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diff --git a/chapter07/content_ch07.tex b/chapter07/content_ch07.tex index 648e5e2..75b5c3b 100644 --- a/chapter07/content_ch07.tex +++ b/chapter07/content_ch07.tex @@ -136,16 +136,123 @@ A third party who has no knowledge of neither the existence of the spread spectr \subsection{Direct-Sequence Spread Spectrum} -\todo{pseudorandom number} +A simple method for increasing the bandwidth systematically re-encoding the symbols using new symbols at a higher symbol rate. +\begin{itemize} + \item The input symbols are at the rate of $f_{sym}$. + \item The \emph{spreading} is achieved by re-encoding each symbol by $M$ new symbols, called \index{chip} \textbf{chips}. + \item After re-encoding, the rate of the new symbols -- the \emph{chips} -- is $f_{chp}$. $f_{chp}$ is the \index{chip rate} \textbf{chip rate}. + \item $M$ is the \index{spreading factor} \textbf{spreading factor}. + \begin{equation} + M = \frac{f_{chp}}{f_{sym}} + \end{equation} +\end{itemize} -\todo{Processing Gain} +\begin{example}{\acs{IEEE} 802.11b} + \acs{IEEE} 802.11b is an early \ac{WLAN} standard from 1999, defining data rates of \SI{1}{Mbit/s} to \SI{11}{Mbit/s}.\footnote{In comparison to that modern \ac{WLAN} standards like \acs{IEEE} 802.11ax have data rates of several \si{Gbit/s}. But they use other spread spectrum technologies.} Usually, the data rate is proportional to the transmission bandwidth. But, \acs{IEEE} 802.11b uses \ac{DSSS} to implement an adaptive data rate whilst retaining a constant bandwidth (approximately \SI{22}{MHz}). + + \begin{figure}[H] + \centering + \begin{circuitikz} + \node[mixer](Spreader){}; + \node[draw,block,below=1.5cm of Spreader](Code){Code\\ generator}; + \node[draw,block,right=3cm of Spreader](Mod){\acs{BPSK}\\ modulator}; + + \node[above=3mm of Spreader,align=center]{Multiplier as\\ the spreader}; + + \draw[-o] (Spreader.west) node[inputarrow]{} -- ++(-1cm,0) node[left,align=right]{Input data\\ at $f_{sym}$}; + \draw (Code.north) -- node[midway,right,align=left]{Spreading code\\ at $f_{chip}$} (Spreader.south) node[inputarrow,rotate=90]{}; + \draw (Spreader.east) -- node[midway,above,align=center]{Chip\\ sequence} (Mod.west) node[inputarrow]{}; + \draw (Mod.east) -- ++(1cm,0) node[inputarrow]{} node[right,align=left]{Baseband\\ signal}; + \end{circuitikz} + \caption{Simplified spreading and \acs{PSK} modulation signal chain of an \acs{IEEE} 802.11b conforming transmitter} + \end{figure} + + Data is represented by bits. At a data rate of $f_{sym} = \SI{1}{Mbit/s}$, the data is spread by a factor $M = 11$. The data symbols are multiplied by the \emph{spreading code}. Because of the multiplication, the same block symbol as the mixer is used. The code has a \emph{chip rate} of $f_{chp} = M f_{sym} = \SI{11}{Mchp/s}$.\footnote{Physically, the units \si{bit} and \si{chp} are dimension-less. Therefore, $\SI{1}{Mbit/s} = \SI{1}{Mchp/s} = \SI{1}{MHz}$. However, the units shall refer to the quantity being considered.} + + \vspace{0.5em} + + For $M = 11$, the \index{Barker code} \textbf{Barker code} with $\vect{C}_{11} = \left[+1, +1, +1, -1, -1, -1, +1, -1, -1, +1, -1\right]$ is used as the spreading code. Let's consider a bit stream of $(10)_2$ as th data, which is encoded as the symbols $\vect{D} = \left[-1, +1\right]$. The spread sequence is: + \begin{equation} + \begin{split} + \vect{S} &= \vect{D} \otimes \vect{C}_{11} \\ + &= \left[\underbrace{-1, -1, -1, +1, +1, +1, -1, +1, +1, -1, +1}_{\text{Spread symbol } -1},\right. \\ &\qquad \left. \underbrace{+1, +1, +1, -1, -1, -1, +1, -1, -1, +1, -1}_{\text{Spread symbol } +1}\right] + \end{split} + \end{equation} + + The \emph{chip sequence} $\vect{S}$ is at a rate of \SI{11}{Mchp/s}. In this case, the \emph{chip} have two discrete states $-1$ or $+1$. They can be modulated by a \ac{BPSK} modulator. It baseband is then mixed to the \ac{RF} band of \SI{2.4}{GHz} using an IQ modulator. + + \vspace{0.5em} + + At a data rate of \SI{11}{Mbit/s}, the data is \underline{not} spread ($M = 1$). The spreading code is $\vect{C}_{1} = \left[+1\right]$. The \emph{chip rate} equals the data bit rate. The \emph{chip rate} remains constant at \SI{11}{Mchp/s}. + + \vspace{0.5em} + + \textbf{Why adaptive data rate?} Spreading by $M = 11$ increases the \ac{SNR}. Each symbol is effectively repeated 11 times. This circumstance can be used to achieve a processing gain and increase noise immunity. The higher data rate at a lower spreading factor of $M = 1$ comes at the drawback of decreased noise immunity. +\end{example} + +The \index{direct-sequence spread spectrum} \textbf{\acf{DSSS}} spreads the symbols by multiplying each data symbol with the whole spreading code. +\begin{itemize} + \item The data symbol sequence is represented by the vector $\vect{D}$. A single data symbol is $D_n$. + \item The data comes at the rate of $f_{sym}$. + \item The code sequence is represented by the $M$-dimensional vector $\vect{C}_M$. $M$ is the length of the code. A single chip of the code is $C_m$. + \item The code repeats at $f_{sym}$. The rate of the single code chips is $f_{chp} = M f_{sym}$. +\end{itemize} +The process of multiplying each data symbol $D_n$ with the whole code sequence $\left[C_0, C_1, \dots, C_{M-1}\right]$ is represented by the \index{Kronecker product} \textbf{Kronecker product} $\otimes$. \nomenclature[Fk]{$\otimes$}{Kronecker product} +\begin{equation} + \begin{split} + \vect{S} &= \vect{D} \otimes \vect{C}_{M} \\ + S_{n M + m} &= D_n \cdot C_m \qquad \forall 0 \leq m < M + \end{split} +\end{equation} +$\vect{S}$ is the sequence of output chips -- the \emph{spread spectrum signal}. The chips in $\vect{S}$ are at the chip rate of $f_{chp} = M f_{sym}$. + +\begin{figure}[H] + \centering + \begin{circuitikz} + \node[draw,block](Spreader){Spreader\\ $\vect{D} \otimes \vect{C}_{M}$}; + \node[draw,block,below=1.5cm of Spreader](Code){Code\\ generator}; + \node[draw,block,right=2.5cm of Spreader](Mod){Modulation\\ (\acs{BPSK}, \acs{QPSK},\\ \acs{QAM}, ...)}; + + \draw[-o] (Spreader.west) node[inputarrow]{} -- ++(-1cm,0) node[left,align=right]{Input symbol\\ sequence $\vect{D}$}; + \draw (Code.north) -- node[midway,right,align=left]{Spreading\\ code $\vect{C}_{M}$} (Spreader.south) node[inputarrow,rotate=90]{}; + \draw (Spreader.east) -- node[midway,above,align=center]{Chip\\ sequence\\ $\vect{S}$} (Mod.west) node[inputarrow]{}; + \draw (Mod.east) -- ++(1cm,0) node[inputarrow]{} node[right,align=left]{Baseband\\ signal}; + \end{circuitikz} + \caption{Abstract \acs{DSSS}} + \label{fig:ch07:abstract_dsss} +\end{figure} + +Figure \ref{fig:ch07:abstract_dsss} depicts an abstract view on \ac{DSSS}. The \emph{spreading code} is a \index{pseudorandom code} \textbf{pseudorandom code}. +\begin{itemize} + \item The code is generated by an algorithm and is thereby predictable if the algorithm is known. + \item For a receiver that does not know the code generation algorithm, the code sequence is random. It is noise-like. + \item \textit{The fact, that the code must be known to the receiver, can be used to implement data encryption.} +\end{itemize} + +Example usage of \ac{DSSS}: +\begin{itemize} + \item \acs{IEEE} 802.11b specification for \ac{WLAN} + \item \ac{GPS} +\end{itemize} \subsection{Frequency-Hopping Spread Spectrum} +Example usage of \ac{FHSS}: +\begin{itemize} + \item \acs{IEEE} 802.15.1 (Bluetooth) +\end{itemize} + \subsection{Time-Hopping Spread Spectrum} +Example usage of \ac{THSS}: +\begin{itemize} + \item \ac{UWB} systems conforming to \acs{IEEE} 802.15.4 +\end{itemize} + \subsection{Symbol Reconstruction} +\todo{Processing Gain} + \todo{Reception under noise} \todo{Cross-correlation} |