with equal spacing to the neighbouring symbols in the complex plane.
+\end{itemize}
+
+\begin{fact}
+	Because the phase is affected, coherent demodulation is required.
+\end{fact}
+
+\begin{figure}[H]
+	\centering
+	\begin{adjustbox}{scale=1}
+		\begin{tikzpicture}[scale=1]
+			\draw[-latex] (-2.9,0) -- (2.9,0) node[below right, align=left]{$\Re\left\{\underline{X}_{QAM}(t)\right\}$};
+			\draw[-latex] (0,-2.9) -- (0,2.9) node[left, align=right]{$\Im\left\{\underline{X}_{QAM}(t)\right\}$};
+			
+			\draw[black,thick,fill=gray!60] (2.25,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{0};
+			\draw[black,thick,fill=gray!60] (2.25,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{1};
+			\draw[black,thick,fill=gray!60] (0.75,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{2};
+			\draw[black,thick,fill=gray!60] (0.75,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{3};
+			
+			\draw[black,thick,fill=gray!60] (2.25,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{4};
+			\draw[black,thick,fill=gray!60] (2.25,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{5};
+			\draw[black,thick,fill=gray!60] (0.75,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{6};
+			\draw[black,thick,fill=gray!60] (0.75,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{7};
+			
+			\draw[black,thick,fill=gray!60] (-2.25,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{8};
+			\draw[black,thick,fill=gray!60] (-2.25,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{9};
+			\draw[black,thick,fill=gray!60] (-0.75,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{10};
+			\draw[black,thick,fill=gray!60] (-0.75,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{11};
+			
+			\draw[black,thick,fill=gray!60] (-2.25,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{12};
+			\draw[black,thick,fill=gray!60] (-2.25,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{13};
+			\draw[black,thick,fill=gray!60] (-0.75,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{14};
+			\draw[black,thick,fill=gray!60] (-0.75,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{15};
+		\end{tikzpicture}
+	\end{adjustbox}
+	\caption{Example constellation diagram of a $16$-\acs{QAM}}
+\end{figure}
 
-\todo{Data to symbol mapping}
+\begin{excursus}{Modulation of \ac{QAM} signals}
+	The IQ modulator (see Figure \ref{fig:ch05:iq_up_circuit}) can be used to modulate the \ac{QAM} symbols.
+	\begin{itemize}
+		\item The \underline{real part of the symbol} phasor $\Re\left\{\underline{X}_{PSK}(t)\right\}$ is converted to an analogue signal by a \ac{DAC} and \underline{directly used as the \ac{I} baseband signal}.
+		\item The \underline{imaginary part of the symbol} phasor $\Im\left\{\underline{X}_{PSK}(t)\right\}$ is converted to an analogue signal by a \ac{DAC} and \underline{directly used as the \ac{Q} baseband signal}.
+	\end{itemize}
 
-\todo{signal chain: S/P -> constellation diagram -> iFFT -> IQ}
+	\vspace{0.5em}
+	
+	The IQ demodulation works vice versa.
+\end{excursus}
 
-\todo{IQ Imbalance}
+The number $K_m$ of the $K_m$-\acs{QAM} selects the number of possible symbols in the constellation diagram.
+\begin{itemize}
+	\item A higher value of $K_m$ enlarges the set of symbols.
+	\begin{itemize}
+		\item More data can be encoded in one symbol.
+		\item The data rate increases while the symbol period is kept constant.
+		\item The bandwidth of the transmitted signal remains constant while the data rate increases.
+	\end{itemize}
+	\item As a drawback, higher values of $K_m$ reduce the noise immunity. Signals need a higher \ac{SNR} to keep the data error after the demodulation low.
+	\item Lower values of $K_m$
+	\begin{itemize}
+		\item are more immune to noise,
+		\item are capable of dealing with a poor \ac{SNR} better,
+		\item but can encode less data in one symbol, and
+		\item therefore have a lower data rate.
+	\end{itemize}
+\end{itemize}
 
-%\subsection{Coherent and Non-Coherent Demodulation}
+\begin{excursus}{Signal bandwidth}
+	The bandwidth is matters!
+	\begin{itemize}
+		\item The electromagnetic spectrum is shared with many other applications and services.
+		\item \textbf{The electromagnetic spectrum is a sparse resource.}
+		\item Its important to use it efficiently. Therefore, each symbol should encode as much data as possible to obtain a high data rate at a moderately narrow bandwidth.
+		\item A trade-off must be made between
+		\begin{itemize}
+			\item a reasonable high value of $K_m$ increasing the data rate and
+			\item a reasonable low value of $K_m$ increasing the immunity to noise.
+		\end{itemize}
+		\item Modern digital communication system are capable of adapting the value of $K_m$ to the current propagation conditions to archive optimal results.
+	\end{itemize}
+\end{excursus}
 
 \subsection{Inter-Symbol Interference}
 
-\todo{Cyclic Prefixes? No -> OFDM}
+\begin{figure}[H]
+	\centering
+	\begin{tikzpicture}
+		\begin{axis}[
+			height={0.15\textheight},
+			width=0.7\linewidth,
+			scale only axis,
+			xlabel={$t$},
+			ylabel={$x_{sym}(t)$},
+			%grid style={line width=.6pt, color=lightgray},
+			%grid=both,
+			grid=none,
+			legend pos=outer 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=-0.5,
+			xmax=8.5,
+			ymin=0,
+			ymax=1.7,
+			%xtick={0,0.125,...,1},
+			%xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$},
+			%ytick={0},
+		]
+			\draw[blue] (axis cs:0,0) -- (axis cs:0,1) -- (axis cs:2,1) -- (axis cs:2,0);
+			\draw[red] (axis cs:2,0) -- (axis cs:2,0.55) -- (axis cs:4,0.55) -- (axis cs:4,0);
+			\draw[green] (axis cs:4,0) -- (axis cs:4,0.85) -- (axis cs:6,0.85) -- (axis cs:6,0);
+			\draw[olive] (axis cs:6,0) -- (axis cs:6,0.7) -- (axis cs:8,0.7) -- (axis cs:8,0);
+			
+			\draw[dashed] (axis cs:2,0) -- (axis cs:2,1.2);
+			\draw[dashed] (axis cs:4,0) -- (axis cs:4,1.2);
+			\draw[latex-latex] (axis cs:2,1.1) -- node[midway,above,align=center]{Symbol period $T_{sym}$} (axis cs:4,1.1);
+		\end{axis}
+	\end{tikzpicture}
+	\caption{Series of four ideal symbols}
+\end{figure}
+
+\begin{itemize}
+	\item \eqref{eq:ch05:sym_rect} defined the symbols as ideal rectangle shapes.
+	\item Real implementations this ideal shape does not exist.
+	\begin{itemize}
+		\item The Fourier transform of the rectangle function is a sinc-function, which indefinitely expands in the frequency domain.
+		\item Real system limit the bandwidth. Remember that \acp{LPF} should be applied after mixers and before \ac{ADC}.
+	\end{itemize}
+	\item Due to the bandwidth-limitation, the rectangle shape is flattened. The ideally steep edges become flattened.
+\end{itemize}
+
+\begin{figure}[H]
+	\centering
+	\begin{tikzpicture}
+		\begin{axis}[
+			height={0.15\textheight},
+			width=0.7\linewidth,
+			scale only axis,
+			xlabel={$t$},
+			ylabel={$x_{sym}(t)$},
+			%grid style={line width=.6pt, color=lightgray},
+			%grid=both,
+			grid=none,
+			legend pos=outer 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=-0.5,
+			xmax=8.5,
+			ymin=0,
+			ymax=1.7,
+			%xtick={0,0.125,...,1},
+			%xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$},
+			%ytick={0},
+		]
+			\draw[blue,smooth] (axis cs:-0.25,0) -- (axis cs:0.25,1) -- (axis cs:1.75,1) -- (axis cs:2.25,0);
+			\draw[red,smooth] (axis cs:1.75,0) -- (axis cs:2.25,0.55) -- (axis cs:3.75,0.55) -- (axis cs:4.25,0);
+			\draw[green,smooth] (axis cs:3.75,0) -- (axis cs:4.25,0.85) -- (axis cs:5.75,0.85) -- (axis cs:6.25,0);
+			\draw[olive,smooth] (axis cs:5.75,0) -- (axis cs:6.25,0.7) -- (axis cs:7.75,0.7) -- (axis cs:8.25,0);
+			
+			\draw[dashed] (axis cs:2,0) -- (axis cs:2,1.2);
+			\draw[dashed] (axis cs:4,0) -- (axis cs:4,1.2);
+			\draw[latex-latex] (axis cs:2,1.1) -- node[midway,above,align=center]{Symbol period $T_{sym}$} (axis cs:4,1.1);
+		\end{axis}
+	\end{tikzpicture}
+	\caption[Series of four non-ideal symbols showing overlap]{Series of four non-ideal symbols where the edges are flattened. The symbols overlap due to \acs{ISI}.}
+\end{figure}
+
+\begin{itemize}
+	\item Due to the flattened shape, the symbols overlap.
+	\item This effect is the \index{inter-symbol interference} \textbf{\acf{ISI}}.
+	\item To mitigate the \ac{ISI} a \index{guard interval} \textbf{guard interval} should be inserted between the symbols.
+	\begin{itemize}
+		\item The guard interval adds a spacing between the symbol pulses.
+		\item The guard interval reduces the \ac{ISI}.
+		\item The symbol period is prolonged. It is the sum of the guard interval and the symbol width (which used to equal the symbol period for ideal rectangle shapes)-
+	\end{itemize}
+	\item Reducing the symbol width by maintaining the symbol period $T_{sym}$ is not feasible, because the bandwidth of the signal would be increased.
+\end{itemize}
+
+\begin{figure}[H]
+	\centering
+	\begin{tikzpicture}
+		\begin{axis}[
+			height={0.15\textheight},
+			width=0.7\linewidth,
+			scale only axis,
+			xlabel={$t$},
+			ylabel={$x_{sym}(t)$},
+			%grid style={line width=.6pt, color=lightgray},
+			%grid=both,
+			grid=none,
+			legend pos=outer 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=-0.5,
+			xmax=10,
+			ymin=0,
+			ymax=1.9,
+			%xtick={0,0.125,...,1},
+			%xticklabels={$- \omega_S$, $- \frac{\omega_S}{2}$, $0$, $\frac{\omega_S}{2}$, $\omega_S$},
+			%ytick={0},
+		]
+			\draw[blue,smooth] (axis cs:-0.25,0) -- (axis cs:0.25,1) -- (axis cs:1.75,1) -- (axis cs:2.25,0);
+			\draw[red,smooth] (axis cs:2.25,0) -- (axis cs:2.75,0.55) -- (axis cs:4.25,0.55) -- (axis cs:4.75,0);
+			\draw[green,smooth] (axis cs:4.75,0) -- (axis cs:5.25,0.85) -- (axis cs:6.75,0.85) -- (axis cs:7.25,0);
+			\draw[olive,smooth] (axis cs:7.25,0) -- (axis cs:7.75,0.7) -- (axis cs:9.25,0.7) -- (axis cs:9.75,0);
+			
+			\draw[dashed] (axis cs:2.5,0) -- (axis cs:2.5,1.6);
+			\draw[dashed] (axis cs:4.5,0) -- (axis cs:4.5,1.4);
+			\draw[dashed] (axis cs:5,0) -- (axis cs:5,1.6);
+			\draw[latex-latex] (axis cs:2.5,1.5) -- node[midway,above,align=center]{Prolonged symbol period $T_{sym}$} (axis cs:5,1.5);
+			\draw[latex-latex] (axis cs:2.5,1.3) node[left,align=right]{Symbol width} -- (axis cs:4.5,1.3);
+			\draw[latex-latex] (axis cs:4.5,1.3) -- (axis cs:5,1.3) node[right,align=left]{Guard interval};
+		\end{axis}
+	\end{tikzpicture}
+	\caption[Series of four non-ideal symbols with additional spacing]{Series of four non-ideal symbols where the edges are flattened. An additional spacing between the symbols (guard interval) is added to mitigate \acs{ISI}. The symbol period is prolonged by the guard interval.}
+\end{figure}
+
+\subsection{IQ Imbalance}
+
+So far, we assumed ideal conditions for the IQ modulation and IQ demodulation (see Figure \ref{fig:ch05:iq_up_circuit} and \ref{fig:ch05:iq_down_circuit}).
+\begin{itemize}
+	\item The phase-shift of the \ac{LO} signal was perfectly \SI{90}{\degree}.
+	\item The phasor of the \ac{LO} signal fed into the \ac{I}-mixer was ideally parallel to the real axis in the complex plane ($\arg\left(\underline{X}_{LO,I}\right) = 0$).
+	\item The phasor of the phase-shifted \ac{LO} signal fed into the \ac{Q}-mixer was ideally parallel to the imaginary axis in the complex plane ($\arg\left(\underline{X}_{LO,Q}\right) = \frac{\pi}{2}$).
+	\item Both phasors were perfectly orthogonal.
+\end{itemize}
+
+\begin{figure}[H]
+	\centering
+	
+	\subfloat[Example constellation diagram of a $16$-\acs{QAM} under ideal conditions]{
+		\centering
+		\begin{adjustbox}{scale=0.9}
+			\begin{tikzpicture}[scale=1]
+				\draw[-latex] (-2.9,0) -- (2.9,0) node[below right, align=left]{$\Re\left\{\underline{X}_{QAM}(t)\right\}$};
+				\draw[-latex] (0,-2.9) -- (0,2.9) node[left, align=right]{$\Im\left\{\underline{X}_{QAM}(t)\right\}$};
+				
+				\pgftransformcm{1}{0}{0}{1}{\pgfpoint{0cm}{0cm}}
+				
+				\foreach \x in {-2.25,-0.75,0.75,2.25}{
+					\draw[dashed] (\x,-2.25) -- (\x,2.25);
+					\draw[dashed] (-2.25,\x) -- (2.25,\x);
+				}
+				
+				
+				\draw[black,thick,fill=gray!60] (2.25,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{0};
+				\draw[black,thick,fill=gray!60] (2.25,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{1};
+				\draw[black,thick,fill=gray!60] (0.75,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{2};
+				\draw[black,thick,fill=gray!60] (0.75,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{3};
+				
+				\draw[black,thick,fill=gray!60] (2.25,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{4};
+				\draw[black,thick,fill=gray!60] (2.25,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{5};
+				\draw[black,thick,fill=gray!60] (0.75,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{6};
+				\draw[black,thick,fill=gray!60] (0.75,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{7};
+				
+				\draw[black,thick,fill=gray!60] (-2.25,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{8};
+				\draw[black,thick,fill=gray!60] (-2.25,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{9};
+				\draw[black,thick,fill=gray!60] (-0.75,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{10};
+				\draw[black,thick,fill=gray!60] (-0.75,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{11};
+				
+				\draw[black,thick,fill=gray!60] (-2.25,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{12};
+				\draw[black,thick,fill=gray!60] (-2.25,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{13};
+				\draw[black,thick,fill=gray!60] (-0.75,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{14};
+				\draw[black,thick,fill=gray!60] (-0.75,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{15};
+				
+				\draw[-latex,red,thick] (0,0) -- (2.6,0) node[above right, align=left]{$\underline{X}_{LO,I}$};
+				\draw[-latex,red,thick] (0,0) -- (0,2.6) node[above right, align=left]{$\underline{X}_{LO,Q}$};
+				\draw[red] (0:0.4) arc(0:90:0.4) node[midway,right,align=left]{$\Delta \varphi_{IQ}$};
+			\end{tikzpicture}
+		\end{adjustbox}
+	}
+	\hfill
+	\subfloat[Example constellation diagram of a $16$-\acs{QAM} subject to IQ imbalance]{
+		\centering
+		\begin{adjustbox}{scale=0.9}
+			\begin{tikzpicture}[scale=1]
+				\draw[-latex] (-2.9,0) -- (2.9,0) node[below right, align=left]{$\Re\left\{\tilde{\underline{X}}_{QAM}(t)\right\}$};
+				\draw[-latex] (0,-2.9) -- (0,2.9) node[left, align=right]{$\Im\left\{\tilde{\underline{X}}_{QAM}(t)\right\}$};
+				
+				\pgftransformcm{1}{0.1}{0.1}{1}{\pgfpoint{0cm}{0cm}}
+				
+				\foreach \x in {-2.25,-0.75,0.75,2.25}{
+					\draw[dashed] (\x,-2.25) -- (\x,2.25);
+					\draw[dashed] (-2.25,\x) -- (2.25,\x);
+				}
+				
+				
+				\draw[black,thick,fill=gray!60] (2.25,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{0};
+				\draw[black,thick,fill=gray!60] (2.25,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{1};
+				\draw[black,thick,fill=gray!60] (0.75,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{2};
+				\draw[black,thick,fill=gray!60] (0.75,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{3};
+				
+				\draw[black,thick,fill=gray!60] (2.25,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{4};
+				\draw[black,thick,fill=gray!60] (2.25,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{5};
+				\draw[black,thick,fill=gray!60] (0.75,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{6};
+				\draw[black,thick,fill=gray!60] (0.75,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{7};
+				
+				\draw[black,thick,fill=gray!60] (-2.25,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{8};
+				\draw[black,thick,fill=gray!60] (-2.25,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{9};
+				\draw[black,thick,fill=gray!60] (-0.75,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{10};
+				\draw[black,thick,fill=gray!60] (-0.75,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{11};
+				
+				\draw[black,thick,fill=gray!60] (-2.25,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{12};
+				\draw[black,thick,fill=gray!60] (-2.25,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{13};
+				\draw[black,thick,fill=gray!60] (-0.75,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{14};
+				\draw[black,thick,fill=gray!60] (-0.75,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{15};
+				
+				\draw[-latex,red,thick] (0,0) -- (2.6,0) node[above right, align=left]{$\underline{X}_{LO,I}$};
+				\draw[-latex,red,thick] (0,0) -- (0,2.6) node[above right, align=left]{$\underline{X}_{LO,Q}$};
+				\draw[red] (0:0.4) arc(0:90:0.4) node[midway,right,align=left]{$\Delta \varphi_{IQ}$};
+			\end{tikzpicture}
+		\end{adjustbox}
+	}
+
+	\caption{Example constellation diagram of a $16$-\acs{QAM} under ideal conditions and while being subject to IQ imbalance}
+\end{figure}
+
+Under real conditions, the phase-shift of the \ac{LO} signal is not precisely \SI{90}{\degree}.
+\begin{itemize}
+	\item The phase-shift between the \ac{I} and \ac{Q} component of the \ac{LO} signal $\Delta \varphi_{IQ}$ differs from \SI{90}{\degree}.
+	\item The difference $\SI{90}{\degree} - \Delta \varphi_{IQ}$ is the \index{IQ imbalance} \textbf{IQ imbalance}.
+	\item The IQ imbalance will disturb the data detection, which reconstructs the symbols from the IQ demodulated \ac{I} and \ac{Q} baseband signals.
+	\item Data error might a corollary of the IQ imbalance.
+\end{itemize}
+
+Countermeasures:
+\begin{itemize}
+	\item The IQ imbalance is a linear coordinate transformation.
+	\begin{equation}
+		\underbrace{\left[\begin{matrix}\Re\left\{\tilde{\underline{X}}_{QAM}(t)\right\}\\ \Im\left\{\tilde{\underline{X}}_{QAM}(t)\right\}\end{matrix}\right]}_{\text{IQ imbalance applied}} = \mat{T}_{IQ} \cdot \underbrace{\left[\begin{matrix}\Re\left\{\underline{X}_{QAM}(t)\right\}\\ \Im\left\{\underline{X}_{QAM}(t)\right\}\end{matrix}\right]}_{\text{Ideal constellation}}
+	\end{equation}
+	where $\mat{T}_{IQ}$ is a $2 \times 2$ transformation matrix.
+	\item The IQ imbalance must be estimated to obtain knowledge about $\mat{T}_{IQ}$.
+	\item The IQ imbalance can then be compensated by applying the inverse transformation matrix $\mat{T}_{IQ}^{-1}$.
+\end{itemize}
 
 \subsection{Synchronization 2: Carrier Recovery}
 
-\todo{Frequency and phase offset}
+\subsubsection{Phase Offset}
+
+Another problem is that the phase of the \ac{RF} signal is not known.
+\begin{itemize}
+	\item The phases of the transmitter \ac{LO} and the receiver \ac{LO} are not synchronized.
+	\item They are phase-shifted by a \emph{phase offset} of $\Delta \varphi_{C}$ (``C'' stands for carrier).
+	\item A corollary is that the \ac{RF} carrier, which is synchronous to the transmitter \ac{LO}, is phase-shifted by $\Delta \varphi_{C}$ in relation to the receiver \ac{LO}.
+	\item The whole constellation diagram is rotated by $\Delta \varphi_{C}$.
+\end{itemize}
+
+\textit{The IQ imbalance is neglected for the following considerations.}
+
+\begin{figure}[H]
+	\centering
+	\begin{adjustbox}{scale=1}
+		\begin{tikzpicture}[scale=1]
+			\draw[-latex] (-2.9,0) -- (2.9,0) node[below right, align=left]{$\Re\left\{\underline{X}_{QAM}(t)\right\}$};
+			\draw[-latex] (0,-2.9) -- (0,2.9) node[left, align=right]{$\Im\left\{\underline{X}_{QAM}(t)\right\}$};
+			
+			\begin{scope}[rotate=30]
+				\foreach \x in {-2.25,-0.75,0.75,2.25}{
+					\draw[dashed] (\x,-2.25) -- (\x,2.25);
+					\draw[dashed] (-2.25,\x) -- (2.25,\x);
+				}
+		
+				\draw[black,thick,fill=gray!60] (2.25,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{0};
+				\draw[black,thick,fill=gray!60] (2.25,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{1};
+				\draw[black,thick,fill=gray!60] (0.75,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{2};
+				\draw[black,thick,fill=gray!60] (0.75,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{3};
+				
+				\draw[black,thick,fill=gray!60] (2.25,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{4};
+				\draw[black,thick,fill=gray!60] (2.25,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{5};
+				\draw[black,thick,fill=gray!60] (0.75,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{6};
+				\draw[black,thick,fill=gray!60] (0.75,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{7};
+				
+				\draw[black,thick,fill=gray!60] (-2.25,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{8};
+				\draw[black,thick,fill=gray!60] (-2.25,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{9};
+				\draw[black,thick,fill=gray!60] (-0.75,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{10};
+				\draw[black,thick,fill=gray!60] (-0.75,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{11};
+				
+				\draw[black,thick,fill=gray!60] (-2.25,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{12};
+				\draw[black,thick,fill=gray!60] (-2.25,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{13};
+				\draw[black,thick,fill=gray!60] (-0.75,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{14};
+				\draw[black,thick,fill=gray!60] (-0.75,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{15};
+				
+				\draw[-latex,red,thick] (0,0) -- (2.6,0) node[above right, align=left]{$\underline{X}_{LO,Tx,I}$};
+				\draw[-latex,red,thick] (0,0) -- (0,2.6) node[above right, align=left]{$\underline{X}_{LO,Tx,Q}$};
+			\end{scope}
+			
+			\draw[-latex,green,thick] (0,0) -- (2.6,0) node[above right, align=left]{$\underline{X}_{LO,Rx,I}$};
+			\draw[-latex,green,thick] (0,0) -- (0,2.6) node[above right, align=left]{$\underline{X}_{LO,Rx,Q}$};
+			
+			\draw[blue] (0:0.4) arc(0:30:0.4) node[midway,right,align=left]{$\Delta \varphi_{C}$};
+		\end{tikzpicture}
+	\end{adjustbox}
+	\caption{Example constellation diagram of a $16$-\acs{QAM} rotated by a phase offset of $\Delta \varphi_{C} = \SI{30}{\degree}$}
+\end{figure}
+
+\textbf{The data detection will fail, because the detector expects a perfectly adjusted symbol constellation.}
+
+\subsubsection{Frequency Offset}
+
+In addition to the phase offset, the frequency of the transmitter \ac{LO} and receiver \ac{LO} might differ.
+\begin{itemize}
+	\item The difference is the \emph{frequency offset} $\Delta \omega_{C}$.
+	\item The \ac{RF} carrier is synchronized to the transmitter \ac{LO}. Its frequency differs by $\Delta \omega_{C}$ from the receiver \ac{LO}.
+	\item A corollary is that the phase offset integrates over time.
+	\begin{equation}
+		\Delta \varphi_{C}(t) = \underbrace{\Delta \varphi_{C,0}}_{\text{Initial offset}} + \int\limits_{t_0}^{t} \omega_{C} \, \mathrm{d} t
+	\end{equation}
+	\item The constellation diagram rotates at $\omega_{C}$ and constantly changes its angle.
+\end{itemize}
+
+\begin{figure}[H]
+	\centering
+	\begin{adjustbox}{scale=1}
+		\begin{tikzpicture}[scale=1]
+			\draw[-latex] (-2.9,0) -- (2.9,0) node[below right, align=left]{$\Re\left\{\underline{X}_{QAM}(t)\right\}$};
+			\draw[-latex] (0,-2.9) -- (0,2.9) node[left, align=right]{$\Im\left\{\underline{X}_{QAM}(t)\right\}$};
+			
+			\begin{scope}[rotate=30]
+				\foreach \x in {-2.25,-0.75,0.75,2.25}{
+					\draw[dashed] (\x,-2.25) -- (\x,2.25);
+					\draw[dashed] (-2.25,\x) -- (2.25,\x);
+				}
+				
+				\draw[black,thick,fill=gray!60] (2.25,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{0};
+				\draw[black,thick,fill=gray!60] (2.25,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{1};
+				\draw[black,thick,fill=gray!60] (0.75,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{2};
+				\draw[black,thick,fill=gray!60] (0.75,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{3};
+				
+				\draw[black,thick,fill=gray!60] (2.25,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{4};
+				\draw[black,thick,fill=gray!60] (2.25,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{5};
+				\draw[black,thick,fill=gray!60] (0.75,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{6};
+				\draw[black,thick,fill=gray!60] (0.75,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{7};
+				
+				\draw[black,thick,fill=gray!60] (-2.25,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{8};
+				\draw[black,thick,fill=gray!60] (-2.25,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{9};
+				\draw[black,thick,fill=gray!60] (-0.75,2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{10};
+				\draw[black,thick,fill=gray!60] (-0.75,0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{11};
+				
+				\draw[black,thick,fill=gray!60] (-2.25,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{12};
+				\draw[black,thick,fill=gray!60] (-2.25,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{13};
+				\draw[black,thick,fill=gray!60] (-0.75,-2.25) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{14};
+				\draw[black,thick,fill=gray!60] (-0.75,-0.75) ++(0,-0.2) arc(-90:270:0.2) node[below,align=center]{15};
+				
+				\draw[-latex,red,thick] (0,0) -- (2.6,0) node[above right, align=left]{$\underline{X}_{LO,Tx,I}$};
+				\draw[-latex,red,thick] (0,0) -- (0,2.6) node[above right, align=left]{$\underline{X}_{LO,Tx,Q}$};
+				
+				\draw[-latex,red,thick] (70:1.2) arc(70:110:1.2) node[left, align=right]{$\omega_{C}$};
+			\end{scope}
+			
+			\draw[-latex,green,thick] (0,0) -- (2.6,0) node[above right, align=left]{$\underline{X}_{LO,Rx,I}$};
+			\draw[-latex,green,thick] (0,0) -- (0,2.6) node[above right, align=left]{$\underline{X}_{LO,Rx,Q}$};
+			
+			\draw[blue] (0:0.4) arc(0:30:0.4) node[midway,right,align=left]{$\Delta \varphi_{C}$};
+		\end{tikzpicture}
+	\end{adjustbox}
+	\caption{Example constellation diagram of a $16$-\acs{QAM} rotating at a frequency offset of $\Delta \omega_{C}$}
+\end{figure}
+
+\textbf{The frequency offset and the rotation symbol constellation makes a data detection impossible.}
+
+\subsubsection{Carrier Recovery}
+
+\begin{fact}
+	The receiver \ac{LO} must be synchronized to the \ac{RF} carrier phase and frequency.
+\end{fact}
+
+The synchronization is called \index{carrier recovery} \textbf{carrier recovery} and is usually implemented as a closed control loop.
+\begin{itemize}
+	\item The control loop adjusts the \ac{LO} phase so that it is synchronous to the \ac{RF} carrier ($\Delta \omega_{C} \rightarrow 0$).
+	\item A synchronization usually requires data (pilot, preamble) to estimate the phase offset from the current symbol constellation.
+	\item The phase offset correction is usually sufficient, because the frequency offset is related to that error via the integral. Because of the frequency offset, the synchronization is only short-term stable and must be repeated regularly.
+	\item The estimated phase error is than fed via a loop filter to the \ac{LO} (hybrid approach).
+	\item An alternative approach is all-digital: The digital \ac{I} and \ac{Q} signals are digitally mixed to compensate the phase and frequency offset.
+\end{itemize}
+
+\textit{Remark:} In contrast to the similar timing recovery, which adjusts the sampling clock, the carrier recovery adjusts the \ac{LO}.
+
+\begin{figure}[H]
+	\centering
+	\begin{adjustbox}{scale=0.6}
+		\begin{circuitikz}
+			\node[draw, block] at(0,0) (Mixer){Mixer};
+			\node[oscillator, below=4cm of Mixer](LO){};
+			\node[draw, block, right=of Mixer](Sampler){Sampler};
+			\node[oscillator, below=of Sampler](Clock){};
+			\node[draw, block, right=of Clock] (Filter){Loop filter};
+			\node[draw, block, right=of Filter] (Pred){Timing error\\ estimator};
+			\node[draw, block, right=5cm of LO] (CarrierFilter){Loop filter};
+			\node[draw, block, right=of CarrierFilter] (CarrierPred){Phase error\\ estimator};
+			
+			\draw[dashed] (Sampler.north) -- ++(0, 2cm) node[below left, align=right]{Analogue\\ domain} node[below right, align=left]{Digital\\ domain};
+			\node[left=2mm of LO, align=right]{\acs{LO}};
+			\node[left=2mm of Clock, align=right]{Sampling\\ clock};
+			
+			\draw[o->] (-2.5,0) node[left,align=right]{Input} -- (Mixer.west);
+			\draw[->] (Mixer.east) -- (Sampler.west);
+			\draw[->] (Sampler.east) -- ++(11,0) node[right,align=left]{Further signal\\ processing};
+			
+			\draw[*->] ([xshift=9cm] Sampler.east) |- (Pred.east);
+			\draw[->] (Pred.west) -- (Filter.east);
+			\draw[->] (Filter.west) -- (Clock.east);
+			
+			\draw[*->] ([xshift=9.5cm] Sampler.east) |- (CarrierPred.east);
+			\draw[->] (CarrierPred.west) -- (CarrierFilter.east);
+			\draw[->] (CarrierFilter.west) -- (LO.east);
+			
+			\draw[->] (LO.north) -- (Mixer.south);
+			\draw[->] (Clock.north) -- (Sampler.south);
+		\end{circuitikz}
+	\end{adjustbox}
+	\caption{Hybrid timing recovery and carrier recovery}
+\end{figure}
+
+\begin{figure}[H]
+	\centering
+	\begin{adjustbox}{scale=0.6}
+		\begin{circuitikz}
+			\node[draw, block] at(0,0) (Mixer){Mixer};
+			\node[oscillator, below=3cm of Mixer](LO){};
+			\node[draw, block, right=of Mixer](Sampler){Sampler};
+			\node[oscillator, below=of Sampler](Clock){};
+			\node[draw, block, right=of Sampler] (DigiMix){Digital\\ mixer};
+			\node[oscillator, below=4.5cm of DigiMix](NCO){};
+			\node[draw, block, right=of DigiMix] (Resampler){Interpolation\\ and resampling};
+			\node[draw, block, below=of Resampler] (Filter){Loop filter};
+			\node[draw, block, right=of Filter] (Pred){Timing error\\ estimator};
+			\node[draw, block, right=of NCO] (CarrierFilter){Loop filter};
+			\node[draw, block, right=of CarrierFilter] (CarrierPred){Phase error\\ estimator};
+			
+			\draw[dashed] (Sampler.north) -- ++(0, 2cm) node[below left, align=right]{Analogue\\ domain} node[below right, align=left]{Digital\\ domain};
+			\node[below=2mm of LO, align=center]{Free-running\\ \acs{LO}};
+			\node[below=2mm of Clock, align=center]{Free-running\\ sampling clock};
+			\node[left=2mm of NCO, align=right]{Digital\\ oscillator};
+			
+			\draw[o->] (-2.5,0) node[left,align=right]{Input} -- (Mixer.west);
+			\draw[->] (Mixer.east) -- (Sampler.west);
+			\draw[->] (Sampler.east) -- (DigiMix.west);
+			\draw[->] (DigiMix.east) -- (Resampler.west);
+			\draw[->] (Resampler.east) -- ++(6,0) node[right,align=left]{Further signal\\ processing};
+			
+			\draw[*->] ([xshift=5cm] Resampler.east) |- (Pred.east);
+			\draw[->] (Pred.west) -- (Filter.east);
+			\draw[->] (Filter.north) -- (Resampler.south);
+			
+			\draw[*->] ([xshift=5.5cm] Resampler.east) |- (CarrierPred.east);
+			\draw[->] (CarrierPred.west) -- (CarrierFilter.east);
+			\draw[->] (CarrierFilter.west) -- (NCO.east);
+			
+			\draw[->] (LO.north) -- (Mixer.south);
+			\draw[->] (Clock.north) -- (Sampler.south);
+			\draw[->] (NCO.north) -- (DigiMix.south);
+		\end{circuitikz}
+	\end{adjustbox}
+	\caption{Digital timing recovery and carrier recovery}
+\end{figure}
 
 
 \phantomsection
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