From 57db9761a58a730e575ab49f18195ca6f5b8a359 Mon Sep 17 00:00:00 2001
From: Philipp Le <philipp-le-prviat@freenet.de>
Date: Tue, 2 Jun 2020 01:22:07 +0200
Subject: WIP Chapter 5: Ideal Mixer

---
 chapter05/content_ch05.tex | 314 ++++++++++++++++++++++++++++++++++++++++++++-
 1 file changed, 309 insertions(+), 5 deletions(-)

(limited to 'chapter05/content_ch05.tex')

diff --git a/chapter05/content_ch05.tex b/chapter05/content_ch05.tex
index 0351261..9350d57 100644
--- a/chapter05/content_ch05.tex
+++ b/chapter05/content_ch05.tex
@@ -8,7 +8,7 @@
 % Please find the full copy of the licence at:
 % https://creativecommons.org/licenses/by-sa/4.0/legalcode
 
-\chapter{Modulation}
+\chapter{Modulation and Mixing}
 
 \begin{refsection}
 	
@@ -28,6 +28,10 @@ Example: Voice transmission
 	\index{modulation} \textbf{Modulation} is the process of altering a signal -- the \index{carrier} \textbf{carrier} -- so that it contains the information to be transmitted.
 \end{definition}
 
+\begin{definition}{Demodulation}
+	\index{demodulation} \textbf{Demodulation} is the inverse process of modulation. The information is extracted from the carrier.
+\end{definition}
+
 In the previous example, the voice was the information-carrying signal. This can be transferred to any kind of information. In this chapter, we will discuss techniques to modulate data on carriers which can be transmitted over wired and wireless channels.
 
 \todo{Block diagram modulator}
@@ -563,14 +567,311 @@ Like the \ac{DSB-TC}, the transmission bandwidth of the \ac{DSB-SC} is $2 \omega
 
 \subsection{Phase Modulation}
 
+\todo{Phase Modulation}
 
 \section{Frequency mixer}
 
-\todo{Modulation vs. Mixing}
+The information-carrying signal and the carrier are usually at different frequencies. A communication system must therefore be able to shift between multiple frequencies. Such communication systems are called \index{heterodyne} \textbf{heterodyne}.
+
+\begin{itemize}
+	\item The modulation and demodulation happen at a low, near-zero frequency. The modulated signal is the \index{baseband signal} \textbf{baseband signal}.
+	\item A device called \index{mixer} \textbf{mixer} converts one frequency to another. The modulated information is kept intact.
+\end{itemize}
+
+\begin{figure}[H]
+	\centering
+	
+	\subfloat[Heterodyne transmitter without \acs{IF} stage]{
+		\centering
+		\begin{adjustbox}{scale=0.8}
+			\begin{circuitikz}
+				\node[block, draw](Baseband){Baseband\\ modulation};
+				\node[mixer, right=2.5cm of Baseband](Mixer){};
+				\node[ampshape, right=4cm of Mixer](Amplifier){};
+				
+				\draw (Mixer.south) node[below,align=center,yshift=-5mm]{Mixer};
+				\draw (Amplifier.south) node[below,align=center,yshift=-5mm]{\acs{RF}\\ amplifier};
+				
+				\draw[-latex] (Baseband.east) -- node[midway,above,align=center]{Baseband\\ signal} (Mixer.west);
+				\draw[-latex] (Mixer.east) to[bandpass] ++(2cm,0) -- node[midway,above,align=center]{\acs{RF}\\ signal} (Amplifier.west);
+				\draw (Amplifier.east) to[bandpass] ++(2cm,0) -- ++(0,0) node[txantenna]{};
+			\end{circuitikz}
+		\end{adjustbox}
+	}
+
+	\subfloat[Heterodyne receiver without \acs{IF} stage]{
+		\centering
+		\begin{adjustbox}{scale=0.8}
+			\begin{circuitikz}
+				\node[ampshape](RFAmplifier){};
+				\node[mixer, right=2cm of RFAmplifier](Mixer){};
+				\node[ampshape, right=2.5cm of Mixer](BBAmplifier){};
+				\node[block, draw, right=2.5cm of BBAmplifier](Baseband){Baseband\\ demodulation};
+				
+				\draw (RFAmplifier.south) node[below,align=center,yshift=-5mm]{\acs{RF}\\ amplifier};
+				\draw (Mixer.south) node[below,align=center,yshift=-5mm]{Mixer};
+				\draw (BBAmplifier.south) node[below,align=center,yshift=-5mm]{Baseband\\ amplifier};
+				
+				\draw (RFAmplifier.west) to[bandpass] ++(-2cm,0) node[rxantenna,xscale=-1]{};
+				
+				\draw[-latex] (RFAmplifier.east) -- node[midway,above,align=center]{\acs{RF}\\ signal} (Mixer.west);
+				\draw[-latex] (Mixer.east) to[bandpass] ++(2cm,0) -- (BBAmplifier.west);
+				\draw[-latex] (BBAmplifier.east) -- node[midway,above,align=center]{Baseband\\ signal} (Baseband.west);
+			\end{circuitikz}
+		\end{adjustbox}
+	}
+
+	\subfloat[Superheterodyne receiver]{
+		\centering
+		\begin{adjustbox}{scale=0.6}
+			\begin{circuitikz}
+				\node[ampshape](RFAmplifier){};
+				\node[mixer, right=2cm of RFAmplifier](RFMixer){};
+				\node[ampshape, right=2.5cm of RFMixer](IFAmplifier){};
+				\node[mixer, right=2cm of IFAmplifier](IFMixer){};
+				\node[ampshape, right=2.5cm of IFMixer](BBAmplifier){};
+				\node[block, draw, right=2.5cm of BBAmplifier](Baseband){Baseband\\ demodulation};
+				
+				\draw (RFAmplifier.south) node[below,align=center,yshift=-5mm]{\acs{RF}\\ amplifier};
+				\draw (RFMixer.south) node[below,align=center,yshift=-5mm]{\acs{RF}\\ Mixer};
+				\draw (IFAmplifier.south) node[below,align=center,yshift=-5mm]{\acs{IF}\\ amplifier};
+				\draw (IFMixer.south) node[below,align=center,yshift=-5mm]{\acs{IF}\\ Mixer};
+				\draw (BBAmplifier.south) node[below,align=center,yshift=-5mm]{Baseband\\ amplifier};
+				
+				\draw (RFAmplifier.west) to[bandpass] ++(-2cm,0) ++(0,0) node[rxantenna,xscale=-1]{};
+				
+				\draw[-latex] (RFAmplifier.east) -- node[midway,above,align=center]{\acs{RF}\\ signal} (RFMixer.west);
+				\draw[-latex] (RFMixer.east) to[bandpass] ++(2cm,0) -- (IFAmplifier.west);
+				\draw[-latex] (IFAmplifier.east) -- node[midway,above,align=center]{\acs{IF}\\ signal} (IFMixer.west);
+				\draw[-latex] (IFMixer.east) to[bandpass] ++(2cm,0) -- (BBAmplifier.west);
+				\draw[-latex] (BBAmplifier.east) -- node[midway,above,align=center]{Baseband\\ signal} (Baseband.west);
+			\end{circuitikz}
+		\end{adjustbox}
+	}
+
+	\caption{A selection of heterodyne architectures}
+\end{figure}
+
+Definitions:
+\begin{itemize}
+	\item The \index{radio-frequency signal} \textbf{\ac{RF} signal}: The \ac{RF} signal is the high-frequency signal transmitted as an electromagnetic wave. This is the frequency used for the communication over the wired or wireless channel.
+	\item The \index{baseband signal} \textbf{baseband signal}: The near-zero signal is processed by the modulator or demodulator. In digital communication system, this is the signal used by the digital signal processing. It is generated by a \ac{DAC} in a transmitter and digitized by an \ac{ADC} in a receiver.
+	\item The \index{intermediate-frequency signal} \textbf{\ac{IF} signal}: Both transmitter and receiver may optionally have an \ac{IF} stage. The \ac{IF} is usually fixed, so that high quality filters can be implemented to achieve good selectivity and high \ac{SNR}. The baseband is a special kind of \ac{IF} signal with an \ac{IF} of zero or close to zero.
+\end{itemize}
+
+\subsection{Mixing Principle}
+
+A \emph{frequency mixer}, or in short \index{mixer} \textbf{mixer}, is a device which produces a new frequency from two input frequencies. The goal is a frequency shift of the input signal frequency to the desired output signal frequency.
+
+\begin{figure}[H]
+	\centering
+	\begin{circuitikz}
+		\node[mixer](Mix){};
+		\node[oscillator,below=of Mix](LO){};
+		
+		\draw (LO.south) node[below,align=center,yshift=-5mm]{\acs{LO}};
+		
+		\draw[latex-o] (Mix.west) -- (-2cm,0) node[left,align=right]{Input\\ signal $x_i(t)$};
+		\draw[-latex] (Mix.east) -- (2cm,0) node[right,align=left]{Output\\ signal $x_o(t)$};
+		\draw[-latex] (LO.north) -- node[midway,right,align=left]{$u_{LO}(t)$} (Mix.south);
+	\end{circuitikz}
+	\caption{A mixer with input, output and \acs{LO}}
+\end{figure}%
+\nomenclature[Bm]{\begin{circuitikz}[baseline={(current bounding box.center)}]\node[mixer](Mix){};\end{circuitikz}}{Mixer}
+
+The block digram of the mixer points out its implementation. The X stands for the multiplication. An ideal mixer is a multiplier:
+\begin{equation}
+	u_o(t) = u_i(t) \cdot u_{LO}(t)
+\end{equation}
+
+The input signals are
+\begin{enumerate}
+	\item the input signal which should be shifted in frequency and
+	\item a sinusoidal \acf{LO} signal.
+\end{enumerate}
+
+In fact, the ideal mixer implements a \ac{DSB-SC} \ac{AM}. Its mathematical considerations can be reused. However, the input signal of the \ac{DSB-SC} \ac{AM} was close to zero. The input signal frequency is arbitrary.
+
+The \ac{LO} is another device generating a sinusoidal signal.
+\begin{equation}
+	x_{LO}(t) = \cos\left(\omega_{LO} + \varphi_{LO}\right)
+	\label{eq:ch05_ideal_mixing_timedomain}
+\end{equation}
+Its frequency can be fixed or configurable. Configurable frequency \acp{LO} are mainly used to tune the communication system to the transmission or reception frequency.
+
+The Fourier transform of the \ac{LO} signal is:
+\begin{equation}
+	\begin{split}
+		x_{LO}(t) &= \cos\left(\omega_{LO} + \varphi_{LO}\right) \\
+		 &= \frac{1}{2}\left(e^{j\left(\omega_{LO} + \varphi_{LO}\right)} + e^{-j\left(\omega_{LO} + \varphi_{LO}\right)}\right) \\
+		 &= \frac{1}{2}\left(e^{j \varphi_{LO}} e^{j \omega_{LO}} + e^{-j \varphi_{LO}} e^{-j \omega_{LO}}\right) \\
+		\underline{X}_{LO}\left(j\omega\right) &= \pi \left( e^{j \varphi_{LO}} \delta\left(\omega - \omega_{LO}\right) + e^{-j \varphi_{LO}} \delta\left(\omega + \omega_{LO}\right) \right)
+	\end{split}
+\end{equation}
+%\textit{Remark:} For simplicity, the phase $\varphi_{LO} = 0$ is zero is most considerations.
 
-\todo{Baseband}
+The multiplication of \eqref{eq:ch05_ideal_mixing_timedomain} becomes a convolution in the frequency-domain:
+\begin{equation}
+	\begin{split}
+		\underline{X}_{o}\left(j\omega\right) &= \underline{X}_{i}\left(j\omega\right) * \underline{X}_{LO}\left(j\omega\right) \\
+		 &= \pi \left( e^{j \varphi_{LO}} \underbrace{\underline{X}_{i}\left(j\omega - j \omega_{LO}\right)}_{\text{Positive frequency-shift}} + e^{-j \varphi_{LO}} \underbrace{\underline{X}_{i}\left(j\omega + j \omega_{LO}\right)}_{\text{Negative frequency-shift}} \right)
+	\end{split}
+\end{equation}
+
+An import property of the input signal is that it is real-valued in the time-domain. Its spectrum is symmetric:
+\begin{equation}
+	\underline{X}_{i}\left(j\omega\right) = \overline{\underline{X}_{i}\left(-j\omega\right)}
+\end{equation}
+So, the input signal's spectrum consists of a positive and a negative part:
+\begin{equation}
+	\begin{split}
+		\underline{X}_{i}^{+}\left(j\omega\right) &= \begin{cases} \underline{X}_{i}\left(j\omega\right) &\quad \text{if } \omega \geq 0,\\ 0  &\quad \text{if } \omega < 0\end{cases} \\
+		\underline{X}_{i}^{-}\left(j\omega\right) &= \begin{cases} \underline{X}_{i}\left(j\omega\right) &\quad \text{if } \omega \leq 0,\\ 0  &\quad \text{if } \omega > 0\end{cases} \\
+		\underline{X}_{i}^{+}\left(j\omega\right) &= \overline{\underline{X}_{i}^{-}\left(-j\omega\right)} \\
+		\underline{X}_{i}\left(j\omega\right) &= \underline{X}_{i}^{+}\left(j\omega\right) + \underline{X}_{i}^{-}\left(j\omega\right)
+	\end{split}
+\end{equation}
+
+\begin{equation}
+	\begin{split}
+		\underline{X}_{o}\left(j\omega\right) &= \underline{X}_{i}\left(j\omega\right) * \underline{X}_{LO}\left(j\omega\right) \\
+		 &= \pi \left( \underbrace{e^{j \varphi_{LO}} \underline{X}_{i}^{+}\left(j\omega - j \omega_{LO}\right) + e^{j \varphi_{LO}} \underline{X}_{i}^{-}\left(j\omega - j \omega_{LO}\right) }_{\text{Positive frequency-shift}} \right. \\ & \qquad \left. + \underbrace{e^{-j \varphi_{LO}} \underline{X}_{i}^{+}\left(j\omega + j \omega_{LO}\right) + e^{-j \varphi_{LO}} \underline{X}_{i}^{-}\left(j\omega + j \omega_{LO}\right)}_{\text{Negative frequency-shift}} \right) \\
+		 &\quad \text{Reordering:} \\
+		 &= \pi \left( \underbrace{e^{j \varphi_{LO}} \underline{X}_{i}^{+}\left(j\omega - j \omega_{LO}\right) + e^{-j \varphi_{LO}} \underline{X}_{i}^{-}\left(j\omega + j \omega_{LO}\right) }_{\text{Output signal 1: Sum of \acs{LO} and input frequencies}} \right. \\ & \qquad \left. + \underbrace{e^{-j \varphi_{LO}} \underline{X}_{i}^{+}\left(j\omega + j \omega_{LO}\right) + e^{j \varphi_{LO}} \underline{X}_{i}^{-}\left(j\omega - j \omega_{LO}\right)}_{\text{Output signal 2: Difference of \acs{LO} and input frequencies}} \right) \\
+		 &= \underline{X}_{o,1}\left(j\omega\right) + \underline{X}_{o,2}\left(j\omega\right)
+	\end{split}
+\end{equation}
+
+\begin{itemize}
+	\item Both the positive $\underline{X}_{i}^{+}$ and negative $\underline{X}_{i}^{-}$ part of the input signals are frequency-shifted in both the positive $+ \omega_{LO}$ and the negative $- \omega_{LO}$ direction. This becomes more evident in Figure \ref{fig:ch05:ideal_mixing_freqdomain}.
+	\item This resembles the \ac{DSB-SC} \ac{AM} with the difference that the input signal's frequencies are not close to zero.
+	\item The output signal is the superposition of two output signal:
+	\begin{itemize}
+		\item Output signal 1: The \ac{LO} frequency $\omega_{LO}$ is added to all input signals frequencies.
+		\item Output signal 2: The \ac{LO} frequency $\omega_{LO}$ is subtracted all input signals frequencies.
+	\end{itemize}
+	\item Both output signal parts 1 and 2 are real-valued in the time-domain when they are considered isolated. Both signals are Hermitian and therefore fulfil the symmetry rules.
+	\item Both output signal parts contain the identical information. They are called \index{mirror frequencies} \textbf{mirror frequencies}.
+\end{itemize}
+
+\begin{definition}{Mirror frequencies}
+	Assuming the input signal is represented by the angular frequency $\omega_i$ (although it is usually a frequency band),
+	\begin{itemize}
+		\item The angular frequency of output signal 1 is $\omega_{o,1} = \omega_{i} + \omega_{LO}$.
+		\item The angular frequency of output signal 2 is $\omega_{o,2} = \omega_{i} - \omega_{LO}$.
+	\end{itemize}
+
+	\vspace{0.5em}
+
+	The frequencies $\omega_{o,1}$ and $\omega_{o,2}$ are called \index{mirror frequencies} \textbf{mirror frequencies}.
+\end{definition}
+
+\begin{attention}
+	Because of the mirror frequency issue, a filter (\ac{LPF}, \ac{BPF}, etc.) must follow or precede a mixer to eliminate the unwanted mirror frequency.
+\end{attention}
+
+
+\begin{figure}[H]
+	\subfloat[Input and \acs{LO} signals in the frequency-domain] {
+		\centering
+		\begin{tikzpicture}
+			\begin{axis}[
+				height={0.15\textheight},
+				width=0.9\linewidth,
+				scale only axis,
+				xlabel={$\omega$},
+				ylabel={$|\underline{X}|$},
+				%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=-3.7,
+				xmax=3.7,
+				ymin=0,
+				ymax=1.2,
+				xtick={-2, -0.8, 0, 0.8, 2},
+				xticklabels={$- \omega_i$, $- \omega_{LO}$, $0$, $\omega_{LO}$, $\omega_i$},
+				ytick={0},
+			]
+				\draw[red, thick] (axis cs:-1.8,0) -- (axis cs:-2,0.7) node[above,align=center]{$\underline{X}_{i}^{-}$} -- (axis cs:-2.5,0);
+				\draw[red, thick] (axis cs:1.8,0) -- (axis cs:2,0.7) node[above,align=center]{$\underline{X}_{i}^{+}$} -- (axis cs:2.5,0);
+				
+				\draw[-latex, blue, very thick] (axis cs:-0.8,0) -- (axis cs:-0.8,1) node[above,align=center]{\acs{LO}};
+				\draw[-latex, blue, very thick] (axis cs:0.8,0) -- (axis cs:0.8,1) node[above,align=center]{\acs{LO}};
+			\end{axis}
+		\end{tikzpicture}
+	}
+	
+	\subfloat[Output signals in the frequency-domain] {
+		\centering
+		\begin{tikzpicture}
+			\begin{axis}[
+				height={0.15\textheight},
+				width=0.9\linewidth,
+				scale only axis,
+				xlabel={$\omega$},
+				ylabel={$|\underline{X}|$},
+				%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=-3.7,
+				xmax=3.7,
+				ymin=0,
+				ymax=1.2,
+				xtick={-2.8, -2, -1.3, 0, 1.3, 2, 2.8},
+				xticklabels={$- \omega_{o,1}$, $- \omega_i$, $- \omega_{o,2}$, $0$, $\omega_{o,2}$, $\omega_i$, $\omega_{o,1}$},
+				ytick={0},
+			]
+				% X-
+				\draw[red, thick] (axis cs:-2.6,0) -- (axis cs:-2.8,0.7) -- (axis cs:-3.3,0);
+				\draw[red, thick] (axis cs:-1.0,0) -- (axis cs:-1.3,0.7) -- (axis cs:-1.8,0);
+				
+				% X+
+				\draw[red, thick] (axis cs:2.6,0) -- (axis cs:2.8,0.7) -- (axis cs:3.3,0);
+				\draw[red, thick] (axis cs:1.0,0) -- (axis cs:1.3,0.7) -- (axis cs:1.8,0);
+				
+				\draw[dashed] (1.3,0) -- (1.3,1);
+				\draw[dashed] (2,0) -- (2,1);
+				\draw[dashed] (2.8,0) -- (2.8,1);
+				\draw[latex-latex] (1.3,0.9) -- node[midway,above,align=center]{$\omega_{i} - \omega_{LO}$} (2,0.9);
+				\draw[latex-latex] (2,0.9) -- node[midway,below,align=center]{$\omega_{i} + \omega_{LO}$} (2.8,0.9);
+			\end{axis}
+		\end{tikzpicture}
+	}
+	
+	\caption{Ideal mixing in the frequency-domain}
+	\label{fig:ch05:ideal_mixing_freqdomain}
+\end{figure}
+
+\begin{example}{Mirror frequencies in a transmitter}
+	A signal of \SI{1500}{MHz} is mixed with an \ac{LO} signal of \SI{900}{MHz}. The resulting output frequencies are \SI{2400}{MHz} and \SI{600}{MHz}. For a \ac{WLAN} transmitter, only the \SI{2400}{MHz} signal is desired. The \SI{600}{MHz} signal is eliminated by a \ac{BPF} with a centre frequency of \SI{2400}{MHz} after the mixer.
+\end{example}
+
+\begin{example}{Mirror frequencies in a receiver}
+	A receiver shall receive at a frequency of \SI{868}{MHz}. It has an \ac{IF} of \SI{1100}{MHz}. The \ac{LO} is configured to \SI{232}{MHz}. Due to the mirror frequency issue, the \ac{IF} mixer converts both \SI{868}{MHz} and the mirror frequency \SI{1332}{MHz} to \SI{1100}{MHz}. The receiver receives therefore on both \SI{868}{MHz} and \SI{1332}{MHz}. A signal on \SI{1332}{MHz} would disturb the reception of the \SI{868}{MHz} signal and must therefore be eliminated by a \ac{BPF} with a centre frequency of \SI{868}{MHz} before the mixer.
+\end{example}
 
-\subsection{Mirror Frequencies}
 
 \subsection{Technical Realization of Mixers}
 
@@ -580,8 +881,12 @@ Like the \ac{DSB-TC}, the transmission bandwidth of the \ac{DSB-SC} is $2 \omega
 
 \subsection{Zero-Intermediate-Frequency}
 
+\todo{coherency}
+
 \subsection{Mixing Complex-Valued Baseband Signals}
 
+\todo{IQ Modulator}
+
 
 \section{Digital Modulation Techniques}
 
@@ -597,7 +902,6 @@ Like the \ac{DSB-TC}, the transmission bandwidth of the \ac{DSB-SC} is $2 \omega
 
 \todo{QAM}
 
-\subsection{IQ Modulator}
 
 \todo{signal chain: S/P -> constellation diagram -> iFFT -> IQ}
 
-- 
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