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Update convolution_discrete_slides.tex
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‎convolution_dt/convolution_discrete_slides.tex‎

Lines changed: 79 additions & 33 deletions
Original file line numberDiff line numberDiff line change
@@ -22,19 +22,19 @@
2222
\usepackage{pgfplots}
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\pgfplotsset{compat=1.18}
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25-
\usepackage{../sig_sys_macros}
25+
\usepackage{../tutorial_latex_deu/sig_sys_macros}
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\title[Discrete-Time Convolution]{Discrete-Time Convolution}
28-
\author[SigSys Tutorial]{Frank Schultz}
29-
\date[Summer Term 2024]{Signals and Systems Tutorial, Summer Term 2024}
28+
\author[SigSys]{Frank Schultz}
29+
\date[Summer Term 2025]{Signals and Systems, Summer Term 2025}
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\institute[]{Prof. Sascha Spors, Institute of Communications Engineering\\
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Faculty of Computer Science and Electrical Engineering, University of Rostock, Germany}
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\begin{document}
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\maketitle
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\begin{frame}{Discrete-Time Convolution / Zeitdiskrete Faltung / Faltung von Folgen}
37-
Task 8.2
37+
cf. \url{sig_sys_ex.pdf} Task 8.2
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%% Signal x[k], h[k]
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\begin{center}
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\begin{tikzpicture}[scale=0.5]
@@ -44,26 +44,28 @@
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\draw[stem] plot coordinates{(0,0) (1,1) (2,1) (3,2) (4,-1) (5,0) };
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\foreach \y in {-1, 1, 2}{\draw (\tic,\y) -- (-\tic,\y) node[left]{$\y$};};
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\draw (1,\tic) -- (1,-\tic) node[below]{$1$};
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\draw (2,\tic) -- (2,-\tic) node[below]{$2$};
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\begin{scope}[xshift=11cm]
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\draw[->] (-1.5,0) -- (4,0) node[right]{$k$};
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\draw[->] (0,-1.5) -- (0,3) node[above]{$h[k]$};
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\foreach \y in {-1,1,2}{\draw (\tic,\y) -- (-\tic,\y) node[left]{$\y$};};
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\draw[stem] plot coordinates{ (-1,0) (0,2) (1,1) (2,-1) (3,0)};
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\draw (1,\tic) -- (1,-\tic) node[below]{$1$};
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\draw (2,\tic) -- (2,-\tic) node[above]{$2$};
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\end{scope}
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\end{tikzpicture}
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\end{center}
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57-
Calculate the convolution of two finite-length signals / two sequences $x[k]$ and $h[k]$ as
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Calculate the linear convolution of the two finite-length sequences $x[k]$ and $h[k]$ as
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$$y[k] = x[k] \ast h[k]$$
59-
$$y[k] = \sum_{\kappa = -\infty}^{+\infty} x[\kappa] \ast h[-\kappa + k] =
60-
\sum_{\kappa = -\infty}^{+\infty} x[-\kappa+k] \ast h[\kappa]
61+
$$y[k] = \sum_{\textcolor{C1}{\kappa} = -\infty}^{+\infty} x[\textcolor{C1}{\kappa}] \ast h[\textcolor{C1}{-\kappa} + k] =
62+
\sum_{\textcolor{C1}{\kappa} = -\infty}^{+\infty} x[\textcolor{C1}{-\kappa}+k] \ast h[\textcolor{C1}{\kappa}]
6163
$$
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\end{frame}
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66-
\begin{frame}{L\"osungsweg I, k=0}
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\begin{frame}{L\"osungsweg I, k=0 (keine \"Uberlappung)}
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%% k = 0
6870
\begin{center}
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\begin{tikzpicture}[scale=0.4]
@@ -82,22 +84,22 @@
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\draw[->] (0,-2.5) -- (0,4) node[above]{$x[\kappa]\cdot h[-\kappa+\k]$};
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\foreach \y in {-2,...,3}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C3] plot coordinates{(0,0) (1,0) (2,0) (3,0) (4,0) (5,0) };
85-
\node at (13,4.5){$y[\k] = \sum_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +0$};
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\node at (13,-4.5){$y[\k] = \sum\limits_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +0$};
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\end{scope}
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\begin{scope}[yshift=-7cm, xshift=10cm]
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\draw[->] (-0.5,0) -- (8,0) node[right]{$k$};
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\draw[->] (0,-3.2) -- (0,5) node[above]{$y[k]$};
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\foreach \y in {-3,...,4}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C2!10] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
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\draw[stem, C2, dashed] plot coordinates{(0,0)};
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\draw[stem, C5, dashed] plot coordinates{(0,0)};
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\end{scope}
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\end{tikzpicture}
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\end{center}
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\end{frame}
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100-
\begin{frame}{L\"osungsweg I, k=1}
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\begin{frame}{L\"osungsweg I, k=1 (teilweise \"Uberlappung)}
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%% k = 1
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\begin{center}
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\begin{tikzpicture}[scale=0.4]
@@ -116,23 +118,23 @@
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\draw[->] (0,-2.5) -- (0,4) node[above]{$x[\kappa]\cdot h[-\kappa+\k]$};
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\foreach \y in {-2,...,3}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C3] plot coordinates{(0,0) (1,2) (2,0) (3,0) (4,0) (5,0) };
119-
\node at (13,4.5){$y[\k] = \sum_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +2$};
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\node at (13,-4.5){$y[\k] = \sum\limits_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +2$};
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\end{scope}
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\begin{scope}[yshift=-7cm, xshift=10cm]
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\draw[->] (-0.5,0) -- (8,0) node[right]{$k$};
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\draw[->] (0,-3.2) -- (0,5) node[above]{$y[k]$};
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\foreach \y in {-3,...,4}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C2!10] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
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\draw[stem, C2!75] plot coordinates{(0,0) (1,2)};
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\draw[stem, C2, dashed] plot coordinates{(1,2)};
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\draw[stem, C5, dashed] plot coordinates{(1,2)};
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\end{scope}
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\end{tikzpicture}
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\end{center}
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\end{frame}
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135-
\begin{frame}{L\"osungsweg I, k=2}
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\begin{frame}{L\"osungsweg I, k=2 (teilweise \"Uberlappung)}
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%% k=2
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\begin{center}
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\begin{tikzpicture}[scale=0.4]
@@ -151,23 +153,23 @@
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\draw[->] (0,-2.5) -- (0,4) node[above]{$x[\kappa]\cdot h[-\kappa+\k]$};
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\foreach \y in {-2,...,3}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C3] plot coordinates{(0,0) (1,1) (2,2) (3,0) (4,0) (5,0) };
154-
\node at (13,4.5){$y[\k] = \sum_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +3$};
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\node at (13,-4.5){$y[\k] = \sum\limits_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +3$};
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\end{scope}
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\begin{scope}[yshift=-7cm, xshift=10cm]
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\draw[->] (-0.5,0) -- (8,0) node[right]{$k$};
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\draw[->] (0,-3.2) -- (0,5) node[above]{$y[k]$};
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\foreach \y in {-3,...,4}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C2!10] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
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\draw[stem, C2!75] plot coordinates{(0,0) (1,2) (2,3) };
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\draw[stem, C2, dashed] plot coordinates{(2,3)};
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\draw[stem, C5, dashed] plot coordinates{(2,3)};
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\end{scope}
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\end{tikzpicture}
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\end{center}
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\end{frame}
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170-
\begin{frame}{L\"osungsweg I, k=3}
172+
\begin{frame}{L\"osungsweg I, k=3 (vollst\"andige \"Uberlappung)}
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%% k=3
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\begin{center}
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\begin{tikzpicture}[scale=0.4]
@@ -186,23 +188,23 @@
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\draw[->] (0,-2.5) -- (0,4) node[above]{$x[\kappa]\cdot h[-\kappa+\k]$};
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\foreach \y in {-2,...,3}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C3] plot coordinates{(0,0) (1,-1) (2,1) (3,4) (4,0) (5,0) };
189-
\node at (13,4.5){$y[\k] = \sum_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +4$};
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\node at (13,-4.5){$y[\k] = \sum\limits_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +4$};
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\end{scope}
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\begin{scope}[yshift=-7cm, xshift=10cm]
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\draw[->] (-0.5,0) -- (8,0) node[right]{$k$};
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\draw[->] (0,-3.2) -- (0,5) node[above]{$y[k]$};
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\foreach \y in {-3,...,4}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C2!10] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
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\draw[stem, C2!75] plot coordinates{(0,0) (1,2) (2,3) (3,4) };
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\draw[stem, C2, dashed] plot coordinates{(3,4)};
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\draw[stem, C5, dashed] plot coordinates{(3,4)};
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\end{scope}
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\end{tikzpicture}
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\end{center}
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\end{frame}
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204206

205-
\begin{frame}{L\"osungsweg I, k=4}
207+
\begin{frame}{L\"osungsweg I, k=4 (vollst\"andige \"Uberlappung)}
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%% k=4
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\begin{center}
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\begin{tikzpicture}[scale=0.4]
@@ -221,23 +223,23 @@
221223
\draw[->] (0,-2.5) -- (0,4) node[above]{$x[\kappa]\cdot h[-\kappa+\k]$};
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\foreach \y in {-2,...,3}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C3] plot coordinates{(0,0) (1,0) (2,-1) (3,2) (4,-2) (5,0) };
224-
\node at (13,4.5){$y[\k] = \sum_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = -1$};
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\node at (13,-4.5){$y[\k] = \sum\limits_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = -1$};
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\end{scope}
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\begin{scope}[yshift=-7cm, xshift=10cm]
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\draw[->] (-0.5,0) -- (8,0) node[right]{$k$};
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\draw[->] (0,-3.2) -- (0,5) node[above]{$y[k]$};
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\foreach \y in {-3,...,4}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C2!10] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
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\draw[stem, C2!75] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1)};
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\draw[stem, C2, dashed] plot coordinates{(4,-1)};
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\draw[stem, C5, dashed] plot coordinates{(4,-1)};
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\end{scope}
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\end{tikzpicture}
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\end{center}
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\end{frame}
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239241

240-
\begin{frame}{L\"osungsweg I, k=5}
242+
\begin{frame}{L\"osungsweg I, k=5 (teilweise \"Uberlappung)}
241243
%% k=5
242244
\begin{center}
243245
\begin{tikzpicture}[scale=0.4]
@@ -256,23 +258,23 @@
256258
\draw[->] (0,-2.5) -- (0,4) node[above]{$x[\kappa]\cdot h[-\kappa+\k]$};
257259
\foreach \y in {-2,...,3}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C3] plot coordinates{(0,0) (1,0) (2,0) (3,-2) (4,-1) (5,0) };
259-
\node at (13,4.5){$y[\k] = \sum_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = -3$};
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\node at (13,-4.5){$y[\k] = \sum\limits_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = -3$};
260262
\end{scope}
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\begin{scope}[yshift=-7cm, xshift=10cm]
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\draw[->] (-0.5,0) -- (8,0) node[right]{$k$};
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\draw[->] (0,-3.2) -- (0,5) node[above]{$y[k]$};
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\foreach \y in {-3,...,4}{\draw (\tic,\y) -- (-\tic,\y);};
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\draw[stem, C2!10] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
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\draw[stem, C2!75] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3)};
267-
\draw[stem, C2, dashed] plot coordinates{(5,-3)};
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\draw[stem, C5, dashed] plot coordinates{(5,-3)};
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\end{scope}
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\end{tikzpicture}
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\end{center}
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\end{frame}
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273275

274276

275-
\begin{frame}{L\"osungsweg I, k=6}
277+
\begin{frame}{L\"osungsweg I, k=6 (teilweise \"Uberlappung)}
276278
%% k=6
277279
\begin{center}
278280
\begin{tikzpicture}[scale=0.4]
@@ -291,23 +293,23 @@
291293
\draw[->] (0,-2.5) -- (0,4) node[above]{$x[\kappa]\cdot h[-\kappa+\k]$};
292294
\foreach \y in {-2,...,3}{\draw (\tic,\y) -- (-\tic,\y);};
293295
\draw[stem, C3] plot coordinates{(0,0) (1,0) (2,0) (3,0) (4,1) (5,0) };
294-
\node at (13,4.5){$y[\k] = \sum_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +1$};
296+
\node at (13,-4.5){$y[\k] = \sum\limits_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +1$};
295297
\end{scope}
296298
\begin{scope}[yshift=-7cm, xshift=10cm]
297299
\draw[->] (-0.5,0) -- (8,0) node[right]{$k$};
298300
\draw[->] (0,-3.2) -- (0,5) node[above]{$y[k]$};
299301
\foreach \y in {-3,...,4}{\draw (\tic,\y) -- (-\tic,\y);};
300302
\draw[stem, C2!10] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
301303
\draw[stem, C2!75] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1)};
302-
\draw[stem, C2, dashed] plot coordinates{(6,1)};
304+
\draw[stem, C5, dashed] plot coordinates{(6,1)};
303305
\end{scope}
304306
\end{tikzpicture}
305307
\end{center}
306308
\end{frame}
307309

308310

309311

310-
\begin{frame}{L\"osungsweg I, k=7}
312+
\begin{frame}{L\"osungsweg I, k=7 (keine \"Uberlappung)}
311313
%% k=7
312314
\begin{center}
313315
\begin{tikzpicture}[scale=0.4]
@@ -326,26 +328,70 @@
326328
\draw[->] (0,-2.5) -- (0,4) node[above]{$x[\kappa]\cdot h[-\kappa+\k]$};
327329
\foreach \y in {-2,...,3}{\draw (\tic,\y) -- (-\tic,\y);};
328330
\draw[stem, C3] plot coordinates{(0,0) (1,0) (2,0) (3,0) (4,0) (5,0) };
329-
\node at (13,4.5){$y[\k] = \sum_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +0$};
331+
\node at (13,-4.5){$y[\k] = \sum\limits_{\kappa} x[\kappa]\cdot h[-\kappa+\k] = +0$};
330332
\end{scope}
331333
\begin{scope}[yshift=-7cm, xshift=10cm]
332334
\draw[->] (-0.5,0) -- (8,0) node[right]{$k$};
333335
\draw[->] (0,-3.2) -- (0,5) node[above]{$y[k]$};
334336
\foreach \y in {-3,...,4}{\draw (\tic,\y) -- (-\tic,\y);};
335337
\draw[stem, C2!10] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
336338
\draw[stem, C2!75] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
337-
\draw[stem, C2, dashed] plot coordinates{(7,0)};
339+
\draw[stem, C5, dashed] plot coordinates{(7,0)};
338340
\end{scope}
339341
\end{tikzpicture}
340342
\end{center}
341343
\end{frame}
342344

345+
\begin{frame}{}
346+
Beginn der Signale
347+
$$k_{x,\text{Start}}=1, \quad k_{h,\text{Start}}=0 \rightarrow k_{y,\text{Start}} = k_{x,\text{Start}} + k_{h,\text{Start}} = 1$$
348+
Länge der Signale
349+
$$N_{x} = 4, \quad N_{h} = 3 \rightarrow N_{y} = N_{x} + N_{h} -1 = 6$$
350+
bedeutet für $y[k]$
351+
$$y[k_{y,\text{Start}} \leq k \leq k_{y,\text{Start}} + N_y - 1] = ...$$
352+
$$y[k<k_{y,\text{Start}}=1]=0,\quad y[k>k_{y,\text{Start}}+N_y-1=6] = 0$$
353+
\begin{center}
354+
\begin{tikzpicture}[scale=0.4]
355+
\def\tic{0.1};
356+
\draw[->] (-0.5,0) -- (6.5,0) node[right]{$k$};
357+
\draw[->] (0,-1.5) -- (0,3) node[above]{$x[k]$};
358+
\draw[stem] plot coordinates{(0,0) (1,1) (2,1) (3,2) (4,-1) (5,0) };
359+
\foreach \y in {-1, 1, 2}{\draw (\tic,\y) -- (-\tic,\y) node[left]{$\y$};};
360+
\draw (1,\tic) -- (1,-\tic) node[below]{$1$};
361+
\draw (2,\tic) -- (2,-\tic) node[below]{$2$};
362+
\draw (3,\tic) -- (3,-\tic) node[below]{$3$};
363+
\draw (4,\tic) -- (4,-\tic) node[above]{$4$};
364+
\begin{scope}[xshift=9.75cm]
365+
\draw[->] (-1.5,0) -- (4,0) node[right]{$k$};
366+
\draw[->] (0,-1.5) -- (0,3) node[above]{$h[k]$};
367+
\foreach \y in {-1,1,2}{\draw (\tic,\y) -- (-\tic,\y) node[left]{$\y$};};
368+
\draw[stem] plot coordinates{ (-1,0) (0,2) (1,1) (2,-1) (3,0)};
369+
\draw (1,\tic) -- (1,-\tic) node[below]{$1$};
370+
\draw (2,\tic) -- (2,-\tic) node[above]{$2$};
371+
\end{scope}
372+
\begin{scope}[yshift=0, xshift=16cm]
373+
\draw[->] (-0.25,0) -- (8,0) node[right]{$k$};
374+
\draw[->] (0,-3.2) -- (0,5) node[right]{$y[k]=x[k] \ast h[k]$};
375+
%\foreach \y in {-3,...,4}{\draw (\tic,\y) -- (-\tic,\y);};
376+
\foreach \y in {-3,-2,...,4}{\draw (\tic,\y) -- (-\tic,\y) node[left]{$\y$};};
377+
\draw[stem, C2!10] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
378+
\draw[stem, C2!75] plot coordinates{(0,0) (1,2) (2,3) (3,4) (4,-1) (5,-3) (6,1) (7,0)};
379+
\draw (1,\tic) -- (1,-\tic) node[below]{$1$};
380+
\draw (2,\tic) -- (2,-\tic) node[below]{$2$};
381+
\draw (3,\tic) -- (3,-\tic) node[below]{$3$};
382+
\draw (4,\tic) -- (4,-\tic) node[above]{$4$};
383+
\draw (5,\tic) -- (5,-\tic) node[above]{$5$};
384+
\draw (6,\tic) -- (6,-\tic) node[below]{$6$};
385+
\end{scope}
386+
\end{tikzpicture}
387+
\end{center}
343388

389+
\end{frame}
344390

345391

346392

347393
\begin{frame}{Discrete-Time Convolution / Zeitdiskrete Faltung / Faltung von Folgen}
348-
Task 8.2
394+
cf. \url{sig_sys_ex.pdf} Task 8.2
349395
%% Signal x[k], h[k]
350396
\begin{center}
351397
\begin{tikzpicture}[scale=0.5]
@@ -365,7 +411,7 @@
365411
\end{tikzpicture}
366412
\end{center}
367413

368-
Calculate the convolution of two finite-length signals / two sequences $x[k]$ and $h[k]$ as
414+
Calculate the convolution of the two finite-length signals $x[k]$ and $h[k]$ as
369415
$$y[k] = x[k] \ast h[k]$$
370416
$$y[k] = \sum_{\kappa = -\infty}^{+\infty} x[\kappa] \ast h[-\kappa + k] =
371417
\sum_{\kappa = -\infty}^{+\infty} x[-\kappa+k] \ast h[\kappa]

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