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0A-02-ACA-Fundamentals-Convolution.tex
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0A-02-ACA-Fundamentals-Convolution.tex
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% move all configuration stuff into includes file so we can focus on the content
\input{include}
\subtitle{module A.2: fundamentals~---~convolution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
% generate title page
\input{include/titlepage}
\section[overview]{lecture overview}
\begin{frame}{introduction}{overview}
\begin{block}{corresponding textbook section}
%\href{http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6331119&}{Chapter 2~---~Fundamentals}: pp.~14--18\\
%\href{http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6331114}{Appendix A~---~Convolution Properties}: pp.~181--183
appendix~A.2
\end{block}
\begin{itemize}
\item \textbf{lecture content}
\begin{itemize}
\item LTI systems
\item convolution
\item filter examples
\end{itemize}
\bigskip
\item<2-> \textbf{learning objectives}
\begin{itemize}
\item basic understanding of linearity and time-invariance
\item basic understanding of the convolution operation
\item ability to implement simple filters
\end{itemize}
\end{itemize}
\inserticon{directions}
\end{frame}
\section[systems]{systems}
\begin{frame}{systems}{introduction}
a system:
\begin{itemize}
\item any process producing an output signal in response to an input signal
\end{itemize}
\begin{figure}
\centering
\begin{picture}(80,35)
%boxes
%\put(0,30){\ovalbox{\footnotesize{\parbox{20mm}{\vspace{2mm}\centering{Composer}\vspace{2mm}}}}}
\put(30,30){\ovalbox{\footnotesize{\parbox{20mm}{\vspace{2.3mm}\centering{System}\vspace{2.3mm}}}}}
%\put(60,30){\ovalbox{\footnotesize{\parbox{20mm}{\vspace{2mm}\centering{Recipient}\vspace{2mm}}}}}
% horizontal
\put(22.4,30.6){\vector(1,0){7.8}}
\put(52.4,30.6){\vector(1,0){7.8}}
\put(15,30){\text{$x(t)$}}
\put(60,30){\text{$y(t)$}}
\end{picture}
\end{figure}
\vspace{-25mm}
\question{name examples for systems in signal processing}
\begin{itemize}
\item filters, effects
\item vocal tract
\item room
\item (audio) cable
\item \ldots
\end{itemize}
\end{frame}
\begin{frame}{systems}{LTI systems}
\begin{block}{LTI: Linear Time-Invariant Systems}
are a great model for many real-world systems
\end{block}
\bigskip
\begin{itemize}
\item \textbf{linearity}
\begin{enumerate}
\item \textit{homogeneity}:
$f(ax) = a f(x)$
\smallskip
\item \textit{superposition} (additivity):
$f(x+y) = f(x) + f(y)$
\end{enumerate}
\bigskip
\item \textbf{time invariance}:
$f\left(x(t-\tau)\right) = f(x)(t-\tau)$
\end{itemize}
\end{frame}
\section[convolution]{convolution}
\begin{frame}{convolution}{introduction}
\begin{block}{convolution}
convolution operation describes the \textbf{output of an LTI system}:
\end{block}
\bigskip
\begin{eqnarray*}
y(t) = (x \ast h)(t) &:=& \int\limits_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau\\
y(i) = (x \ast h)(i) &:=& \sum\limits_{j=-\infty}^{\infty}x(j)h(i-j)
\end{eqnarray*}
\end{frame}
\begin{frame}{convolution}{animation}
\vspace{-3mm}
\includeanimation
{Convolution}
{000}
{600}
{10}
%\vspace{-5mm}
%\begin{center}
%\animategraphics[autoplay,loop]{10}{animateConvolution/Convolution-}{000}{600}
%\end{center}
%\addreference{matlab source: \href{https://github.com/alexanderlerch/ACA-Slides/blob/master/matlab/animateConvolution.m}{matlab/animateConvolution.m}}
\inserticon{video}
\end{frame}
\begin{frame}{convolution}{properties}
\begin{itemize}
\item<1-> \textbf{identity}:
\begin{footnotesize}\begin{equation*}
x(i) = \delta(i)\ast x(i)
\end{equation*}\end{footnotesize}
\item<2-> \textbf{commutativity}:
\begin{footnotesize}\begin{equation*}
h(i) \ast x(i) = x(i) \ast h(i)
\end{equation*}\end{footnotesize}
\item<3-> \textbf{associativity}:
\begin{footnotesize}\begin{equation*}
\big(g(i) \ast h(i)\big) \ast x(i) = g(i) \ast \big(h(i) \ast x(i)\big)
\end{equation*}\end{footnotesize}
\item<4-> \textbf{distributivity}:
\begin{footnotesize}\begin{equation*}
g(i) \ast \big(h(i) + x(i)\big) = \big(g(i) \ast h(i)\big) + \big(g(i) \ast x(i)\big)
\end{equation*}\end{footnotesize}
\item<5-> \textbf{circularity}:\\
\begin{footnotesize}
$h(i)$ periodic $\Rightarrow y(i) = h(i) \ast x(i)$ periodic
\end{footnotesize}
\end{itemize}
\end{frame}
\section{filter examples}
\begin{frame}{filter}{example 1: Moving Average}
\begin{equation*}
y(i) = \sum\limits_{j=0}^{\mathcal{J}-1}{b(j)\cdot x(i-j)}
\end{equation*}
\begin{itemize}
\item replaces current sample with average of $\mathcal{J}$ samples
\item smooths a signal (low pass)
\item IR: rectangular
\item linear phase, but inefficient for many coefficients
\item Finite Impulse Response (FIR)
\end{itemize}
\end{frame}
\begin{frame}{filter}{example 2: Single-Pole}
\begin{equation*}
y(i) = (1-\alpha)\cdot x(i) + \alpha\cdot y(i-1)
\end{equation*}
\begin{itemize}
\item \textbf{recursive system}: output depends on previous \textit{output}
\item the larger alpha, the less the current input is taken into account (low pass)
\item alpha from 0\ldots 1
\item efficient, but non-linear phase
\item Infinite Impulse Response (IIR)
\end{itemize}
\end{frame}
\section{summary}
\begin{frame}{summary}{lecture content}
\begin{itemize}
\item \textbf{LTI system}
\begin{itemize}
\item good model for many real-world system
\item linear (homogeneity, superposition) and time-invariant
\item impulse response reflects all system properties
\end{itemize}
\bigskip
\item \textbf{convolution}
\begin{itemize}
\item operation that computes the output of an LTI system from the input
\end{itemize}
\bigskip
\item \textbf{low pass filters}
\begin{itemize}
\item often used to smooth a signal
\item typical examples are moving average (FIR) and single pole (IIR)
\end{itemize}
\end{itemize}
\inserticon{summary}
\end{frame}
\end{document}