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R2653/teaching/COM303/lectures/16 - interpolation - alt not/1_analogworld.tex
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\documentclass[aspectratio=169]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
\begin{document}
\begin{frame} \frametitle{Two views of the world}
\begin{columns}
\begin{column}{0.5\paperwidth}
\centering
\includegraphics[width=0.4\paperwidth]{Picture_world_analog.eps}
\end{column}
\begin{column}{0.5\paperwidth}
\centering
\includegraphics[width=0.4\paperwidth]{Picture_world_digital.eps}
\end{column}
\end{columns}
\vspace{2em}
\centering Analog/continuous versus discrete/digital
\end{frame}
\begin{frame} \frametitle{Two views of the world}
\begin{columns}
\begin{column}{0.3\paperwidth}
analog worldview:
\vspace{1em}
\begin{itemize}
\item calculus
\item distributions
\item system theory
\item electronics
\end{itemize}
\end{column}
\begin{column}{0.3\paperwidth}
\only<2>{
digital worldview:
\vspace{1em}
\begin{itemize}
\item arithmetic
\item combinatorics
\item computer science
\item DSP
\end{itemize}}
\end{column}
\end{columns}
\end{frame}
\begin{frame} \frametitle{Two views of the world}
\begin{columns}
\begin{column}{0.4\paperwidth}
digital worldview:
\vspace{1em}
\begin{itemize}
\item countable integer index $n$
\item sequences $x[n] \in \ell_2(\mathbb{Z})$
\item frequency $\omega \in [-\pi, \pi]$
\item DTFT: $\ell_2(\mathbb{Z}) \mapsto L_2([-\pi,\pi])$
\end{itemize}
\end{column}
\begin{column}{0.4\paperwidth}
\only<2>{
analog worldview:
\vspace{1em}
\begin{itemize}
\item real-valued time $t$ (sec)
\item functions $x(t) \in L_2(\mathbb{R})$
\item frequency $f \in \mathbb{R}$ (Hz)
\item FT: $L_2(\mathbb{R}) \mapsto L_2(\mathbb{R})$
\end{itemize}}
\end{column}
\end{columns}
\end{frame}
\begin{frame} \frametitle{Bridging the gap: interpolation}
\setbeamercovered{invisible}
\begin{center}
\begin{figure}
\begin{dspBlocks}{2}{1}
$x[n]~~$ & \BDfilter{sound card} & \raisebox{-1.2em}{\includegraphics[height=6em]{speaker.eps}} \\
& [mnode=circle] $T_s$ & \hspace{-5cm} {\em system clock} \\
\ncline{->}{2,2}{1,2}
\ncline{->}{1,1}{1,2}\ncline{->}{1,2}{1,3}
\end{dspBlocks}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Bridging the gap: sampling}
\setbeamercovered{invisible}
\begin{center}
\begin{figure}
\begin{dspBlocks}{2}{1}
\raisebox{-1.5em}{\includegraphics[height=5em]{mike.eps}} & \BDfilter{sound card} & ~~$x[n]$\\
& [mnode=circle] $T_s$ & \hspace{-3cm} {\em system clock} \\
\ncline{->}{2,2}{1,2}
\ncline{->}{1,1}{1,2}\ncline{->}{1,2}{1,3}
\end{dspBlocks}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Bridging the gap}
\begin{figure}
\center
\psset{unit=3.5cm}%
\begin{pspicture}(-1,-1)(1,1)%
\uput[0](0.6,0){$x[n]$}
\uput[90](0,0.6){\color{darkred} sampling}
\uput[180](-0.6,0){$x(t)$}
\uput[270](0,-0.6){\color{darkred} interpolation}
\psarc[linewidth=3pt,linecolor=gray]{<-}(0,0){0.73}{10}{65}
\psarc[linewidth=3pt,linecolor=gray]{<-}(0,0){0.73}{115}{170}
\psarc[linewidth=3pt,linecolor=gray]{<-}(0,0){0.73}{190}{235}
\psarc[linewidth=3pt,linecolor=gray]{<-}(0,0){0.73}{305}{350}
\end{pspicture}
\end{figure}
\end{frame}
\begin{frame} \frametitle{Today, processing is as digital as possible}
\begin{itemize}
\item analog to digital
\item digital to analog
\item analog to digital to analog
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Digital processing of signals from the analog world}
\begin{itemize}
\item input is continuous-time: $x(t)$
\item output is discrete-time: $y[n]$
\item processing is on sequences: $x[n], y[n]$
\end{itemize}
\vspace{1em}
\centering
\begin{figure}
\begin{dspBlocks}{1.5}{.5}
analog world~~ & \BDfilterMulti{processing \\ for analysis} & ~~digital world \\
\ncline{->}{1,1}{1,2}\ncline{->}{1,2}{1,3}
\end{dspBlocks}
\end{figure}
examples: storage and compression (MP3, JPG), control systems, monitoring
\end{frame}
\begin{frame} \frametitle{Digital processing of signals to the analog world}
\begin{itemize}
\item input is discrete-time: $x[n]$
\item output is continuous-time: $y(t)$
\item processing is on sequences: $x[n], y[n]$
\end{itemize}
\vspace{1em}
\centering
\begin{figure}
\begin{dspBlocks}{1.5}{.5}
digital world~~ & \BDfilterMulti{processing \\ for synthesis} & ~~analog world \\
\ncline{->}{1,1}{1,2}\ncline{->}{1,2}{1,3}
\end{dspBlocks}
\end{figure}
examples: music synthesizers, computer graphics, video games
\end{frame}
\begin{frame} \frametitle{Digital processing of signals from/to the analog world}
\begin{itemize}
\item input is continuous-time: $x(t)$
\item output is continuous-time: $y(t)$
\item processing is on sequences: $x[n], y[n]$
\end{itemize}
\vspace{1em}
\centering
\begin{figure}
\begin{dspBlocks}{1.5}{.5}
analog world~~ & \BDfilter{processing} & ~~analog world \\
\ncline{->}{1,1}{1,2}\ncline{->}{1,2}{1,3}
\end{dspBlocks}
\end{figure}
examples: telephony, VOIP, sound effects, digital photography
\end{frame}
\intertitle{continuous-time signal processing}
\begin{frame} \frametitle{About continuous time}
\begin{itemize}[<+->]
\item time: real variable $t$
\item signal $x(t)$: complex functions of a real variable
\item finite energy: $x(t) \in L_2(\mathbb{R})$ (square integrable functions)
\item inner product in $L_2(\mathbb{R})$
\[
\langle x(t), y(t) \rangle = \int_{-\infty}^{\infty} x^*(t)y(t) dt
\]
\item energy: $||x(t)||^2 = \langle x(t), x(t) \rangle$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Analog LTI filters}
\begin{figure}
\center
\begin{dspBlocks}{2.5}{1}
$x(t)$~~ & \rnode{F}{\BDfilter{$\mathcal{H}$}} & ~~$y(t)$
\ncline{->}{1,1}{1,2}\ncline{->}{1,2}{1,3}
\end{dspBlocks}
\end{figure}
\vspace{1em}
\begin{align*}
y(t) &= (x \ast h)(t) \\ \pause
&= \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau \\ \pause
&= \langle h^*(t-\tau), x(\tau) \rangle
\end{align*}
\end{frame}
\begin{frame} \frametitle{Real-world frequency}
\begin{center}
frequency: number of repetitions per \textit{second}
\end{center}
\vspace{2em}
\begin{itemize}
\item $f$ expressed in Hz (1/sec)
\item alternatively, angular frequency in rad/s: $\Omega = 2\pi f$
\item period for periodic signals is $T = \displaystyle \frac{1}{f} = \frac{2\pi}{\Omega}$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Fourier analysis}
\begin{itemize}[<+->]
\item in discrete time max angular frequency is $\pm\pi$
\item in continuous time no max frequency: $f \in \mathbb{R}$
\item concept is the same: similarity to sinusoidal components
\begin{align*}
X(f) &= \langle e^{j2\pi f t}, x(t) \rangle \\
&= \int_{-\infty}^{\infty} x(t) e^{-j2\pi f t} dt \qquad \leftarrow\mbox{\em not periodic!} \\ \\
x(t) &= \int_{-\infty}^{\infty} X(f) e^{j2\pi f t} df
\end{align*}
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Fourier analysis (in rad/s)}
\begin{align*}
X(j\Omega) &= \langle e^{j\Omega t}, x(t) \rangle \\
&= \int_{-\infty}^{\infty} x(t) e^{-j\Omega t} dt \qquad \leftarrow\mbox{\em not periodic!} \\ \\
x(t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty} X(j\Omega) e^{j\Omega t} d\Omega
\end{align*}
\begin{itemize}
\item Laplace transform computed on the imaginary axis
\end{itemize}
\end{frame}
\def\s{10 }
\begin{frame} \frametitle{Example}
\begin{center}
$x(t) = e^{-at^2}$
\begin{dspPlot}[xticks=10,xout=true,sidegap=0,ylabel={$x(t)$}]{-50,50}{0,1.1}
\moocStyle
\dspFunc{x dup mul neg 2 \s \s mul mul div 2.718281828459046 exch exp}
\end{dspPlot}
\end{center}
\end{frame}
\begin{frame} \frametitle{Example}
\begin{center}
$X(f) = \sqrt{\pi / a}\, e^{-\frac{\pi^2}{a}f^2}$
\begin{dspPlot}[xticks=10,xout=true,sidegap=0,yticks=custom,ylabel={$X(f)$}]{-50,50}{0,26}
\moocStyle
\dspFunc{x dup mul neg 2 \s \s mul mul 2 div div 2.718281828459046 exch exp \s 2.506628274631 mul mul}
\dspCustomTicks[axis=y]{25 $\sqrt{\pi / a}$}
\end{dspPlot}
\end{center}
\end{frame}
\def\s{10 }
\begin{frame} \frametitle{Example}
\begin{center}
$x(t) = \cos(2\pi f_0 t)$
\begin{dspPlot}[xticks=10,xout=true,sidegap=0,ylabel={$x(t)$}]{-4,4}{-1.1,1.1}
\moocStyle
\dspFunc{x 360 mul cos}
\dspCustomTicks[axis=x]{1 $1/f_0$ 2 $2/f_0$ 3 $3/f_0$ -1 $-1/f_0$}
\end{dspPlot}
\end{center}
\end{frame}
\begin{frame} \frametitle{Example}
\begin{center}
$X(f) = (1/2)\delta(f \pm f_0)$
\begin{dspPlot}[xout=true,xticks=10,yticks=1,sidegap=0,ylabel={$X(f)$}]{-2,2}{0,1.1}
\moocStyle
\dspDiracs{1 1 -1 1}
\dspCustomTicks[axis=x]{1 $f_0$ -1 $-f_0$}
\end{dspPlot}
\end{center}
\end{frame}
\begin{frame} \frametitle{Convolution theorem}
\begin{figure}
\center
\begin{dspBlocks}{1}{1}
$x(t)$~~ & \rnode{F}{\BDfilter{$\mathcal{H}$}} & ~~$y(t)$
\ncline{->}{1,1}{1,2}\ncline{->}{1,2}{1,3}
\end{dspBlocks}
\end{figure}
\vspace{2em}
\[
Y(f) = X(f)\,H(f)
\]
\end{frame}
\begin{frame} \frametitle{A new concept: bandlimited functions}
\centering
a continuous-time signal is bandlimited if there exists a frequency $F_s$ such that:
\[
X(f) = 0 \quad \mbox{for } |f| > F_s/2
\]
\end{frame}
\begin{frame} \frametitle{Prototypical bandlimited function}
\center
\begin{figure}
\begin{dspPlot}[xtype=freq,xticks=custom,yticks=custom]{-1.5,1.5}{0,1.4}
\moocStyle
\dspFunc{x \dspRect{0}{1}}
\dspCustomTicks[axis=x]{0 0 -0.5 $-F_s/2$ 0.5 $F_s/2$}
\dspCustomTicks[axis=y]{1 $G$}%{1 $\pi/\Omega_N$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{The prototypical bandlimited function}
\note<1>{We will normalize so that a) area is $2\pi$ \\ or (equivalent) b) max time-domain \\ value is 1}
\[
\Phi(f) = G\, \rect\left(\frac{f}{F_s}\right)
\]
\vspace{1em}
\pause
\begin{align*}
\varphi(t) &= \int_{-\infty}^{\infty} \Phi(f) e^{j2\pi f t} df \\
&= \ldots \\
&= G F_s\sinc\left(t F_s\right)
\end{align*}
\end{frame}
\begin{frame} \frametitle{The prototypical bandlimited function}
\begin{itemize}
\item total bandwidth: $F_s$
\item define $T_s = 1/F_s$
\item normalization: $G = 1/F_s = T_s$
\end{itemize}
\end{frame}
\begin{frame} \frametitle{The prototypical bandlimited function}
\begin{align*}
\Phi(f) &= \frac{1}{F_s} \mbox{rect}\left(\frac{f}{F_s}\right) \\ \\
\varphi(t) &= \mbox{sinc}\left(\frac{t}{T_s}\right)
\end{align*}
\end{frame}
\begin{frame} \frametitle{The prototypical bandlimited function}
\center
$F_s = 1/T_s$
\begin{figure}
\begin{dspPlot}[xtype=freq,xticks=custom,yticks=custom]{-1.5,1.5}{0,1.4}
\moocStyle
\dspFunc{x \dspRect{0}{1}}
\dspCustomTicks[axis=x]{0 0 -0.5 $-F_s/2$ 0.5 $F_s/2$}
\dspCustomTicks[axis=y]{1 $1/F_s$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{The prototypical bandlimited function}
\center
\begin{figure}
\begin{dspPlot}[xticks=custom,sidegap=0,xout=true]{-8,8}{-0.3,1.2}
\moocStyle
\dspFunc{x \dspSinc{0}{1}}
\dspCustomTicks[axis=x]{0 0 1 $T_s$ -1 $-T_s$ 2 $2T_s$ 3 $3T_s$ 4 $4T_s$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{When $T_s = 1$}
\begin{align*}
\Phi(f) &= \mbox{rect}\left( f \right) \\ \\
\varphi(t) &= \mbox{sinc}(t)
\end{align*}
\end{frame}
\begin{frame} \frametitle{The prototypical bandlimited function ($T_s=1$)}
\center
\begin{figure}
\begin{dspPlot}[xtype=freq,xticks=custom,yticks=custom]{-1.5,1.5}{0,1.4}
\moocStyle
\dspFunc{x \dspRect{0}{1}}
\dspCustomTicks[axis=x]{0 0 -0.5 $-1/2$ 0.5 $1/2$}
\dspCustomTicks[axis=y]{1 $1$}
\end{dspPlot}
\end{figure}
\end{frame}
\begin{frame} \frametitle{The prototypical bandlimited function ($T_s=1$)}
\center
\begin{figure}
\begin{dspPlot}[xticks=1,sidegap=0,xout=true]{-8,8}{-0.3,1.2}
\moocStyle
\dspFunc{x \dspSinc{0}{1}}
\end{dspPlot}
\end{figure}
\end{frame}
\end{document}
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