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2_bandwidth.tex
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2_bandwidth.tex
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\documentclass[aspectratio=169]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
\begin{document}
\begin{frame}
\frametitle{First problem: the bandwidth constraint}
\intro{we need to squeeze the spectral support over the allotted bandwidth. We need multirate processing to do so}
\centering
\uncover<2>{$a[n]$ white: $P_a(e^{j\omega}) = \sigma_a^2$}
\begin{figure}
\begin{dspPlot}[height=3cm,xtype=freq,xticks=1,yticks=custom]{-1,1}{0,1.2}
\moocStyle
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](.25,-.05)(.6,0.05)
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](-.25,-.05)(-.6,0.05)
\only<2>{\dspFunc{1}}
\dspCustomTicks[axis=x]{.25 $\omega_{\min}$ .6 $\omega_{\max}$}
\dspCustomTicks[axis=y]{0 0 1 $\sigma_a^2$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Shaping the bandwidth}
Our problem:
\begin{itemize}[<+->]
\item bandwidth constraint requires us to control the spectral support of a signal
\item we need to be able to ``shrink'' the support of a full-band signal
\item the answer is {\em multirate}\/ techniques
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{ideal digital interpolator}
\center
\begin{figure}
\begin{dspBlocks}{1.3}{0.6}
$x[n]$ & \BDupsmp{N} & \BDfilter{LP$\{\pi/N\}$} & $y[n]$
\psset{arrows=->,linewidth=1.5pt}
\ncline{1,1}{1,2} \ncline{-}{1,2}{1,3}
\ncline{1,3}{1,4}
\end{dspBlocks}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Upsampling by $K=3$}
\setbeamercovered{invisible}
\begin{center}
\begin{figure}
\begin{dspPlot}[height=1.25cm,xtype=freq,xticks=2,yticks=1,ylabel={$X(e^{j\omega})$}]{-1,1}{0,1.2}
\moocStyle
\dspFunc{x \dspPeriodize \dspPorkpie{0}{.75}}
\dspCustomTicks[axis=x]{.75 $3\pi/4$}
\end{dspPlot}
\uncover<2->{
\begin{dspPlot}[height=1.25cm,xtype=freq,xticks=1,yticks=1,ylabel={$X(e^{j\omega})$}]{-5,5}{0,1.2}
\moocStyle
\dspFunc{x \dspPeriodize \dspPorkpie{0}{.75}}
\dspMainPeriod
\pnode(-3,0){leftA}\pnode(3,0){rightA}
\end{dspPlot}}
\uncover<3->{
\begin{dspPlot}[height=1.25cm,xtype=freq,xticks=2,yticks=1,ylabel={$X_{U}(e^{j\omega})$}]{-1,1}{0,1.2}
\pnode(-1,1.2){leftB}\pnode( 1,1.2){rightB}
\psset{linecolor=orange}
\ncline[linewidth=2pt]{leftA}{rightA}
\ncline[linewidth=1pt,linestyle=dashed]{->}{leftA}{leftB}
\ncline[linewidth=1pt,linestyle=dashed]{->}{rightA}{rightB}
\moocStyle
\only<4-5>{\dspFunc{x 3 mul \dspPeriodize \dspPorkpie{0}{.75}}}
\only<5>{\dspFunc[linecolor=green,linestyle=dashed]{x \dspRect{0}{0.66666}}}
\only<6->{\dspFunc{x 3 mul \dspPorkpie{0}{.75}}\dspCustomTicks[axis=x]{.25 $\pi/4$}}
\end{dspPlot}}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Fulfilling the bandwidth constraint}
\begin{figure}
\begin{dspPlot}[height=3cm,xticks=custom,yticks=none,sidegap=0]{0,5}{0,1.1}
\moocStyle
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](1,0)(2.4,0.5)
\dspCustomTicks[axis=x]{0 0 1 $F_{\min}$ 2.4 $F_{\max}$}
\dspCustomTicks[axis=x]{4 ${\color{red}F_s/2}$}
\psbrace[braceWidth=0.5pt,braceWidthOuter=3pt,rot=90,nodesepB=10pt](1,-.3)(2.4,-.3){$W$}%
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Fulfilling the bandwidth constraint}
\begin{figure}
\begin{dspPlot}[height=3cm,xticks=1,xtype=freq,yticks=none]{0,1}{0,1.1}
\moocStyle
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](.25,0)(.6,0.5)
\dspCustomTicks[axis=x]{.25 $\omega_{\min}$ .6 $\omega_{\max}$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Here's a neat trick}
let $W = F_{\max}-F_{\min}$; pick $F_s$ so that:
\begin{itemize}[<+->]
\item $F_s > 2F_{\max}$ (obviously)
\item $F_s = KW$, $K\in \mathbb{N}$
\vspace{2em}
\item $\displaystyle \omega_{\max}-\omega_{\min} = 2\pi\frac{W}{F_s} = \frac{2\pi}{K}$
\item we can simply upsample by $K$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Baud rate}
\begin{itemize}[<+->]
\item upsampling does not change the {\em data}\/ rate, only the sample rate
\item we produce (and transmit) $W$ symbols per second
\item $W$ is sometimes called the Baud rate of the system and is equal to the available bandwidth
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Fulfilling the bandwidth constraint}
\begin{figure}
\begin{dspPlot}[height=3cm,xtype=freq,xticks=1,yticks=1]{-1,1}{0,1.2}
\moocStyle
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](.25,-.05)(.6,0.05)
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](-.25,-.05)(-.6,0.05)
\only<2>{\dspFunc{1}}
\only<3>{\dspFunc{x \dspRect{0}{.35} }}
\only<4>{\dspFunc{x \dspRect{.425}{.35} x \dspRect{-.425}{.35} add 2 div}}
\dspCustomTicks[axis=x]{.25 $\omega_{\min}$ .425 $\omega_c$ .6 $\omega_{\max}$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Transmitter design, continued}
\begin{figure}
\psset{linearc=0.2}
\begin{dspBlocks}{1.0}{0.8}
\parbox{4ex}{\small \tt \bf ..01100\\ 01010...}\hspace{5ex} &
\BDfilter{Scrambler} & \BDfilter{Mapper} & [name=A,mnode=circle] $K \uparrow$ \\
& &
\ncline{->}{1,1}{1,2} \ncline{->}{1,2}{1,3}
\ncline{->}{1,3}{1,4}^{~~~~$a[n]$}
\end{dspBlocks}
\begin{dspBlocks}{1.2}{0.4}
& [name=B] \BDlowpass[0.5em] & \BDmul & \BDfilter{D/A} & $s(t)$ \\
& & $\cos\omega_c n$ & $F_s=KW$
\ncline{->}{1,2}{1,3}^{$b[n]$} \ncline{->}{1,3}{1,4}^{$s[n]$}
\ncline{->}{1,4}{1,5} \ncline{->}{1,5}{1,6} \ncline{->}{1,6}{1,7}
\ncline{->}{2,3}{1,3}
\end{dspBlocks}
\ncbarr[angleA=0,arm=1cm,linewidth=1.2pt]{A}{B}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Raised Cosine}
\[
H_{K,\beta}(e^{j\omega}) = \begin{cases} \displaystyle
1 & |\omega| \le \displaystyle\frac{\pi(1-\beta)}{K} \\
\frac{1}{2}\left(1 + \cos\left(\displaystyle\frac{K|\omega| - (1-\beta)\pi}{\beta}\right)\right) & \displaystyle\frac{\pi(1-\beta)}{K} \le |\omega| \le \displaystyle\frac{\pi(1+\beta)}{K} \\
0 & |\omega| > \displaystyle\frac{\pi(1+\beta)}{K}
\end{cases}
\]
\end{frame}
\begin{frame} \frametitle{Raised Cosine}
\center
$H_{2,\beta}(e^{j\omega})$
\begin{figure}
\begin{dspPlot}[xtype=freq]{-1,1}{0,1.1}
\smallStems
\only<1->{\dspFunc[linecolor=red]{x \dspRaisedCos{0.5}{0.1}}}
\only<2->{\dspFunc[linecolor=green]{x \dspRaisedCos{0.5}{0.2}}}
\only<3->{\dspFunc[linecolor=blue]{x \dspRaisedCos{0.5}{0.3}}}
\only<4->{\dspFunc[linecolor=cyan]{x \dspRaisedCos{0.5}{0.4}}}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Raised Cosine}
\[
h_{K,\beta}[nK] = \begin{cases}
1 & n = 0 \\
0 & \mbox{otherwise}
\end{cases}
\]
\vspace{2em}
\[
h_{K,\beta}[nK] \propto \frac{1}{(\beta n)^2}
\]
\end{frame}
\begin{frame} \frametitle{Raised Cosine: $1/n^2$ decay}
\center
$h_{2,\beta}[n]$
\begin{figure}
\begin{dspPlot}{-20,20}{-0.2,0.6}
\moocStyle
\dspTaps{-20 -0.0000 -19 -0.0063 -18 0.0000 -17 0.0088 -16 -0.0000 -15 -0.0120 -14 0.0000 -13 0.0161 -12 -0.0000 -11 -0.0216 -10 0.0000 -9 0.0291 -8 -0.0000 -7 -0.0405 -6 0.0000 -5 0.0600 -4 -0.0000 -3 -0.1039 -2 0.0000 -1 0.3176 0 0.5000 1 0.3176 2 0.0000 3 -0.1039 4 -0.0000 5 0.0600 6 0.0000 7 -0.0405 8 -0.0000 9 0.0291 10 0.0000 11 -0.0216 12 -0.0000 13 0.0161 14 0.0000 15 -0.0120 16 -0.0000 17 0.0088 18 0.0000 19 -0.0063 20 -0.0000}
%\only<2->{\dspFunc[linecolor=green,linestyle=dashed]{x \dspSinc{0}{2} 0.5 mul}}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Spectral shaping with raised cosine}
\begin{figure}
\begin{dspPlot}[height=3cm,xtype=freq,xticks=1,yticks=1]{-1,1}{0,1.2}
\moocStyle
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](.25,-.05)(.6,0.05)
\psframe[fillstyle=vlines,linecolor=green!70,hatchcolor=green!70,hatchangle=20](-.25,-.05)(-.6,0.05)
\dspFunc{x \dspRaisedCosine{.425}{.175}{0.2} x \dspRaisedCosine{-.425}{.175}{0.2} add 2 div}
\dspCustomTicks[axis=x]{.25 $\omega_{\min}$ .6 $\omega_{\max}$}
\end{dspPlot}
\end{figure}
\end{frame}
\end{document}
\begin{figure}
\psset{linearc=0.2}
\begin{dspBlocks}{1.0}{0.8}
\parbox{4ex}{\small \tt \bf ..01100\\ 01010...}\hspace{5ex} &
\BDfilter{Scrambler} & \BDfilter{Mapper} & [name=A,mnode=circle] $K \uparrow$ \\
& &
\ncline{->}{1,1}{1,2} \ncline{->}{1,2}{1,3}
\ncline{->}{1,3}{1,4}^{~~~~$a[n]$}
\end{dspBlocks}
\begin{dspBlocks}{1.2}{0.4}
& [name=B] \BDfilter{$G(z)$} & \BDmul & \BDfilter{Re} & \BDfilter{$I(t)$} & $s(t)$ \\
& & $e^{j\omega_c n}$
\ncline{->}{1,2}{1,3}^{$b[n]$} \ncline{->}{1,3}{1,4}^{$c[n]$}
\ncline{->}{1,4}{1,5}^{$s[n]$} \ncline{->}{1,5}{1,6} \ncline{->}{1,6}{1,7}
\ncline{->}{2,3}{1,3}
\end{dspBlocks}
\ncbarr[angleA=0,arm=1cm,linewidth=1.2pt]{A}{B}
\end{figure}
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