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R2653/teaching/COM202/lectures/03 - DT signals/4_operators.tex
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\documentclass[aspectratio=169]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com202}
\begin{document}
\intertitle{Basic discrete-time operations}
\begin{frame} \frametitle{The four basic operators}
\begin{itemize}
\item scaling:
\[
y[n] = \alpha x[n]
\]\pause
\item sum:
\[
y[n] = x[n] + w[n]
\]\pause
\item product:
\[
y[n] = x[n]\cdot w[n]
\]\pause
\item shift by $k$ (delay):
\[
y[n] = x[n - k]
\]
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Border effects}
Finite-length signals require some additional caution:
\vspace{1em}
\begin{itemize}
\item sum and products require signals of equal length
\item (for later) when using filters, data at edges will be treated differently
\item shifts can be performed in two different ways
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Shifting finite-length signals via finite-support embedding}
\note<1>{Shifting a finite-length sequence. We need to embed the sequence into an infinite sequence for a notation of the type $x[n-M]$ to make sense. We have two obvious choices, embedding into a finite support (zeros everywhere) or into a periodic sequence. The trick is to show that it's like a shift register, with two types of fill-in}
\begin{center}
\mbox{\vphantom{$x[n-1234]$}}%
\only<2|handout:2>{$x[n]$}%
\only<3|handout:3>{$\bar{x}[n]$}%
\only<4|handout:4>{$\bar{x}[n-1]$}%
\only<5|handout:5>{$\bar{x}[n-2]$}%
\only<6|handout:6>{$\bar{x}[n-3]$}%
\only<7|handout:7>{$\bar{x}[n-4]$}%
\begin{figure}
\begin{psmatrix}[colsep=0,rowsep=1,mnodesize=1.7em]
\only<1|handout:1>{ & & & & $[x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7]$ & & & &}%
\only<2|handout:2>{ & & & $\ldots$ & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $\ldots$ & & &}%
\only<3|handout:3>{$\ldots$ & 0 & 0 & 0 & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & 0 & 0 & 0 & $\ldots$}%
\only<4|handout:4>{$\ldots$ & 0 & 0 & 0 & 0 & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & 0 & 0 & $\ldots$}%
\only<5|handout:5>{$\ldots$ & 0 & 0 & 0 & 0 & 0 & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & 0 & $\ldots$}%
\only<6|handout:6>{$\ldots$ & 0 & 0 & 0 & 0 & 0 & 0 & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $\ldots$}%
\only<7|handout:7>{$\ldots$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $\ldots$}%
\only<2-|handout:2->{\ncbox[linecolor=red!30,nodesep=0.3em,boxsize=.4,linearc=.2,opacity=0.5]{1,5}{1,12}}
\end{psmatrix}
\end{figure}
\end{center}
\vspace{3em}
\begin{columns}
\begin{column}{.5\paperwidth}
\centering
\begin{dspPlot}[width=5cm,height=2cm,xticks=1]{0,7}{0,1.1}
\moocStyle
\dspSignal{10 x sub 10 div }
\end{dspPlot}
\end{column}
\begin{column}{.5\paperwidth}
\centering
\begin{dspPlot}[width=5cm,height=2cm,xticks=1]{0,7}{0,1.1}
\moocStyle
\only<2-3|handout:2-3>{\dspSignal{10 x sub 10 div }}%
\only<4|handout:4>{\dspSignal{x 1 ge {10 x 1 sub sub 10 div } {0} ifelse}}%
\only<5|handout:5>{\dspSignal{x 2 ge {10 x 2 sub sub 10 div } {0} ifelse}}%
\only<6|handout:6>{\dspSignal{x 3 ge {10 x 3 sub sub 10 div } {0} ifelse}}%
\only<7|handout:7>{\dspSignal{x 4 ge {10 x 4 sub sub 10 div } {0} ifelse}}%
\end{dspPlot}
\end{column}
\end{columns}
\end{frame}
\begin{frame} \frametitle{Shifting finite-length signals via periodization}
\begin{center}
\mbox{\vphantom{$x[n-1234]$}}%
\only<2|handout:2>{$x[n]$}%
\only<3|handout:3>{$\tilde{x}[n]$}%
\only<4|handout:4>{$\tilde{x}[n-1]$}%
\only<5|handout:5>{$\tilde{x}[n-2]$}%
\only<6|handout:6>{$\tilde{x}[n-3]$}%
\only<7|handout:7>{$\tilde{x}[n-4]$}%
\begin{figure}
\begin{psmatrix}[colsep=0,rowsep=1,mnodesize=1.7em]
\only<1|handout:1>{ & & & & $[x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7]$ & & & &}%
\only<2|handout:2>{ & & & $\ldots$ & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $\ldots$ & & &}%
\only<3|handout:3>{$\ldots$ & $x_5$ & $x_6$ & $x_7$ & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $x_0$ & $x_1$ & $x_2$ & $\ldots$}%
\only<4|handout:4>{$\ldots$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $x_0$ & $x_1$ & $\ldots$}%
\only<5|handout:5>{$\ldots$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $x_0$ & $\ldots$}%
\only<6|handout:6>{$\ldots$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $\ldots$}%
\only<7|handout:7>{$\ldots$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $x_0$ & $x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $\ldots$}%
\only<2-|handout:2->{\ncbox[linecolor=red!30,nodesep=0.3em,boxsize=.4,linearc=.2,opacity=0.5]{1,5}{1,12}}
\end{psmatrix}
\end{figure}
\end{center}
\vspace{3em}
\begin{columns}
\begin{column}{.5\paperwidth}
\centering
\begin{dspPlot}[width=5cm,height=2cm,xticks=1]{0,7}{0,1.1}
\moocStyle
\dspSignal{10 x sub 10 div }
\end{dspPlot}
\end{column}
\begin{column}{.5\paperwidth}
\centering
\begin{dspPlot}[width=5cm,height=2cm,xticks=1]{0,7}{0,1.1}
\moocStyle
\only<2-3|handout:2-3>{\dspSignal{x cvi 8 mod 10 exch sub 10 div }}%
\only<4|handout:4>{\dspSignal{x 7 add cvi 8 mod 10 exch sub 10 div }}% mod tricky with negatives
\only<5|handout:5>{\dspSignal{x 6 add cvi 8 mod 10 exch sub 10 div }}%
\only<6|handout:6>{\dspSignal{x 5 add cvi 8 mod 10 exch sub 10 div }}%
\only<7|handout:7>{\dspSignal{x 4 add cvi 8 mod 10 exch sub 10 div }}%
\end{dspPlot}
\end{column}
\end{columns}
\end{frame}
\end{document}
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