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R2653/teaching/COM418/lectures/quantization/adda.tex
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\documentclass[xcolor=dvipsnames,aspectratio=169]{beamer}
%\documentclass[aspectratio=169,handout]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com418}
\begin{document}
\begin{frame} \frametitle{AD and DA conversion}
\begin{itemize}
\item ADC's and DAC's are the gateways between analog and digital
\item crucial components in audio acquisition and processing
\item goals are maximize SNR while reducing cost and power consumption
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Main technologies}
\begin{itemize}
\item ADC's and DAC's are the gateways between analog and digital
\item crucial components in audio acquisition and processing
\item goals are maximize SNR while reducing cost and power consumption
\end{itemize}
\end{frame}
\def\putbox#1#2#3{\makebox[0in][l]{\makebox[#1][l]{}\raisebox{\baselineskip}[0in][0in]{\raisebox{#2}[0in][0in]{#3}}}}
\begin{frame} \frametitle{AD 2-bit Flash Converter}
\note<1>{\vspace{10em} The multi-resistor voltage divider generates equally spaced boundary points (the ``$i_k$'' in our previous discussion). Each op-amp is a comparator that goes to $+V_{cc}$ when the input is greater than the relative boundary point. So the output of the battery of comparators gives the interval number the input belongs to as a series of ``1'' (unary notation). The XOR and diodes form a simple logic network for binary encoding. work out an example with $x = 0.2V$, output is 10}
\begin{figure}
\centering
\begin{minipage}{11.5cm}
\raggedright
\epsfig{file=adL.ps}\\
% translate x=1024 y=944 scale 0.20
\putbox{0.62in}{2.10in}{\em R}%
\putbox{0.66in}{1.36in}{\em R}%
\putbox{0.66in}{0.80in}{\em R}%
\putbox{0.62in}{0.06in}{\em R}%
\putbox{0.06in}{1.97in}{$+V_0$}%
\putbox{0.93in}{2.32in}{$x[n] {\color{red}=0.2V}$}%
\putbox{2.93in}{0.93in}{LSB {\color{red}$=0$}}%
\putbox{2.93in}{1.49in}{MSB {\color{red}$=1$}}%
\putbox{2.14in}{2.23in}{11}%
\putbox{2.14in}{1.66in}{10}%
\putbox{2.19in}{1.10in}{01}%
\putbox{0.32in}{1.62in}{$+0.5V_0$}%
\putbox{0.66in}{1.10in}{$0$}%
\putbox{0.32in}{0.58in}{$-0.5V_0$}%
\putbox{0.10in}{0.23in}{$-V_0$}%
{\putbox{39mm}{48mm}{\color{red}\footnotesize$-$}}%
{\putbox{39mm}{34mm}{\color{red}\footnotesize$+$}}%
{\putbox{39mm}{20mm}{\color{red}\footnotesize$+$}}%
{\putbox{53mm}{40mm}{\color{red}\footnotesize$+$}}%
{\putbox{53mm}{26mm}{\color{red}\footnotesize$-$}}%
\end{minipage}
\end{figure}
\end{frame}
\begin{frame} \frametitle{AD 4-bit Flash Converter}
\end{frame}
\begin{frame} \frametitle{Limits of Flash Converters}
\begin{itemize}
\item require $2^R - 1$ comparators
\item difficult to manufacture for $R > 8$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Scrounging bits via oversampling}
\begin{figure}
\begin{dspBlocks}{0.7}{0.2}
$x(t)$~ & \BDsampler & \BDfilter{$\mathcal{Q}\{\cdot\}$} & ~~~$\hat{x}[n] = x[n] + e[n]$ \\
& $T_s = 1/F_s$
\psset{linewidth=1.5pt}
\ncline{-}{1,1}{1,2} \ncline{1,2}{1,3}
\ncline{->}{1,3}{1,4}
\end{dspBlocks}
\end{figure}
\vspace{2em}
(slightly unrealistic) quantization noise model:
\begin{itemize}
\item $e[n]$ white with PSD $P_e(\omega) = \Delta^2/12$
\item $e[n]$ independent on sampling rate
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Oversampled A/D}
\begin{figure}
$X(e^{j\omega}) = F_s\,X\left(\frac{\omega}{2\pi}F_s \right)$
\begin{dspPlot}[xtype=freq,xticks=4]{-1,1}{0,4.1}
\moocStyle \psset{linecolor=blue!80}
\psline[linewidth=2pt,linecolor=gray](-1,0.4)(1,0.4)
\psframe[linewidth=0pt,fillstyle=vlines,hatchcolor=gray](-1,0.4)(1,0)
\only<1>{\dspFunc{x \dspPorkpie{0}{1} 1.5 mul } \dspText(0.5,3){$F_s=2f_{\max}$}}
\only<2>{\dspFunc{x \dspPorkpie{0}{.5} 2 mul} \dspText(0.5,3){$F_s=4f_{\max}$}}
\only<3>{\dspFunc{x \dspPorkpie{0}{.333333} 3 mul} \dspText(0.5,3){$F_s=6f_{\max}$}}
\only<4->{\dspFunc{x \dspPorkpie{0}{.25} 4 mul} \dspText(0.5,3){$F_s=8f_{\max}$}}
\only<5->{\dspFunc[linecolor=green,linestyle=dashed]{x \dspRect{0}{0.5} } }
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Oversampled A/D}
\begin{figure}
\begin{dspPlot}[xtype=freq,xticks=4]{-1,1}{0,4.1}
\moocStyle \psset{linecolor=blue!80}
\psline[linewidth=2pt,linecolor=gray](-1,0.1)(1,0.1)
\psframe[linewidth=0pt,fillstyle=vlines,hatchcolor=gray](-1,0.1)(1,0)
\dspFunc{x \dspPorkpie{0}{1} 1.5 mul }
\dspText(-.5,3){after downsampling by 4:}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Oversampled A/D}
\begin{figure}
\begin{dspBlocks}{0.7}{0.2}
$x(t)$~ & \BDsampler & \BDfilter{$\mathcal{Q}\{\cdot\}$} & \BDfilter{LP$\{\pi/K\}$} & \BDdwsmp{$K$} & $x[n]$ \\
& $T_s = 1/(2Kf_{\max})$
\psset{linewidth=1.5pt}
\ncline{-}{1,1}{1,2} \ncline{1,2}{1,3}
\ncline{1,3}{1,4}\ncline{1,4}{1,5}
\ncline{->}{1,5}{1,6}
\end{dspBlocks}
\end{figure}
\vspace{1em}
\begin{itemize}
\item $\mbox{SNR}_\text{O} \approx K\,\mbox{SNR}$
\item 3dB per octave (half bit every doubling of $F_s$)
\item diminishing returns: key assumption (independence) breaks down fast...
\item not enough to push an 8-bit flash AD to 16-bit resolution (which, anyway, would require $F_s \approx 3~\text{GHz}$)
\end{itemize}
\end{frame}
\begin{frame} \frametitle{SAR}
\end{frame}
\begin{frame} \frametitle{SAR}
\begin{itemize}
\item $F_s$ up to 5 MHz
\item up to 16 or 18 bps
\item diminishing returns: key assumption (independence) breaks down fast...
\item not enough to push an 8-bit flash AD to 16-bit resolution (which, anyway, would require $F_s \approx 3~\text{GHz}$)
\end{itemize}
\end{frame}
\begin{frame} \frametitle{SAR}
\end{frame}
\begin{frame} \frametitle{ADC comparison}
\centering
\begin{tabular}{c|c|c|c|c}
\textbf{type} & \textbf{max bps} &\textbf{ max $F_s$} & \textbf{power} & \textbf{cost} \\ \hline\hline
\textbf{flash} & up to 8 & up to 10 GHz & high & high \\ \hline
\textbf{SAR} & up to 18 & up to 10 MHz & medium & medium-low \\ \hline
\textbf{sigma-delta} & up to 32 & up to 1 MHz & low & low
\end{tabular}
\end{frame}
\end{document}
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