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R2653/teaching/COM303/2021/lectures/week10/recap_oversampling.tex
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\documentclass[aspectratio=169]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
\begin{document}
\begin{frame} \frametitle{Oversampling}
\begin{itemize}
\item oversampled A/D
\begin{itemize}
\item reduce quantization error
\end{itemize}
\vspace{2em}
\item oversampled D/A
\begin{itemize}
\item use cheaper hardware for interpolation
\end{itemize}
\end{itemize}
\end{frame}
\intertitle{Oversampled A/D}
\begin{frame} \frametitle{Oversampled A/D}
\begin{figure}
\begin{dspBlocks}{0.7}{0.2}
$x(t)$~ & \BDsampler & \BDfilter{$\mathcal{Q}\{\cdot\}$} & ~$\hat{x}[n]$ \\
& $T_s = 1/F_s$
\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}
\[
\hat{x}[n] = x[n] + e[n]
\]
\end{frame}
\begin{frame} \frametitle{Oversampled A/D}
Key assumptions:
\[
e[n] \mbox{~i.i.d. process, {\color{darkred} independent of $x[n]$}}
\]
\[
P_e(e^{j\omega}) = \frac{\Delta^2}{12} \qquad \mbox{over $[-\pi,\pi]$ (no aliasing)}
\]
\vspace{1em}
Key observation:
\[
X(e^{j\omega}) = F_s\,X\left(\frac{\omega}{2\pi}F_s \right)
\]
\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](-1,0.4)(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$\{\pi/N\}$} & \BDdwsmp{$N$} & $x[n]$ \\
& $T_s = 1/(2Nf_{\max})$
\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 $\mbox{SNR}_{O} \approx N\,\mbox{SNR}$
\item 3dB per octave (doubling of $F_s$)
\item but key assumption (independence) breaks down fast...
\end{itemize}
\end{frame}
\intertitle{Oversampled D/A}
\begin{frame} \frametitle{Oversampled D/A}
Sinc interpolation:
\[
X_c(f) = \frac{1}{F_s}\,X(e^{j2\pi f/F_s}) \rect\left(\frac{f}{F_s}\right)
\]
\end{frame}
\begin{frame} \frametitle{Sinc interpolation}
\begin{center}
\begin{figure}
\only<1->{
\begin{dspPlot}[height=2.1cm,xtype=freq,yticks=1,ylabel={$X(e^{j\omega})$}]{-1,1}{0,1.2}
\moocStyle \psset{linecolor=blue!80}
\dspFunc{x \dspPorkpie{0}{0.95}}
\end{dspPlot}}
\only<2->{
\begin{dspPlot}[height=2.1cm,xtype=freq,xticks=custom,yticks=1,]{-6,6}{0,1.2}
\moocStyle
\dspCustomTicks[axis=x]{0 0 1 $F_s/2$ 2 $F_s$ 4 $2F_s$ 6 $3F_s$}
\only<2-3>{\dspFunc{x \dspPeriodize \dspPorkpie{0}{0.95}} \dspText(0,-.6){$X(e^{j2\pi f / F_s})$} }
\only<3>{\dspFunc[linecolor=green,linestyle=dashed]{x \dspRect{0}{2}} }
\only<4->{\dspFunc{x \dspPorkpie{0}{0.95}} \dspText(0,-.6){$X_c(f)$} }
\end{dspPlot}}
\end{figure}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Oversampled D/A}
In general:
\[
X_c(f) = \frac{1}{F_s}\,X(e^{j2\pi f/F_s})\, I\left(\frac{f}{F_s}\right)
\]
The cheapest (hence most common) interpolator is the zero-order hold:
\[
i_0(t) = \rect(t)
\]
\[
I_0(f) = \sinc(f)
\]
\end{frame}
\begin{frame} \frametitle{ZOH interpolation}
\begin{center}
\begin{figure}
\only<1->{
\begin{dspPlot}[height=2.1cm,xtype=freq,yticks=1,ylabel={$X(e^{j\omega})$}]{-1,1}{0,1.2}
\moocStyle \psset{linecolor=blue!80}
\dspFunc{x \dspPorkpie{0}{0.95}}
\end{dspPlot}}
\only<2->{
\begin{dspPlot}[height=2.1cm,xtype=freq,xticks=custom,yticks=1,]{-6,6}{0,1.2}
\moocStyle
\dspCustomTicks[axis=x]{0 0 1 $F_s/2$ 2 $F_s$ 4 $2F_s$ 6 $3F_s$}
\only<2-3>{\dspFunc{x \dspPeriodize \dspPorkpie{0}{0.95}} \dspText(0,-.6){$X(e^{j2\pi f/F_s})$} }
\only<3>{\dspFunc[linecolor=green!80,linestyle=dashed]{x \dspSinc{0}{2} abs } }
\only<4->{\dspFunc{x \dspPeriodize \dspPorkpie{0}{0.95} x \dspSinc{0}{2} abs mul} \dspText(0,-.6){$X_c(f)$} }
\only<5->{\dspFunc[linecolor=orange!80,linestyle=dashed]{x abs 1 ge {0} {1 x \dspSinc{0}{2} abs div 0.7 mul} ifelse} }
\end{dspPlot}}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Oversampled D/A}
\center
\begin{dspBlocks}{1.2}{0.3}
$x[n]$~ & \BDupsmp{$N$} & \BDfilter{$F(e^{j\omega})$} & \BDfilter{$I_0(f)$} & $x(t)$ \\
& & & $F_o = NF_s$
\end{dspBlocks}
\psset{linewidth=1pt}
\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{frame}
\begin{frame} \frametitle{Oversampled D/A, using a ZOH ($N=2$)}
\begin{center}
\begin{figure}
\only<1->{
\begin{dspPlot}[height=2.1cm,xtype=freq,yticks=1,ylabel=%
\only<1-2>{{$X_{2\uparrow}(e^{j\omega})$}}%
\only<3->{{$X_{o}(e^{j\omega})$}}%
]{-1,1}{0,1.2}
\smallStems \psset{linecolor=blue}
\only<1-2>{\dspFunc[linecolor=lightgray]{x 2 mul \dspPeriodize \dspPorkpie{0}{0.95}} \dspFunc{x 2 mul \dspPorkpie{0}{0.95}}}
\only<2>{\dspFunc[linecolor=darkgreen!60,linestyle=dashed]{x abs 0.5 ge {0} {1 x \dspSinc{0}{1} abs div 0.7 mul} ifelse} }
\only<3->{\dspFunc{x abs 0.5 ge {0} {x 2 mul \dspPorkpie{0}{0.95} x \dspSinc{0}{1} abs div } ifelse} }
\end{dspPlot}}
\only<4->{
\begin{dspPlot}[height=2.1cm,xtype=freq,xticks=custom,yticks=1,ylabel={$X_o(f)$}]{-6,6}{0,1.2}
\smallStems \moocStyle
\dspCustomTicks[axis=x]{0 0 1 $F_s/2$ 2 $F_o/2$ 4 $F_o$}
\only<4>{
\dspFunc{x \dspPeriod{2} \dspPorkpie{0}{0.95}}
\dspFunc[linecolor=green,linestyle=dashed]{x \dspSinc{0}{4} abs } }
\only<5->{\dspFunc{x \dspPeriod{2} \dspPorkpie{0}{0.95} x \dspSinc{0}{4} abs mul} }
\only<6->{\dspFunc[linecolor=orange,linestyle=dashed]{x 0.8 mul 4 exp 1 add 1 exch div}}
\end{dspPlot}}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Oversampled A/D}\
key points:
\begin{itemize}
\item we can pre-compensate the in-band distortion in the digital domain
\item we can interpolate with a cheap ZOH
\item the higher the upsampling, the cheaper the analog lowpass needed to eliminate out-of-band distortion
\item only price: higher D/A rate
\end{itemize}
\end{frame}
\end{document}
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