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3_adsl.tex
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\documentclass[aspectratio=169]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
\begin{document}
\def\sp{ }
\begin{frame} \frametitle{Overview:}
\intro{DSLAM=Digital subscriber line access multiplexer}
\begin{itemize}[<+->]
\item Channel
\item Signaling strategy
\item Discrete Multitone Modulation (DMT)
\end{itemize}
\end{frame}
\def\phone{\raisebox{-0.5em}{\includegraphics[height=1.5em]{phone.eps}}}
\def\home{\raisebox{-2em}{\includegraphics[height=5em]{home.eps}}}
\def\dslam{\raisebox{0em}{\begin{tabular}{c}DSLAM \\ \includegraphics[height=6em]{dslam.eps}\end{tabular}}}
% Digital subscriber line access multiplexer
\begin{frame} \frametitle{The telephone network today}
\begin{figure}[t]
\psset{linearc=0.2,angleB=180}
\begin{dspBlocks}{0.8}{0}
& & & & & \phone \\
& & \BDfilterMulti{voice \\ network} & \BDfilter{CO} & \phone \\
& & & & & \phone \\
\home & \BDfilter{CO} & \\
& & \dslam & [mnode=circle] internet & \\
\end{dspBlocks}
\ncline{->}{4,1}{4,2}
\ncline{->}{2,3}{2,4}
\ncdiag[lineAngle=80]{->}{4,2}{2,3} \ncdiag[lineAngle=100]{-}{4,2}{5,3}
\ncdiag[lineAngle=-80,linestyle=dotted]{->}{2,4}{3,6}
\ncline{->}{5,3}{5,4}\ncline[linestyle=dotted]{->}{5,4}{5,5}
\ncline[linestyle=dotted]{->}{2,4}{2,5}\ncdiag[lineAngle=80,linestyle=dotted]{->}{2,4}{1,6}
\only<2->{\ncline[linecolor=darkred,linewidth=3pt]{->}{4,1}{4,2}\uput[0](-125,10){{\color{darkred}``last mile''}}}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{The last mile}
\begin{itemize}[<+->]
\item copper wire (twisted pair) between home and nearest CO
\item very large bandwidth (well over 1MHz)
\item very uneven spectrum: noise, attenuation, interference, etc.
\end{itemize}
\end{frame}
% ADSL2 can reach up to 12Mbps while ADSL can only reach 8Mbps.
\begin{frame} \frametitle{The ADSL channel}
\begin{figure}
\begin{dspPlot}[height=3cm,xtype=freq,xticks=custom,yticks=none]{0,1.06}{0,1.2}
\moocStyle
\psframe[fillstyle=vlines,linecolor=red!70,hatchcolor=red!70,hatchangle=20](0,0)(.04,1)
\psframe[fillstyle=vlines,linecolor=cyan!70,hatchcolor=cyan!70,hatchangle=20](.1,0)(.3,1)
\psframe[fillstyle=vlines,linecolor=blue!70,hatchcolor=blue!70,hatchangle=20](.35,0)(1,1)
\dspCustomTicks[axis=x]{0 0 1 1MHz}
\dspLegend(0.35,1.8){red!70 POTS cyan!70 upstream blue!70 downstream}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{The ADSL channel}
\begin{figure}
\begin{dspPlot}[height=3cm,xtype=freq,xticks=custom,yticks=none]{0,1.06}{0,1.2}
\moocStyle
\psframe[fillstyle=vlines,linecolor=red!20,hatchcolor=red!20,hatchangle=20](0,0)(.04,1)
\psframe[fillstyle=vlines,linecolor=cyan!20,hatchcolor=cyan!20,hatchangle=20](.1,0)(.3,1)
\psframe[fillstyle=vlines,linecolor=blue!20,hatchcolor=blue!20,hatchangle=20](.35,0)(1,1)
\dspFuncData[plotstyle=curve]{0.05 0 .35 1 .75 .5 .83 0 .9 .3 1.05 0}
\pscustom[fillstyle=solid,fillcolor=gray,linestyle=none]{%
\dspFuncData[plotstyle=curve]{0.2 0 0.25 .2 0.3 .25 0.33 .1 0.35 .12 0.4 .13 0.51 .2 .55 0}}
\dspDiracs{0.7 1}
\dspCustomTicks[axis=x]{0 0 1 $F_{\max}$}
%\dspLegend(0.35,1.8){red!70 POTS cyan!70 upstream blue!70 downstream}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Idea: split the band into independent subchannels}
\begin{figure}
\begin{dspPlot}[height=3cm,xtype=freq,xticks=custom,yticks=none]{0,1.06}{0,1.2}
\moocStyle
\psframe[fillstyle=solid,linestyle=none,fillcolor=yellow!30](0,0)(0.5,1.2)%
\multido{\n=0+1}{10}{%
\ifodd\n\psframe[fillstyle=solid,linestyle=none,fillcolor=yellow!30](! \n\sp 0.1 mul 0.05 add 0)(! \n\sp 0.1 mul 0.16 add 1.2)%
\else\psframe[fillstyle=solid,linestyle=none,fillcolor=magenta!30](! \n\sp 0.1 mul 0.05 add 0)(! \n\sp 0.1 mul 0.16 add 1.2)\fi}%
\psframe[fillstyle=vlines,linecolor=red!20,hatchcolor=red!20,hatchangle=20](0,0)(.04,1)
\psframe[fillstyle=vlines,linecolor=cyan!20,hatchcolor=cyan!20,hatchangle=20](.1,0)(.3,1)
\psframe[fillstyle=vlines,linecolor=blue!20,hatchcolor=blue!20,hatchangle=20](.35,0)(1,1)
\dspFuncData[plotstyle=curve]{0.05 0 .35 1 .75 .5 .83 0 .9 .3 1.05 0}
\pscustom[fillstyle=solid,fillcolor=gray,linestyle=none]{%
\dspFuncData[plotstyle=curve]{0.2 0 0.25 .2 0.3 .25 0.33 .1 0.35 .12 0.4 .13 0.51 .2 .55 0}}
\dspDiracs{0.7 1}
\dspCustomTicks[axis=x]{0 0 1 $F_{\max}$}
%\dspLegend(0.35,1.8){red!70 POTS cyan!70 upstream blue!70 downstream}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Subchannel structure}
\begin{itemize}[<+->]
\item allocate $N$ subchannels over the total positive bandwidth
\item equal subchannel bandwidth $W = F_{\max}/N$
\item equally spaced subchannels with center frequency $kF_{\max}/N$, $k=0,\ldots,N-1$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{The digital design}
\begin{itemize}[<+->]
\item pick $F_s = 2F_{\max}$ ($F_s = 2NW = 2F_{\max}$. i.e. $F_s$ multiple of $W$)
\item center frequency for each subchannel $\displaystyle \omega_k = 2\pi\frac{kW}{F_s} = \frac{2\pi}{2N}k$
\item bandwidth of each subchannel $\displaystyle\frac{2\pi}{2N}$
\item to send symbols over a subchannel: upsampling factor $K = 2N$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{The digital design ($N=3)$}
\begin{figure}
\begin{dspPlot}[height=3cm,xtype=freq,xticks=custom,yticks=none]{-1,1}{0,1.2}
\only<1>{\dspCustomTicks[axis=x]{0 0 1 $\pi$ -1 $-\pi$}}
\only<2->{\dspCustomTicks[axis=x]{0 $\omega_0=0$ 0.3333 $\omega_1=2\pi/6$ 0.6666 $\omega_2=4\pi/6$ -.3333 $-\omega_1$ -.6666 $-\omega_2$ 1 $\pi$ -1 $-\pi$}}
\only<3>{%
\dspFunc[linecolor=green!70]{x \dspRaisedCosine{.6666}{.1666}{.2} x \dspRaisedCosine{-.66666}{.1666}{.2} add 2 div}
\dspFunc[linecolor=magenta!70]{x \dspRaisedCosine{.3333}{.1666}{.2} x \dspRaisedCosine{-.3333}{.1666}{.2} add 2 div}
\dspFunc[linecolor=yellow!70]{x \dspRaisedCosine{0}{.1666}{.2}}}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{The digital design}
\begin{itemize}[<+->]
\item put a QAM modem on each channel
\item decide on constellation size independently
\item noisy or forbidden subchannels send zeros
\end{itemize}
\end{frame}
\def\hich#1#2#3{%
\only<#1>{%
\psframe[linecolor=green!70](! #2\sp 0.1 mul 0)(! #2\sp 0.1 mul 0.1 add 1.2)
\dspText(! #2\sp 0.1 mul 1.3){#3}}}
\begin{frame} \frametitle{Idea: split the band into independent subchannels}
\begin{figure}
\begin{dspPlot}[height=3cm,xtype=freq,xticks=none,yticks=none]{0,1.06}{0,1.2}
\moocStyle
\dspFuncData[plotstyle=curve]{0.05 0 .35 1 .75 .5 .83 0 .9 .3 1.05 0}
\pscustom[fillstyle=solid,fillcolor=gray,linestyle=none]{%
\dspFuncData[plotstyle=curve]{0.2 0 0.25 .2 0.3 .25 0.33 .1 0.35 .12 0.4 .13 0.51 .2 .55 0}}
\dspDiracs{0.7 1}
\hich{1}{3.5}{channel \#5, SNR 20dB}
\hich{2}{5.5}{channel \#7, SNR 50dB}
\hich{3}{6.5}{channel \#8, spike noise, unusable}
\hich{4}{7.8}{channel \#9, too much attenuation, unusable}
\hich{5}{9}{channel \#10, SNR 10dB}
%\dspCustomTicks[axis=x]{0 0 1 1MHz}
%\dspLegend(0.35,1.8){red!70 POTS cyan!70 upstream blue!70 downstream}
\end{dspPlot}
\end{figure}
\end{frame}
\def\modem#1#2#3#4{%
\begin{dspBlocks}{.7}{.4}%
& & & $e^{j(2\pi/2N)#3}$ \\
$a_{#1}[m]$~ & \BDupsmp{#2} & \BDlowpass[.6em] & \BDmul & [name=#4]
\ncline{->}{2,1}{2,2}\ncline{->}{2,2}{2,3}\ncline{->}{2,3}{2,4}^{~~$b_{#1}[n]$}%
\ncline{->}{2,3}{2,4}\ncline{-}{2,4}{2,5}^{~~$c_{#1}[n]$}%
\ncline{->}{1,4}{2,4}%
\end{dspBlocks}}
\def\modemDMT#1#2#3#4{%
\begin{dspBlocks}{.7}{.4}%
& $e^{j(2\pi/2N)#3}$ \\
$a_{#1}[\lfloor n/#2 \rfloor]$~ & \BDmul & [name=#4]
\ncline{->}{2,1}{2,2}\ncline{-}{2,2}{2,3}
\ncline{->}{1,2}{2,2}%
\end{dspBlocks}}
\begin{frame} \frametitle{The subchannel modem}
\centering
\modem{k}{2N}{kn}{nop}
\vspace{2em}
\begin{dspBlocks}{.7}{.4}
\hspace{-4em} rate: $W$ symbols/sec & & & \hspace{5em} $2NW=F_s$ samples/sec
\end{dspBlocks}
\end{frame}
\begin{frame} \frametitle{The bank of modems}
\begin{figure}
\scalebox{0.6}{%
\begin{dspBlocks}{1.2}{0.2}
\modem{0}{2N}{(0\cdot n)}{A} & & \\
\modem{1}{2N}{(1\cdot n)}{B} & & \\
\modem{2}{2N}{(2\cdot n)}{C} & & [name=Z,mnode=circle] + & [name=CC] \BDfilter{Re} & [name=S] $~s[n]$ \\
\hspace{1em}$\ldots$ & & \\
\modem{N-2}{2N}{(N-2)n}{D} & & \\
\modem{N-1}{2N}{(N-1)n}{E} & &
\ncline{->}{A}{Z}\ncline{->}{B}{Z}\ncline{->}{C}{Z}\ncline{->}{D}{Z}\ncline{->}{E}{Z}
\ncline{->}{Z}{CC}^{$c[n]$}\ncline{->}{CC}{S}
\end{dspBlocks}}
\end{figure}
\end{frame}
\def\Gen#1{%
\raisebox{-1.2em}{%
\psset{xunit=1em,yunit=1em,linewidth=1.5pt}%
\pspicture(-1,0)(7.1,-3)%
\rput[r]{0}(0,-1.5){$A_{#1}$}\rput[r]{0}(0,-2.5){$\phi_{#1}$}
\rput[l](1.9,-1.5){\psframebox[framesep=.2]{\parbox{4em}{~~\pscirclebox{\Large$\mathbf{\sim}$}$_{#1}$}}}
\psline[linewidth=0.8pt]{->}(0.1,-1.5)(1.8,-1.5)
\psline[linewidth=0.8pt]{->}(0.1,-2.5)(1.8,-2.5)
\endpspicture}}
\begin{frame} \frametitle{If it looks familiar...}
\centering
remember the DFT reconstruction formula?
\begin{figure}
\scalebox{0.7}{%
\begin{dspBlocks}{1.2}{0.2}
\rnode{A}{\Gen{\,0}} & & \\
\Gen{\,1} & & \\
\Gen{\,2} & & \BDadd & $x[n]$ \\
\hspace{1em}$\ldots$ & & \\
\Gen{N-2} & & \\
\rnode{B}{\Gen{N-1}} & &
\ncline{1,1}{1,2}\ncline{->}{1,2}{3,3}
\ncline{2,1}{2,2}\ncline{->}{2,2}{3,3}
\ncline{3,1}{3,2}\ncline{->}{3,2}{3,3}
\ncline{5,1}{5,2}\ncline{->}{5,2}{3,3}
\ncline{6,1}{6,2}\ncline{->}{6,2}{3,3}
\ncline{->}{3,3}{3,4}
\end{dspBlocks}}
\end{figure}
\end{frame}
\begin{frame} \frametitle{DMT via IFFT}
\begin{itemize}[<+->]
\item we will show that transmission can be implemented efficiently via an IFFT
\item Discrete Multitone Modulation
\end{itemize}
\end{frame}
\begin{frame} \frametitle{The great ADSL trick}
\centering
instead of using a ``semi-ideal'' digital interpolator (e.g. raised cosine), use a zero-order hold:
\[
h[n] = \begin{cases} 1 & \mbox{for $0 \leq n < 2N$} \\ 0 & \mbox{otherwise} \end{cases}
\]
\end{frame}
\begin{frame} \frametitle{Digital ZOH: interval indicator signal}
\begin{center}
\begin{figure}
\begin{dspPlot}[xticks=custom]{-20,20}{-.5, 1.2}
\moocStyle
\dspSignal{x 0 lt {0} {x 9 gt {0} {1} ifelse} ifelse}
\dspCustomTicks[axis=x]{0 0 9 $2N-1$}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\def\sinInt#1#2{ #1 #2 div 2 mul sub 90 mul dup 0 eq { pop 1} {dup #2 mul sin exch dup #1 1 sub mul cos exch sin div mul #2 div} ifelse }
\begin{frame} \frametitle{DTFT of interval signal}
\intro{not really a good filter compared to raised cosine, but it still works (thanks to some clever things we don't have time to go into)}
\begin{figure}
\begin{dspPlot}[xtype=freq,yticks=1,xticks=1,ylabel={$|H(e^{j\omega})|$}]{-1,1}{0, 1.2}
\moocStyle
\dspFunc[linecolor=green!40]{x \dspRaisedCosine{0}{.125}{.2}}
\dspFunc{x \sinInt{0}{8} abs}
\dspCustomTicks[axis=x]{0.25 $\pi/N$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Back to the subchannel modem}
\centering
\modem{k}{2N}{kn}{nop}
\vspace{2em}
\begin{dspBlocks}{.7}{.4}
rate: $W$ symbols/sec & & & \hspace{5em} $2NW=F_s$ samples/sec \\
$m$ changes every $1/W$ sec & & & \hspace{5em} $n$ changes every $1/F_s = (1/W)/(2N)$
\end{dspBlocks}
\end{frame}
\begin{frame} \frametitle{Back to the subchannel modem}
\centering
by using the ZOH interpolator:
\vspace{2em}
\begin{dspBlocks}{.7}{.4}%
& $e^{j(2\pi/2N)nk}$ \\
$a_{k}[\lfloor n/2N \rfloor]$~ & \BDmul & ~$c_{k}[n]$
\ncline{->}{2,1}{2,2}\ncline{->}{2,2}{2,3}
\ncline{->}{1,2}{2,2}%
\end{dspBlocks}
\vspace{2em}
\begin{dspBlocks}{.7}{.4}
rate: $W$ symbols/sec & & & \hspace{5em} $2NW=F_s$ samples/sec
\end{dspBlocks}
\vspace{1em}
$n$ changes every $1/F_s$
\end{frame}
\begin{frame} \frametitle{The bank of modems, revisited}
\begin{figure}
\scalebox{0.6}{%
\begin{dspBlocks}{1.2}{0.2}
\modemDMT{0}{2N}{(0\cdot n)}{A} & & \\
\modemDMT{1}{2N}{(1\cdot n)}{B} & & \\
\modemDMT{2}{2N}{(2\cdot n)}{C} & & [name=Z,mnode=circle] + & [name=CC] \BDfilter{Re} & [name=S] $~s[n]$ \\
\hspace{1em}$\ldots$ & & \\
\modemDMT{N-2}{2N}{(N-2)n}{D} & & \\
\modemDMT{N-1}{2N}{(N-1)n}{E} & &
\ncline{->}{A}{Z}\ncline{->}{B}{Z}\ncline{->}{C}{Z}\ncline{->}{D}{Z}\ncline{->}{E}{Z}
\ncline{->}{Z}{CC}^{$c[n]$}\ncline{->}{CC}{S}
\end{dspBlocks}}
\end{figure}
\begin{dspBlocks}{.7}{.4}
\hspace{4em} rate: $NW = F_{\max}$ symbols/sec & & & \hspace{5em} $2NW=F_s$ samples/sec
\end{dspBlocks}
\end{frame}
\begin{frame} \frametitle{The complex output signal}
\[
c[n] = \sum_{k=0}^{N-1} a_k[\lfloor n/2N \rfloor]e^{j\frac{2\pi}{2N}nk}
\]
\vspace{1em}
\pause
for every successive block of $2N$ output samples:
\begin{itemize}
\item $a_k[\lfloor n/2N \rfloor] = a_k[p]$ doesn't change
\item $c[n] = 2N \cdot \mbox{IDFT}_{2N}\left\{\begin{bmatrix} a_0[p] & a_1[p] & \ldots & a_{N-1}[p] & 0 & 0 & \ldots & 0 \end{bmatrix}\right\}[n] $
\end{itemize}
\end{frame}
\begin{frame} \frametitle{The complex output signal: vector space approach}
for every successive block of $2N$ output samples:
\[
\mathbf{c} = \sum_{k=0}^{N-1} a_k \, \mathbf{w}^{(k)}
\]
\vspace{1em}
\begin{itemize}
\item $\mathbf{c}, \mathbf{w}^{(k)} \in \mathbb{C}^{2N}$
\item $ \mathbf{w}^{(k)}$ Fourier basis vector for $\mathbb{C}^{2N}$
\item orthogonality of basis vector makes detection easy: $a_k = \langle \mathbf{w}^{(k)}, \mathbf{c} \rangle \, /(2N)$
\item can obtain all $a_k$'s with a single DFT
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Neat implementation detail}
\begin{itemize}[<+->]
\item we are interested in $s[n] = \Real\{c[n]\} = (c[n] + c^*[n])/2$
\item it is easy to prove (exercise) that:
\[
\IDFT{\begin{bmatrix} x_0 & x_1 & x_2 & \ldots & x_{N-2} & x_{N-1}\end{bmatrix}}^* =
\IDFT{\begin{bmatrix} x_0 & x_{N-1} & x_{N-2} & \ldots & x_2 & x_1 \end{bmatrix}^*}
\]
\item $c[n] = 2N \cdot \IDFT{\begin{bmatrix} a_0[p] & a_1[p] & \ldots & a_{N-1}[p] & 0 & 0 & \ldots & 0 \end{bmatrix}}[n]$
\item $c^*[n] = 2N \cdot \IDFT{\begin{bmatrix} a_0[p] & 0 & 0 & \ldots & 0 & a^*_{N-1}[p] & \ldots & a^*_1[p] \end{bmatrix}}[n]$
\item therefore
\[
s[n] = N\cdot\IDFT{\begin{bmatrix} 2a_0[p] & a_1[p] & \ldots & a_{N-1}[p] & a^*_{N-1}[p] & a^*_{N-2}[p] & \ldots & a^*_1[p] \end{bmatrix}}[n]
\]
\end{itemize}
\end{frame}
\def\topiter#1#2#3#4{%
\FPupn\t{#3 0.4 mul 1 max 0 trunc}
\multido{\n=0+1}{\t}{\FPupn\c{\n{} 1 #3 - - #2 + 0 trunc} #4}}
%
\def\botiter#1#2#3#4{%
\FPupn\b{#3 0.3 mul 1.0 max 0 trunc}
\multido{\n=1+1}{\b}{\FPupn\c{#2 1 \n{} - + 0 trunc} #4}}
%
\def\slabelsL(#1,#2)#3#4{%
\topiter{#1}{#2}{#3}{\psdot(#1,\c)\rput[r]{0}(#1,\c){$\ifnum\n=0 2\fi#4_{\n}[p]$~~}}
\botiter{#1}{#2}{#3}{\psdot(#1,\c)\rput[r]{0}(#1,\c){$#4_{N-\n}[p]$~~}}}
%
\def\slabelsR(#1,#2)#3#4{%
\topiter{#1}{#2}{#3}{\rput[l]{0}(! #1 0.5 add \c\sp 0.5 add){$#4[2Np\ifnum\n=0\relax\else+\n\fi]$}}
\botiter{#1}{#2}{#3}{\rput[l]{0}(! #1 0.5 add \c\sp 0.5 add){$#4[2Np+N-\n]$}}}
%
\def\sconn(#1,#2)#3{%
\topiter{#1}{#2}{#3}{\psline(#1,\c)(! #1 2 add \c\sp)}
\botiter{#1}{#2}{#3}{\psline(#1,\c)(! #1 2 add \c\sp)}
\psline[linestyle=dotted](! #1 1 add #2 #3 add 1 sub \t\sp sub)(! #1 1 add #2 \b\sp add)}
%
\def\sblock(#1,#2)#3#4#5{%
\psframe(! #1 #2 0.2 sub)(! #1 2 add #2 #3 0.8 sub add)
\rput[B]{#4}(! #1 1 add #2 #3 1 sub 2 div add){#5}}
\begin{frame}
\frametitle{ADSL transmitter}
\begin{figure}
\scalebox{.5}{%
\psset{xunit=1.2cm,yunit=0.8cm,linewidth=2.5pt}
\begin{pspicture}(0,0)(13,15)
\Large
\sconn(0,8){8}\slabelsL(0,8){8}{a}
\sconn(2,8){8}
\sblock(4,0){16}{90}{\Huge IFFT}
\sconn(6,0){16}\sconn(8,0){16}
\slabelsR(6,0){16}{s}
\sblock(10,0){16}{90}{\Large parallel to serial}
%
\pnode(1,14){A}\pnode(4,0){B}
\ncangle[angleA=-90,angleB=180]{A}{B}
\pnode(1.2,13){A}\pnode(4,1){B}
\ncangle[angleA=-90,angleB=180]{A}{B}
\pnode(2.4,9){A}\pnode(4,6){B}
\ncangle[angleA=-90,angleB=180]{A}{B}
\pnode(2.6,8){A}\pnode(4,7){B}
\ncangle[angleA=-90,angleB=180]{A}{B}
\psline[linestyle=dotted](! 2 1 add 1 8 add 1 sub \t\sp sub)(! 2 1 add 0 \b\sp add)
\topiter{2}{0}{8}{\rput[r]{0}(! 2 1.8 add \c\sp 0.3 add){$*$}}
\botiter{2}{0}{8}{\rput[r]{0}(! 2 1.8 add \c\sp 0.3 add){$*$}}
%
\psline(12,7.5)(14,7.5)\psdot(14,7.5)\rput[l]{0}(14,7.5){~~$s[n]$}
\end{pspicture}}
\end{figure}
\vspace{-1em}
\begin{dspBlocks}{.7}{.4}
\hspace{4em} \footnotesize rate: $NW$ symbols/sec & & & & & \hspace{5em}\footnotesize $2NW=F_s$ samples/sec
\end{dspBlocks}
\end{frame}
\begin{frame}
\frametitle{ADSL specs}
\begin{itemize}[<+->]
\item $F_{\max} = 1104$~KHz
\item $N = 256$
\item subchannel width $W = 4312.5$~Hz
\item each QAM can send from 0 to 15 bits per symbol
\item forbidden channels: 0 to 7 (voice)
\item channels 7 to 31: upstream data
\item max theoretical throughput: 14.9~Mbps (downstream)
\end{itemize}
\end{frame}
\end{document}

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