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diff --git a/chapter05/content_ch05.tex b/chapter05/content_ch05.tex index 3579757..7252131 100644 --- a/chapter05/content_ch05.tex +++ b/chapter05/content_ch05.tex @@ -1553,7 +1553,7 @@ The process can be reversed. The \ac{RF} signal is always real-valued and contains the basebased shifted as a whole (including its non-symmetric positive and negative parts) to the \ac{RF} frequency $\omega_{RF}$. -\begin{proof}{} +\begin{proof}{IQ modulation} The complex-valued baseband signal $\underline{x}_{B}(t)$ can be decomposed into its real and imaginary values, the \ac{I} and \ac{Q} components. \begin{equation} \underline{x}_{B}(t) = x_{B,I}(t) + j \cdot x_{B,Q}(t) @@ -1578,38 +1578,404 @@ The \ac{RF} signal is always real-valued and contains the basebased shifted as a \end{align} \end{subequations} - Following conditions must be true to fulfil \eqref{eq:ch05:baseband_tx_freqdom} and the symmetry rules of $\underline{X}_{B,I}\left(j\omega\right)$ and $\underline{X}_{B,Q}\left(j\omega\right)$: + Following conditions must be true to fulfil \eqref{eq:ch05:baseband_tx_freqdom} and the symmetry rules $\underline{X}_{B,I}\left(j\omega\right) = \overline{\underline{X}_{B,I}\left(-j\omega\right)}$ and $\underline{X}_{B,Q}\left(j\omega\right) = \overline{\underline{X}_{B,Q}\left(-j\omega\right)}$: \begin{subequations} \begin{align} - \underline{X}_{B,I}^{+}\left(j\omega\right) = -j \underline{X}_{B,Q}^{+}\left(j\omega\right) &= \begin{cases} + \underline{X}_{B,I}^{+}\left(j\omega\right) = j \underline{X}_{B,Q}^{+}\left(j\omega\right) &= \begin{cases} \frac{1}{2} \underline{X}_{B}\left(j\omega\right) & \quad \text{ if } \omega \geq 0 \\ 0 & \quad \text{ if } \omega \leq 0 \end{cases} \\ - \underline{X}_{B,I}^{-}\left(j\omega\right) = j \underline{X}_{B,Q}^{-}\left(j\omega\right) &= \begin{cases} + \underline{X}_{B,I}^{-}\left(j\omega\right) = -j \underline{X}_{B,Q}^{-}\left(j\omega\right) &= \begin{cases} 0 & \quad \text{ if } \omega \geq 0 \\ - \frac{1}{2} \overline{\underline{X}_{B}\left(j\omega\right)} & \quad \text{ if } \omega \leq 0 + \frac{1}{2} \overline{\underline{X}_{B}\left(-j\omega\right)} & \quad \text{ if } \omega \leq 0 \end{cases} \end{align} \end{subequations} + + The \ac{I} component mixed up to the \ac{RF} band is: + \begin{equation} + \begin{split} + x_{RF,I}(t) &= x_{B,I}(t) \cdot \cos\left(\omega_{RF} t\right) \\ + \underline{X}_{RF,I}\left(j\omega\right) &= \underline{X}_{B,I}\left(j\omega\right) * \left(\delta\left(\omega-\omega_{RF}\right) + \delta\left(\omega+\omega_{RF}\right)\right) \pi \\ + &= \pi \left(\underline{X}_{B,I}^{+}\left(j\omega-j\omega_{RF}\right) + \underline{X}_{B,I}^{-}\left(j\omega+j\omega_{RF}\right)\right. \\ &\qquad \left. + \underline{X}_{B,I}^{-}\left(j\omega-j\omega_{RF}\right) + \underline{X}_{B,I}^{+}\left(j\omega+j\omega_{RF}\right)\right) \\ + &= \pi \left(\frac{1}{2} \underline{X}_{B}\left(j\omega-j\omega_{RF}\right) + \frac{1}{2} \overline{\underline{X}_{B}\left(-j\omega-j\omega_{RF}\right)}\right. \\ &\qquad \left. + \frac{1}{2} \overline{\underline{X}_{B}\left(-j\omega+j\omega_{RF}\right)} + \frac{1}{2} \underline{X}_{B}\left(j\omega+j\omega_{RF}\right)\right) + \end{split} + \end{equation} + + The \ac{Q} component mixed up to the \ac{RF} band is: + \begin{equation} + \begin{split} + x_{RF,Q}(t) &= x_{B,Q}(t) \cdot \underbrace{\cos\left(\omega_{RF} t + \frac{\pi}{2}\right)}_{= \sin\left(\omega_{RF} t\right)} \\ + \underline{X}_{RF,Q}\left(j\omega\right) &= \underline{X}_{B,Q}\left(j\omega\right) * \left(e^{j\frac{\pi}{2}} \delta\left(\omega-\omega_{RF}\right) + e^{-j\frac{\pi}{2}} \delta\left(\omega+\omega_{RF}\right)\right) \pi \\ + &= \pi \left(j \underline{X}_{B,Q}^{+}\left(j\omega-j\omega_{RF}\right) - j \underline{X}_{B,Q}^{-}\left(j\omega+j\omega_{RF}\right)\right. \\ &\qquad \left. + j \underline{X}_{B,Q}^{-}\left(j\omega-j\omega_{RF}\right) - j \underline{X}_{B,Q}^{+}\left(j\omega+j\omega_{RF}\right)\right) \\ + &= \pi \left(\frac{1}{2} \underline{X}_{B}\left(j\omega-j\omega_{RF}\right) + \frac{1}{2} \overline{\underline{X}_{B}\left(-j\omega-j\omega_{RF}\right)}\right. \\ &\qquad \left. - \frac{1}{2} \overline{\underline{X}_{B}\left(-j\omega+j\omega_{RF}\right)} - \frac{1}{2} \underline{X}_{B}\left(j\omega+j\omega_{RF}\right)\right) + \end{split} + \end{equation} + + The sum of both components mixed up to the \ac{RF} band is: + \begin{equation} + \begin{split} + x_{RF}(t) &= x_{RF,I}(t) + x_{RF,Q}(t) \\ + \underline{X}_{RF}\left(j\omega\right) &= \underline{X}_{RF,I}\left(j\omega\right) + \underline{X}_{RF,Q}\left(j\omega\right) \\ + &= \pi \left( \underline{X}_{B}\left(j\omega-j\omega_{RF}\right) + \overline{\underline{X}_{B}\left(-j\omega-j\omega_{RF}\right)} \right) \\ + &= \pi \left( \underbrace{\underline{X}_{B}\left(j\left(\omega-\omega_{RF}\right)\right)}_{\text{Baseband shifted to $+\omega_{RF}$}} + \underbrace{\overline{\underline{X}_{B}\left(-j\left(\omega+\omega_{RF}\right)\right)}}_{\text{Conjugate complex baseband mirrored and shifted to $-\omega_{RF}$}} \right) + \end{split} + \end{equation} + + \begin{itemize} + \item $\underline{X}_{RF}\left(j\omega\right)$ is Hermitian. $x_{RF}(t)$ is therefore real-valued. + \item $\underline{X}_{RF}\left(j\omega\right)$ contains the baseband shifted up to $\omega_{RF}$ as a whole leaving amplitudes and phases of both the positive and negative frequencies of the baseband intact. + \end{itemize} \end{proof} \section{Digital Modulation Techniques} -\subsection{Amplitude-Shift Keying} +\begin{itemize} + \item The goal of a digital communication system is conveying data from the sender to the receiver. + \item The source of the data is somewhere outside the radio. + \item \textbf{The \index{radio} \emph{radio} is the part of a digital communication system concerning with data coding, signal processing and \ac{RF} modulation.} + \item Example sources of data: + \begin{itemize} + \item Higher protocol layers, especially layer 2 (data link layer) + \item Buttons (e.g. remote control) + \item External devices (the radio acts as a modem) + \item \ac{PCM}-encoded voice + \item ... + \end{itemize} + \item In general, digital data is any kind of time-discrete and value-discrete information. + \begin{itemize} + \item In a computer system the smallest chunk of data is one bit. A bit stream is handed over to the radio which modulates it onto a carrier. + \item We will stick to bits to represent information for explanatory reasons. However, keep in mind that data can be any kind of time-discrete and value-discrete information. + \end{itemize} +\end{itemize} + +After we have learnt about the theory of modulation, let's have a look at practical modulation techniques for digital data. + +\subsection{Symbols} + +\subsubsection{Symbol Concept} + +All digital modulation techniques take time-discrete and value-discrete data. +\begin{itemize} + \item The modulation changes a property (amplitude, phase) of the carrier for a certain amount of time. + \item This time is the \index{symbol period} \textbf{symbol period} $T_{sym}$. Its inverse is the \index{symbol rate} \textbf{symbol rate} $f_{sym}$. + \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 The result is a series of symbols $x_{sym}(t)$. + \end{itemize} +\end{itemize} + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.6\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=9.5, + 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}, + ytick={0, 0.25, 0.5, 0.75, 1}, + yticklabels={0, $(00)_2 \mapsto 0.25$, $(01)_2 \mapsto 0.5$, $(10)_2 \mapsto 0.75$, $(11)_2 \mapsto 1$}, + ] + \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); + + \draw[latex-latex] (axis cs:2,1.1) -- node[midway,above,align=center]{Symbol period $T_{sym}$} (axis cs:4,1.1); + +% \draw (1,-0.5) node{11}; +% \draw (3,-0.5) node{00}; +% \draw (5,-0.5) node{10}; +% \draw (7,-0.5) node{01}; + \end{axis} + \end{tikzpicture} + \caption{Series of symbols, each encoding 2 bits} +\end{figure} + +\subsubsection{Symbol Mapping} + +Assume that the data is represented by a $N$-dimensional vector (time-discrete) of information with $K_d$ possible discrete values. +\begin{equation} + \vec{d} = \left[d_0, d_1, d_2, \cdots, d_{N-1}\right]^{\mathrm{T}} +\end{equation} + +However, the modulation is capable of encoding only $K_m$ discrete values. \textbf{So, a mapping of the $K_d$ discrete data states to the $K_m$ modulator symbol states is necessary.} This mapping is called \index{symbol mapping} \textbf{symbol mapping}. + +\begin{table}[H] + \centering + \caption[Example symbol mapping]{Example symbol mapping: There are $K_d = 2$ discrete data states, but $K_m = 4$ modulator symbol states. Two data point are mapped to one symbol. So one symbol carries two data points.} + \begin{tabular}{|l|l|} + \hline + Data & Symbol \\ + \hline + \hline + $\left[0, 0\right]^{\mathrm{T}}$ & 0.25 \\ + \hline + $\left[1, 0\right]^{\mathrm{T}}$ & 0.5 \\ + \hline + $\left[0, 1\right]^{\mathrm{T}}$ & 0.75 \\ + \hline + $\left[1, 1\right]^{\mathrm{T}}$ & 1 \\ + \hline + \end{tabular} +\end{table} + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \node[block,draw](Mapper){Symbol mapping}; + \node[block,draw,right=2cm of Mapper](Mod){Modulation}; + + \draw[latex-o] (Mapper.west) -- +(-2cm,0) node[left,align=right]{Data}; + \draw[-latex] (Mapper.east) -- node[midway,above,align=left]{Symbols} (Mod.west); + \draw[-latex] (Mod.east) -- +(2cm,0) node[right,align=left]{\ac{RF}}; + \end{tikzpicture} + \caption{Abstract transmitter signal chain of symbol mapping and modulation} +\end{figure} + +\subsubsection{Data Detection} + +On the receiver side, the \emph{symbol mapping} must be reversed. This is accomplished by the \index{data detection} \textbf{data detection}. +\begin{itemize} + \item The demodulator delivers a series of symbols. + \item Each symbol is mapped to the data. $K_m$ modulator symbol states are mapped to $K_d$ discrete data states. + \item The mapping is a kind of quantization of the modulator output. +\end{itemize} + +Remember that the received signal is subject to noise (thermal noise, quantization noise, ...). +\begin{itemize} + \item The modulator output is in fact value-discrete. + \item Noise is added to the modulator output. + \item The received symbol may be mapped to a false data value, because the additive noise pushes the signal outside the detection region for the correct data value. +\end{itemize} + +\begin{fact} + Reconstructed data at the receiver side can differ from the true value at the transmitter. This is called \emph{data error}. +\end{fact} + +\begin{itemize} + \item Noise is a stochastic process. Therefore, the data error is random. + \item The probability for the data error increases as the \ac{SNR} decreases. + \item A high \ac{SNR} makes data errors unlikely. + \item Measures for the data error are (which can be defined in relation to the \ac{SNR}) amongst others: + \begin{itemize} + \item The \index{bit error rate} \textbf{\acf{BER}}, measuring the proportion of erroneous bits, when the output data is a bit stream. + \item The \index{packet error rate} \textbf{packet error rate}, measuring the proportion of erroneous packets, where packets are a collection of data processed by higher protocol layers. Higher protocol layers may not be able to process the packet if a number of their data entities are incorrect. + \end{itemize} +\end{itemize} + +\begin{fact} + The number of modulator symbol states $K_m$ influences the probability for data errors. +\end{fact} + +\begin{itemize} + \item Low numbers of modulator symbol states $K_m$ perform better under noisy conditions with a low \ac{SNR}. + \item Higher numbers of modulator symbol states $K_m$ are sensitive to noise, because the spacing between the discrete values gets smaller. The performance is bad under noisy conditions with a low \ac{SNR}. + \item Communication systems may mitigate this problem by adapting the number of modulator symbol states $K_m$ dynamically. + \begin{itemize} + \item The link quality is estimated. + \item If the link quality is good, the modulator is configured to a high $K_m$. The data rate is thereby increased. + \item When the link quality drops, the modulator is configured to a low $K_m$. The data rate must be decreased for the sake of reliability under noisy conditions. + \item The adaption requires some additional overhead in the implementation. Usually, higher protocol layers, especially the data link layer (layer 2), is involved, because the $K_m$ value must be announced to the receiver. + \end{itemize} +\end{itemize} + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \node[block,draw](Det){Symbol mapping}; + \node[block,draw,right=2cm of Det](Demod){Demodulation}; + + \draw[-latex] (Det.west) -- +(-2cm,0) node[left,align=right]{Reconstructed\\ data}; + \draw[latex-] (Det.east) -- node[midway,above,align=left]{Noisy\\ symbols} (Demod.west); + \draw[latex-o] (Demod.east) -- +(2cm,0) node[right,align=left]{\ac{RF}}; + \end{tikzpicture} + \caption{Abstract receiver signal chain of demodulation and data detection} +\end{figure} + +\subsection{Amplitude-Shift Keying and Symbol Mapping} + +The application of the \ac{AM} in digital communication system is the \index{amplitude-shift keying} \acf{ASK}. +\begin{itemize} + \item The amplitude of a carrier is altered to modulate the data. + \item Because only the amplitude carries the information, the demodulator can be coherent or non-coherent. + \item Advantage: Both modulator and demodulator circuits are simple and can be realized at low cost. Therefore, \ac{ASK} is favoured for simple and cost-efficient applications. + \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. +\begin{equation} + x_{ASK}(t) = x_{sym}(t) \underbrace{\cos\left(\omega_{RF} t\right)}_{\text{Carrier}} +\end{equation} + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$x_{ASK}(t)$}, + %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, {sin(deg(2*pi*2*\x))}); + \addplot[red, smooth, domain=2:4, samples=50] plot(\x, {0.2*sin(deg(2*pi*2*\x))}); + \addplot[red, smooth, domain=4:6, samples=50] plot(\x, {sin(deg(2*pi*2*\x))}); + \addplot[red, smooth, domain=6:8, samples=50] plot(\x, {0.2*sin(deg(2*pi*2*\x))}); + \addplot[red, smooth, domain=8:9, samples=50] plot(\x, {sin(deg(2*pi*2*\x))}); + \addlegendentry{\acs{ASK} Signal $x_{ASK}(t)$}; + \draw[olive, dashed] (axis cs:0,1) -- (axis cs:2,1) -- (axis cs:2,0.2) -- (axis cs:4,0.2) -- (axis cs:4,1) -- (axis cs:6,1) -- (axis cs:6,0.2) -- (axis cs:8,0.2) -- (axis cs:8,1) -- (axis cs:9,1); + %\addlegendentry{Envelope of $x_B(t)$}; + + \draw[dashed] (axis cs:2,-1.2) -- (axis cs:2,1.2); + \draw[dashed] (axis cs:4,-1.2) -- (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); + + \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} + \caption{\acs{ASK} with $K_m = 2$ discrete states capable of encoding 1 bit} +\end{figure} + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.6\linewidth, + scale only axis, + xlabel={$t$}, + ylabel={$x_{ASK}(t)$}, + %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, {sin(deg(2*pi*2*\x))}); + \addplot[red, smooth, domain=2:4, samples=50] plot(\x, {0.2*sin(deg(2*pi*2*\x))}); + \addplot[red, smooth, domain=4:6, samples=50] plot(\x, {0.8*sin(deg(2*pi*2*\x))}); + \addplot[red, smooth, domain=6:8, samples=50] plot(\x, {0.5*sin(deg(2*pi*2*\x))}); + \addplot[red, smooth, domain=8:9, samples=50] plot(\x, {sin(deg(2*pi*2*\x))}); + \addlegendentry{\acs{ASK} Signal $x_{ASK}(t)$}; + \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}; + \end{axis} + \end{tikzpicture} + \caption{\acs{ASK} with $K_m = 4$ discrete states capable of encoding 2 bits} +\end{figure} + +\subsubsection{Phasor Representation} + +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{table}[H] + \centering + \caption{Example phasor representation of the symbols} + \begin{tabular}{|l|l|} + \hline + Symbol & Phasor \\ + \hline + \hline + 0 & $0.25 \cdot e^{j 0}$ \\ + \hline + 1 & $0.5 \cdot e^{j 0}$ \\ + \hline + 2 & $0.75 \cdot e^{j 0}$ \\ + \hline + 3 & $1 \cdot e^{j 0}$ \\ + \hline + \end{tabular} +\end{table} + +At each transition to a new symbol, the phasor is switched. The new phasor alters the carrier amplitude. \subsection{Phase-Shift Keying} -\subsection{Constellation Diagrams} +The application of the \ac{PM} in digital communication system is the \index{phase-shift keying} \acf{PSK}. -\todo{What is a symbol?} +\todo{Constellation Diagrams} -\todo{Data to symbol mapping} +\subsection{Quadrature Amplitude Modulation} -\todo{QAM} +\todo{What is a symbol?} +\todo{Data to symbol mapping} \todo{signal chain: S/P -> constellation diagram -> iFFT -> IQ} +\todo{IQ Imbalance} + %\subsection{Coherent and Non-Coherent Demodulation} \subsection{Inter-Symbol Interference} |