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diff --git a/chapter07/content_ch07.tex b/chapter07/content_ch07.tex index cdafe24..9ee374b 100644 --- a/chapter07/content_ch07.tex +++ b/chapter07/content_ch07.tex @@ -343,7 +343,7 @@ Example usage of \ac{DSSS}: Another straightforward method spreading a signal across the frequency spectrum is transmitting each symbol at another frequency. This technique is called \index{frequency-hopping spread spectrum} \textbf{\acf{FHSS}}. \begin{itemize} - \item The frequency band $\Delta f_{FHSS}$ is divided into $M$ \index{sub-band} \textbf{sub-bands} with a bandwidth of $\Delta f_{sub}$. + \item The frequency band $\Delta f_{FHSS}$ is divided into $M$ \index{sub-band} \textbf{sub-bands} with a bandwidth of $\Delta f_{sub}$. \nomenclature[Sf]{$\Delta f_{sub}$}{Bandwidth of a sub-band} \begin{equation} \Delta f_{sub} = \frac{\Delta f_{FHSS}}{M} \label{eq:ch07:fhss_sub_f} @@ -1114,11 +1114,204 @@ The processing gain is not only limited to wide-band (\ac{AWGN}, thermal, quanti \section{Multi-carrier Modulation} +Multi-carrier modulation is a spread spectrum technique, which does not only increase the bandwidth whilst keeping the data rate constant. Multi-carrier modulation is related to \ac{FHSS}, but does not implement a hopping scheme. Instead, all sub-band are used, to transmit $M$ data streams parallelly. + +\begin{itemize} + \item The (serial) sequence of data symbols is parallelized. + \item The $M$ parallel symbol streams are then modulated independently. + \item Each modulated symbol stream is then transmitted in another sub-band. +\end{itemize} + +\begin{figure}[H] + \centering + \begin{adjustbox}{scale=0.8} + \begin{circuitikz} + \node[draw,block,minimum height=6cm](SP){Serial-to-\\ parallel}; + \node[adder,right=8cm of SP](Add){}; + + \draw[-o] (SP.west) node[inputarrow]{} -- ++(-1cm,0) node[left,align=right]{Data stream $\vect{D}$}; + + \foreach \n/\y in {1/2.5, 2/1, M/-2.5}{ + \draw ([yshift={\y cm}]SP.east) -- ++(1cm,0) node[inputarrow]{} node[draw,block,anchor=west](Mod\n){Modulator \n}; + \draw (Mod\n.east) -- ++(1cm,0) node[inputarrow]{} node[mixer,anchor=west](Mix\n){}; + } + + \node[above=5mm of Mix1,align=center]{Sub-band\\ mixers}; + + \draw (Mix1.east) -| (Add.north) node[inputarrow,rotate=-90]{}; + \draw (Mix2.east) -| (Add.north); + \draw (MixM.east) -| (Add.south) node[inputarrow,rotate=90]{}; + + \draw[draw=none] (Mod2.south) -- node[midway]{$\vdots$} (ModM.north); + + \draw (Add.east) -- ++(1cm,0) node[inputarrow]{} node[right,align=left]{Multi-carrier\\ signal}; + \end{circuitikz} + \end{adjustbox} + \caption{Multi-carrier modulator} +\end{figure} + +\begin{itemize} + \item If the rate of the input symbols is $f_{sym}$, the symbol rate in each of the $M$ parallel streams is $f_{sym,M}$. \nomenclature[Sf]{$f_{sym,M}$}{Symbol rate in one of the $M$ sub-bands} + \begin{equation} + f_{sym,M} = \frac{f_{sym}}{M} + \end{equation} + \item The symbol period in each sub-band is $M$ times longer. \nomenclature[Sf]{$T_{sym,M}$}{Symbol period in one of the $M$ sub-bands} + \begin{equation} + T_{sym,M} = M \cdot T_{sym} + \end{equation} + \item The bandwidth of one sub-band is approximately $\Delta f_{sub} \approx f_{sym,M}$. + \item The total bandwidth $\Delta f_{MC}$ of all sub-bands together is approximately $\Delta f_{MC} = M \cdot \Delta f_{sub} \approx f_{sym}$. + \item The duration of one transmitted symbol is $T_{sym,M}$. + \item the \emph{time-bandwidth product} is + \begin{equation} + \Delta f_{MC} \cdot T_{sym,M} = M \gg 1 + \end{equation} + The condition for a spread spectrum signal is fulfilled. +\end{itemize} + +\begin{figure}[H] + \centering + \begin{tikzpicture}[x=1cm,y=1cm] + \draw[-latex] (0,0) -- (6.5,0) node[below right, align=left]{$t = n \cdot T_{sym}$}; + \draw[-latex] (0,0) -- (0,0.4) (0,0.6) -- (0,5.5) node[left, align=right]{$f$}; + \draw (-0.3,0.4) -- (0.3,0.4); + \draw (-0.3,0.6) -- (0.3,0.6); + + \draw (0,3) -- (-0.2,3) node[left,align=right]{$f_c$}; + \draw[latex-latex] (-1,2) -- node[midway,left,align=center,anchor=south,rotate=90]{$\Delta f_{sub}$} (-1,3); + \draw[latex-latex] (-2,1) -- node[midway,left,align=center,anchor=south,rotate=90]{$\Delta f_{MC}$} (-2,5); + \draw[latex-latex] (2,-0.5) -- node[midway,below,align=center]{$T_{sym,M} = M/f_{sym}$} (4,-0.5); + + \foreach \k in {0, 1, 2, 3}{ + \draw[dotted] (0,{\k+1}) -- (8,{\k+1}); + \node[right,align=left] at(7,{\k+1.5}) {\small\itshape Sub-band $k = \k$}; + } + \draw[dotted] (0,5) -- (8,5); + + \foreach \n/\x/\k in {0/0/0, 1/0/1, 2/0/2, 3/0/3, 4/1/0, 5/1/1, 6/1/2, 7/1/3, 8/2/0, 9/2/1, 10/2/2, 11/2/3}{ + \node[fill=gray!50, draw=black, minimum height=1cm, minimum width=2cm, anchor=south west] at({\x*2},{\k+1}) {\footnotesize $D[\n]$}; + } + \end{tikzpicture} + \caption[Time-frequency plot: distribution of symbols in an $M$-ary multi-carrier modulation (with $M = 4$)]{Time-frequency plot: distribution of symbols in an $M$-ary multi-carrier modulation (with $M = 4$). $M$ symbols can be transmitted parallelly. The symbol rate is reduced proportionally.} + \label{fig:ch07:mulcarr_mod_spectrum} +\end{figure} + \subsection{Inter-Carrier Interference} +The \emph{\ac{ISI}} was an issue in the time-domain. +\begin{itemize} + \item Neighbouring symbols interfered if no guard interval was inserted. + \item The reason was the band limitation of the symbols, which flattened the ideal, rectangular slopes of the symbols in the time-domain. +\end{itemize} + +Due to the duality of time-domain and frequency-domain, an analogous problem arises in the frequency domain -- the \index{inter-carrier interference} \textbf{inter-carrier interference}. +\begin{itemize} + \item Symbols are assumed to be ideal. They have a rectangular shape. + \item Their Fourier transform is a sinc-function, whose \ac{PSD} spreads across the infinite frequency range. + \item The side lobes of neighbouring sinc-functions would overlap. A symbol in one sub-band would interfere with its neighbouring sub-bands. + \item \textbf{A \emph{guard band} must be inserted to reduce the interference.} +\end{itemize} + +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + height={0.15\textheight}, + width=0.7\linewidth, + scale only axis, + xlabel={$f$}, + ylabel={Sub-bands}, + %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}, + ] + \addplot[blue,smooth] coordinates {(-1.5,0) (0,0.5) (0.5,0.85) (1,1) (1.5,0.85) (2,0.5) (3.5,0)}; + \addplot[red,smooth] coordinates {(0.5,0) (2,0.5) (2.5,0.85) (3,1) (3.5,0.85) (4,0.5) (5.5,0)}; + \addplot[green,smooth] coordinates {(2.5,0) (4,0.5) (4.5,0.85) (5,1) (5.5,0.85) (6,0.5) (7.5,0)}; + \addplot[olive,smooth] coordinates {(4.5,0) (6,0.5) (6.5,0.85) (7,1) (7.5,0.85) (8,0.5) (9.5,0)}; + + \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]{Sub-band bandwidth $\Delta f_{sub}$} (axis cs:4,1.1); + \end{axis} + \end{tikzpicture} + \caption{Neighbouring sub-bands interfering with each other and thereby causing inter-carrier interference} +\end{figure} + +\begin{fact} + All frequency-division spread spectrum techniques (\ac{FHSS} and multi-carrier) suffer from inter-carrier interference. +\end{fact} + +A draw-back of inserting guard bands is the increased bandwidth of the whole multi-carrier signal. + \subsection{Orthogonal Frequency-Division Multiplex} -\todo{OFDM} +The increased bandwidth makes frequency-division spread spectrum techniques unattractive. Luckily, the inter-carrier interference issue can be mitigated without significantly increasing the bandwidth. +\begin{itemize} + \item The sinc-function has a special property. It has \emph{zeros} at each $f = k \cdot \frac{1}{T_{sym,M}}$ (or as an angular freuqency $\omega = k \cdot \frac{2\pi}{T_{sym,M}}$) for all integer values except zero $k \in \mathbb{Z} \ \left\{0\right\}$. + \item If the centre frequency (sub-carrier frequency) of the neighbouring sub-bands were at these zeros of the sinc-function, the inter-carrier interference would be minimal. + \item Because the sub-carrier frequency is in a zero of the sin-function, \textbf{all sub-carriers are orthogonal}. + \item This means that the optimal spacing between the carriers of the sub-bands $\Delta f_{sc-sc}$ (the \index{sub-carrier spacing} \textbf{sub-carrier spacing}) is + \begin{equation} + \Delta f_{sc-sc} = \frac{1}{T_{sym,M}} = f_{sym,M} + \end{equation}% + \nomenclature[Sf]{$\Delta f_{sc-sc}$}{Sub-carrier spacing in a multi-carrier system} +\end{itemize} + +\todo{Plot sinc-functions with zeros and its neighbouring carriers} + +The total bandwidth occupied is +\begin{equation} + \Delta f_{MC} = M \Delta f_{sc-sc} +\end{equation} +which is the minimum possible value and therefore optimal. + +The optimal sub-carrier spacing makes the sub-carriers orthogonal. The technique is called \index{orthogonal frequency-division multiplex} \textbf{\acf{OFDM}}. + +\subsubsection{OFDM Implementation Using the FFT} + +Please remember Chapter 4, when we discussed the orthogonality of the frequency vectors of a \ac{DFT}. This circumstance is used to implement the \ac{OFDM}. +\begin{itemize} + \item Symbol are parallelized. + \item Each parallel sub-symbol is then modulated (\acs{BPSK}, \acs{QPSK}, \acs{QAM}, ...). The modulator generates a complex-valued IQ output for each sub-band. + \item The complex-valued modulator output is then fed into an \ac{IFFT}. Each sub-carrier is represented by one input frequency-domain sample of the \ac{IFFT}. + \item The \ac{IFFT} transforms the multi-carrier signal to the time-domain. + \item It complex-valued IQ output of the \ac{IFFT} is the complex-valued baseband signal. + \item The complex-valued baseband signal is then converted to an analogue signal is mixed by an IQ modulator to the \ac{RF} band. +\end{itemize} +The \ac{IFFT} is, like the \ac{FFT}, implemented by an efficient algorithm. + +\todo{OFDM Tx block diagram} + +In the receiver, the signal processing chain is reversed: +\begin{itemize} + \item The IO demodulator outputs an complex-valued baseband signal which is digitized. + \item The digitized \ac{I} and \ac{Q} components are given as time-domain samples to an \ac{FFT}. + \item The \ac{FFT} calculates the frequency-domain samples. + \item Each frequency-domain sample represents a sub-band. + \item Each sub-band is demodulated (\acs{BPSK}, \acs{QPSK}, \acs{QAM}, ...) independently. + \item The demodulated, parallel symbols are then serialized. The data stream is reconstructed. +\end{itemize} + +\todo{OFDM Rx block diagram} \section{Multiple Access} |