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R2653/teaching/COM303/lectures/04 - fourier intro/1_preamble.tex
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\documentclass[aspectratio=169]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
%\setbeameroption{show only notes}\def\logoEPFL{}
%\setbeameroption{show notes}
\begin{document}
\begin{frame} \frametitle{The fundamental idea}
\begin{itemize}
\item signals are elements of a vector space
\item vectors can be expressed as linear combinations of basis elements for any basis
\item basis elements are the ``atomic particles'' of a signal
\item the canonical basis is composed of instantaneous time elements
\item the Fourier basis is composed of \textit{oscillatory} elements
\end{itemize}
\end{frame}
\def\timeplot#1#2{
\begin{dspPlot}[width=2.5cm,height=1cm,xticks=1,yticks=1]{0,7}{0,1.2}
\moocStyle
\dspSignal{x #1 eq {1} {0} ifelse}
\dspText(3,1.4){$#2 \times$}
\end{dspPlot}\hspace{-2em}}
\begin{frame} \frametitle{The time domain}
\begin{center}
\begin{dspPlot}[width=5cm,height=1.5cm,xticks=1,yticks=5]{0,7}{0,6}
\moocStyle
\dspTapsAt{0}{1 2 3 4 5 4 3 2}
\end{dspPlot}
\end{center}
\begin{tabular}{cccc}
\timeplot{0}{1}
&
\timeplot{1}{2}
&
\timeplot{2}{3}
&
\timeplot{3}{4}
\\
\timeplot{4}{5}
&
\timeplot{5}{4}
&
\timeplot{6}{3}
&
\timeplot{7}{2}
\end{tabular}
\end{frame}
\def\freqplot#1#2{
\begin{dspPlot}[width=2.5cm,height=1cm,xticks=none,yticks=1]{0,7}{-1.3,1.3}
\moocStyle
\dspSignal{x 360 8 div #1 mul mul cos}
\only<2>{\dspFunc[linewidth=0.3pt,linecolor=blue]{x 360 8 div #1 mul mul cos}}
\dspText(3,1.8){$#2 \times$}
\end{dspPlot}\hspace{-2em}}
\begin{frame} \frametitle{The frequency domain}
\begin{center}
\begin{dspPlot}[width=5cm,height=1.5cm,xticks=,yticks=5]{0,7}{0,6}
\moocStyle
\dspTapsAt{0}{1 2 3 4 5 4 3 2}
\end{dspPlot}
\end{center}
\begin{tabular}{cccc}
\freqplot{0}{3}
&
\freqplot{1}{-0.8536}
&
\freqplot{2}{0}
&
\freqplot{3}{-0.1464}
\\
\freqplot{4}{0}
&
\freqplot{5}{-0.1464}
&
\freqplot{6}{0}
&
\freqplot{7}{-0.8536}
\end{tabular}
\end{frame}
\begin{frame} \frametitle{The Fourier transform}
\begin{center}
Fourier analysis: express a signal as a combination of periodic oscillations:
\[
\mathbf{x} = \sum_{k=0}^{N-1} X_k\mathbf{w}^{(k)}
\]
where $\{\mathbf{w}^{(k)}\}$ is the Fourier oscillatory basis.
\vspace{1em}
Fourier transform: an algorithm to perform a change of basis in the space of signals
\end{center}
\end{frame}
\begin{comment}
\def\onepoint#1#2#3{%
\only<#3>{\dspTaps{#1 #2}\dspText(#1, 1){$#2\,\delta[n-#1]$}}
}
\begin{frame} \frametitle{The time domain}
\begin{center}
discrete-time signals can be expressed as a linear combination of ``atomic'' time units:
\[
x[n] = \sum_{k=0}^{N-1} x[k]\delta[n-k]
\]
\begin{dspPlot}[xticks=1,yticks=1]{0,7}{0,6.5}
\moocStyle
\dspTapsAt[linecolor=lightgray]{0}{3 4 2 3 6 5 5 2}
\onepoint{0}{3}{2}
\onepoint{1}{4}{3}
\onepoint{2}{2}{4}
\onepoint{3}{3}{5}
\onepoint{4}{6}{6}
\onepoint{5}{5}{7}
\onepoint{6}{5}{8}
\onepoint{7}{2}{9}
\end{dspPlot}
\end{center}
\end{frame}
\begin{frame} \frametitle{The time domain}
\begin{center}
in vector notation:
\[
\mathbf{x} = \sum_{k=0}^{N-1} x_k\boldsymbol{\delta}^{(k)}
\]
where $\{\boldsymbol{\delta}^{(k)}\}$ is the canonical basis for $\mathbb{C}^N$; e.g.:
\[
\boldsymbol{\delta}^{(2)} = \begin{bmatrix} 0 & 0 & 1 & 0 & \ldots & 0 \end{bmatrix}^T
\]
\end{center}
\end{frame}
\begin{frame} \frametitle{The frequency domain}
\begin{center}
Fourier analysis: express a signal as a combination of periodic oscillations:
\[
\mathbf{x} = \sum_{k=0}^{N-1} X_k\mathbf{w}^{(k)}
\]
where $\{\mathbf{w}^{(k)}\}$ is the Fourier basis.
\vspace{1em}
\onslide<2->{Fourier transform: a change of basis in the space of discrete-time signals}
\end{center}
\end{frame}
\end{comment}
\begin{frame} \frametitle{Oscillations are everywhere!}
\begin{center}
\begin{figure}
\center
\includegraphics[width=100mm]{oscillatory.eps}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Oscillations are everywhere}
\begin{itemize}
\item sustainable dynamic systems exhibit oscillatory behavior
\item intuitively: things that don't move in circles can't last:
\begin{itemize}
\item bombs
\item rockets
\item human beings...
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Oscillations are everywhere}
\note<1>{prototypical oscillatory behavior is a rotation\\
rotations per second = frequency\\
rotation coordinates: sin and cos}
\begin{center}
\psset{unit=10mm}
\begin{pspicture}(-7,-3)(7,3)
\SpecialCoor%
\qdisk(! 50 cos 2.5 mul 50 sin 2.5 mul){3pt}
\only<2->{
\psarc[linecolor=darkred]{->}{2.5}{50}{49}
\qdisk(! 50 cos 2.5 mul 50 sin 2.5 mul){3pt}
\uput{2.5}[50](0,0){~$\mathbf{x}(t)$}}
\only<3->{
\psline{->}(-5,0)(5,0)
\psline{->}(0,-3)(0,3)
\psplot[linewidth=2pt,plotpoints=100,linejoin=1,linecolor=green!60]{-4.5}{4.5}{x 2.5 div 180 mul sin}
\uput[0](!4.5 4.5 2.5 div 180 mul sin){~$x_0(t)$}
\psplot[linewidth=2pt,plotpoints=100,linejoin=1,linecolor=blue!40]{-4.5}{4.5}{x 2.5 div 180 mul cos}
\uput[0](!4.5 4.5 2.5 div 180 mul cos){~$x_1(t)$}}
\NormalCoor%
\end{pspicture}
\end{center}
\end{frame}
\begin{frame} \frametitle{You too can detect sinusoids!}
\begin{center}
the human body has two receptors for sinusoidal signals:
\vspace{1em}
\begin{columns}
\begin{column}{0.45\paperwidth}
\centering
cochlea (inner ear)
\includegraphics[width=0.4\paperwidth]{cochlea.eps}
\begin{itemize}
\item air pressure oscillations
\item frequencies from 20Hz to 20KHz
\item wavelength from 17m to 1.7cm
\end{itemize}
\end{column}
\begin{column}{0.45\paperwidth}
\centering
rods and cones (retina)
\includegraphics[width=0.4\paperwidth]{retina.eps}
\begin{itemize}
\item electromagnetic oscillations
\item frequencies from 430THz to 790THz
\item wavelength from 740nm to 380nm
\end{itemize}
\end{column}
\end{columns}
\end{center}
\end{frame}
\begin{frame}
\frametitle{The intuition}
\begin{itemize}[<+->]
\item humans analyze complex signals (audio, images) in terms of their sinusoidal components
\item we can build instruments that ``resonate'' at one or multiple frequencies\\ (tuning fork vs piano)
\item the ``frequency domain'' seems to be as important as the time domain
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Fundamental question}
\begin{center}
can we decompose any signal into sinusoidal elements?
\vspace{1em}
\uncover<2->{yes, and Fourier showed us how to do it {\em exactly}!}
\vspace{1em}
\vspace{1em}
\begin{columns}
\begin{column}{0.45\paperwidth}
\uncover<3->{
\centering
analysis
\begin{itemize}
\item from time domain to frequency domain
\item find the contribution of different frequencies
\item discover ``hidden'' signal properties
\end{itemize}}
\end{column}
\begin{column}{0.45\paperwidth}
\uncover<4->{
\centering
synthesis
\begin{itemize}
\item from frequency domain to time domain
\item create signals with known frequency content
\item fit signals to specific frequency regions
\end{itemize}}
\end{column}
\end{columns}
\end{center}
\end{frame}
\begin{frame}
\frametitle{The mathematical setup}
\begin{itemize}[<+->]
\item let's start with finite-length signals (i.e. vectors in $\mathbb{C}^N$)
\item Fourier analysis is a simple change of basis
\item a change of basis is a change of perspective
\item a change of perspective can reveal things (if the basis is good)
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Mystery signal}
\note<1>{We can use\\
the concept of rotation to discover hidden structures \\ in higher-dimensional vector spaces. For instance, \\here is a 1024-point signal that looks like noise. \\ But with a suitable ``rotation'' we can discover what it hides.}
\begin{center}
\begin{figure}
\begin{dspPlot}[yticks=2,xout=true]{0, 1023}{-3, 3}
\moocStyle
\dspFunc{x 1024 div 360 mul 64 mul cos rand 2147483647 div 0.5 sub 2 mul add}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame}
\frametitle{Mystery signal in the Fourier basis}
\begin{center}
\begin{figure}
\begin{dspPlot}[yticks=200,xout=true]{0, 1023}{0, 520}
\moocStyle
\dspFuncFile{mysteryDFT.txt}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\end{document}
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