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R2653/teaching/COM303/2016/lectures/02 - hilbert space/3_normspace.tex
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\documentclass[aspectratio=169]{beamer}
%\documentclass[aspectratio=169,handout]{beamer}
\def\stylepath{../styles}
\usepackage{\stylepath/com303}
\usepackage{pst-3dplot}
%\setbeameroption{show only notes}\def\logoEPFL{}
\setbeameroption{show notes}
\begin{document}
\begin{frame} \frametitle{Inner product}
\begin{center}
$\langle \cdot, \cdot \rangle \,:\, V \times V \rightarrow \mathbb{C}$
\end{center}
\begin{itemize}
\item measure of similarity between vectors
\item inner product is zero? vectors are \textit{orthogonal} (maximally different)
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Formal properties of the inner product}
For $\mathbf{x}, \mathbf{y}, \mathbf{z} \in V$ and $\alpha \in \mathbb{C}$:
\begin{itemize}[<+->]
\item $\langle \mathbf{x} + \mathbf{y}, \mathbf{z} \rangle = \langle \mathbf{x}, \mathbf{z} \rangle + \langle \mathbf{y}, \mathbf{z} \rangle$
\item $\langle \mathbf{x} , \mathbf{y} \rangle = \langle \mathbf{y} , \mathbf{x} \rangle^*$
\item $\langle \alpha\mathbf{x} , \mathbf{y} \rangle = \alpha^* \langle \mathbf{x} , \mathbf{y} \rangle$ \\
$\langle\mathbf{x} , \alpha\mathbf{y} \rangle = \alpha \langle \mathbf{x} , \mathbf{y} \rangle$
\item $\langle \mathbf{x} , \mathbf{x} \rangle \geq 0$
\item $\langle \mathbf{x} , \mathbf{x} \rangle = 0 \Leftrightarrow \mathbf{x} = \mathbf{0}$
\item if $\langle \mathbf{x} , \mathbf{y} \rangle = 0$ and $\mathbf{x}, \mathbf{y} \neq \mathbf{0}$ then $\mathbf{x}$ and $\mathbf{y}$ are called orthogonal
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Inner product in $\mathbb{R}^2$}
\begin{center}
$\langle \mathbf{x}, \mathbf{y} \rangle = x_0 y_0 + x_1 y_1$
\vspace{2em}
\psset{unit=8mm}
\begin{pspicture}(-1,-1)(9,5)
\psgrid[subgriddiv=1,gridlabels=0pt,gridcolor=gray,griddots=10](0,0)(-1,-1)(9,5)
\psaxes[labelFontSize=\scriptstyle]{->}(0,0)(-1,-1)(9,5)
\psline{->}(6,2)\uput[-45](6,2){$\mathbf{x}$}
\psline{->}(3,4)\uput[-45](3,4){$\mathbf{y}$}
\end{pspicture}
\end{center}
\end{frame}
\begin{frame} \frametitle{Inner product in $\mathbb{R}^2$: the norm}
\begin{center}
$\langle \mathbf{x}, \mathbf{x} \rangle = x_0^2 + x_1^2 \only<2>{ = \|\mathbf{x}\|^2}$
\vspace{2em}
\psset{unit=8mm}
\begin{pspicture}(-1,-1)(9,5)
\psgrid[subgriddiv=1,gridlabels=0pt,gridcolor=gray,griddots=10](0,0)(-1,-1)(9,5)
\psaxes[labelFontSize=\scriptstyle]{->}(0,0)(-1,-1)(9,5)
\only<1->{\psline{->}(6,2)\uput[-45](6,2){$\mathbf{x}$}}
\only<2->{\uput{3.2}[33]{0}(0,0){$\|\mathbf{x}\|$}}
\end{pspicture}
\end{center}
\end{frame}
\begin{frame} \frametitle{Inner product in $\mathbb{R}^2$}
\begin{center}
$
\langle \mathbf{x}, \mathbf{y} \rangle = x_0 y_0 + x_1 y_1
\only<2>{ = \|\mathbf{x}\|\,\|\mathbf{y}\|\,\cos\alpha}
$
\vspace{2em}
\psset{unit=8mm}
\begin{pspicture}(-1,-1)(9,5)
\psgrid[subgriddiv=1,gridlabels=0pt,gridcolor=gray,griddots=10](0,0)(-1,-1)(9,5)
\psaxes[labelFontSize=\scriptstyle]{->}(0,0)(-1,-1)(9,5)
\only<1->{
\psline{->}(6,2)\uput[-45](6,2){$\mathbf{x}$}
\psline{->}(3,4)\uput[-45](3,4){$\mathbf{y}$}}
\only<2->{
\psarc[linecolor=darkred]{->}{3}{18.435}{53.13}
\uput{3.2}[33]{0}(0,0){$\alpha$}}
\end{pspicture}
\end{center}
\end{frame}
\begin{frame} \frametitle{Inner product in $\mathbb{R}^2$: orthogonality}
\begin{center}
$\langle \mathbf{x}, \mathbf{y} \rangle = x_0 y_0 + x_1 y_1= \|\mathbf{x}\|\,\|\mathbf{y}\|\,\cos\alpha$
\vspace{2em}
\psset{unit=8mm}
\begin{pspicture}(-1,0)(8,6)
\SpecialCoor%
\def\a{10 }\def\r{6 }
\def\b{100 }\def\s{5 }
\psline{->}(! \a cos \r mul \a sin \r mul)
\uput{\r}[\a](0,0){~$\mathbf{x}$}
\psline{->}(! \b cos \s mul \b sin \s mul)
\uput{\s}[\b](0,0){~$\mathbf{y}$}
\psarc[linecolor=darkred]{->}{3}{\a}{\b}
\uput{3.1}[55]{0}(0,0){$\alpha = \pi/2$}
\uput{5}[30]{0}(0,0){$\langle \mathbf{x}, \mathbf{y} \rangle = 0$}
\NormalCoor%
\end{pspicture}
\end{center}
\end{frame}
\begin{frame} \frametitle{Inner product in $L_2[-1,1]$}
\[
\langle \mathbf{x}, \mathbf{y} \rangle = \int_{-1}^{1} x(t)y(t)\, dt
\]
\end{frame}
\begin{frame} \frametitle{Inner product in $L_2[-1,1]$: the norm}
\begin{center}
$\langle \mathbf{x}, \mathbf{x} \rangle = \|\mathbf{x}\|^2 = \int_{-1}^{1} \sin^2(\pi t) dt = 1$
\vspace{1em}
\begin{figure}
\begin{dspPlot}[xticks=1,xout=true,yticks=2,sidegap=0]{-1, 1}{-1.25, 1.25}
\moocStyle
\only<1>{\dspFunc{x 180 mul sin}}
\only<2->{
\dspFunc[linecolor=lightgray]{x 180 mul sin}
\pscustom[fillstyle=solid,fillcolor=blue!30,linestyle=none]{%
\dspFunc[linecolor=blue]{x 180 mul sin dup mul}}}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Inner product in $L_2[-1,1]$: the norm}
\begin{center}
$ \|\mathbf{y}\|^2 = \int_{-1}^{1} t^2\,dt = 2/3$
\vspace{1em}
\begin{figure}
\begin{dspPlot}[xticks=1,xout=true,yticks=1,sidegap=0]{-1, 1}{-1.25, 1.25}
\moocStyle
\dspFunc{x }
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Inner product in $L_2[-1,1]$}
\begin{center}
$\langle \mathbf{x}, \mathbf{y} \rangle = \int_{-1}^{1} \only<1-2>{x(t)y(t)}\only<3->{\sqrt{3/2}t \sin(\pi t)} dt \only<3>{ = (2/\pi)\sqrt{3/2} \approx 0.78}$
\vspace{1em}
\begin{figure}
\begin{dspPlot}[xticks=1,xout=true,yticks=1,sidegap=0]{-1, 1}{-1.25, 1.25}
\moocStyle
\only<1-2>{\dspFunc{x 180 mul sin}}
\only<2>{\dspFunc[linecolor=orange]{x 1.225 mul}}
\only<3>{
\dspFunc[linecolor=lightgray]{x 180 mul sin}
\dspFunc[linecolor=lightgray]{x 1.225 mul}
\pscustom[fillstyle=solid,fillcolor=blue!30,linestyle=none]{%
\dspFunc[linecolor=blue]{x 180 mul sin x 1.225 mul mul}}}
\only<1->{\dspText(0,-.8){$\mathbf{x} = \sin(\pi t)$}}
\only<2->{\dspText(-.6,-.3){$\mathbf{y} = \sqrt{3/2} t$}}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Inner product in $L_2[-1,1]$}
\begin{center}
$\mathbf{x}, \mathbf{y}$ from orthogonal subspaces:
\only<3>{$\langle \mathbf{x}, \mathbf{y} \rangle = 0$}
\vspace{1em}
\begin{figure}
\begin{dspPlot}[xticks=1,xout=true,yticks=2,sidegap=0]{-1, 1}{-1.25, 1.25}
\moocStyle
\only<1->{\dspText(.2,-.8){$\mathbf{x} = \sin(\pi t)$, antisymmetric}}
\only<2->{\dspText(-.6,.8){$\mathbf{y} = 1-|t|$, symmetric}}
\only<1-2>{\dspFunc{x 180 mul sin}}
\only<2>{\dspFunc[linecolor=orange]{x abs 1 exch sub }}
\only<3>{
\dspFunc[linecolor=lightgray]{x 180 mul sin}
\dspFunc[linecolor=lightgray]{x abs 1 exch sub }
\pscustom[fillstyle=solid,fillcolor=green!30,linestyle=none]{%
\dspFunc[linecolor=green]{x 180 mul sin x abs 1 exch sub mul dup 0 ge {} {pop 0} ifelse}}
\pscustom[fillstyle=solid,fillcolor=red!30,linestyle=none]{%
\dspFunc[linecolor=red]{x 180 mul sin x abs 1 exch sub mul dup 0 ge {pop 0} {} ifelse}}}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Inner product in $L_2[-1,1]$}
\begin{center}
sinusoids with frequencies integer multiples of a fundamental
\vspace{1em}
\begin{figure}
\begin{dspPlot}[xticks=1,xout=true,yticks=2,sidegap=0]{-1, 1}{-1.2, 1.2}
\moocStyle
\only<1-2>{\dspText(1.3,-.5){{\color{darkred} $\mathbf{x} = \sin(4\pi t)$}}}
\only<2-3>{\dspText(1.3,.5){{\color{orange} $\mathbf{y} = \sin(5\pi t)$}}}
\only<1-2>{\dspFunc{x 180 mul 4 mul sin}}
\only<2>{\dspFunc[linecolor=orange]{x 180 mul 5 mul sin}}
\only<3>{
\dspFunc[linecolor=lightgray]{x 180 mul 4 mul sin}
\dspFunc[linecolor=lightgray]{x 180 mul 5 mul sin}}
\only<3-4>{
\pscustom[fillstyle=solid,fillcolor=green!30,linestyle=none]{%
\dspFunc[linecolor=green]{x 180 mul 4 mul sin x 180 mul 5 mul sin mul dup 0 ge {} {pop 0} ifelse}}
\pscustom[fillstyle=solid,fillcolor=red!30,linestyle=none]{%
\dspFunc[linecolor=red]{x 180 mul 4 mul sin x 180 mul 5 mul sin mul dup 0 ge {pop 0} {} ifelse}}}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{Norm vs Distance}
\begin{itemize}[<+->]
\item inner product defines a norm: $\| \mathbf{x} \| = \sqrt{\langle \mathbf{x} , \mathbf{x} \rangle}$
\item norm defines a distance: $d(\mathbf{x}, \mathbf{y}) = \| \mathbf{x} - \mathbf{y}\|$
\vspace{2em}
\end{itemize}
\end{frame}
\begin{frame} \frametitle{Norm and distance in $\mathbb{R}^2$}
\begin{center}
\only<1>{$\|\mathbf{x}\| = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle} = \sqrt{x_0^2 + x_1^2}$}
\only<2>{$\|\mathbf{y}\| = \sqrt{\langle \mathbf{y}, \mathbf{y} \rangle} = \sqrt{y_0^2 + y_1^2}$}
\only<3>{$\|\mathbf{x} - \mathbf{y}\| = \sqrt{(x_0 - y_0)^2 + (x_1 - y_1)^2}$}
\psset{unit=8mm}
\begin{pspicture}(-1,0)(8,6)
\only<1->{
\psline{->}(6,1)\uput[-45](3,0.5){$\|\mathbf{x}\|$}}
\only<2->{
\psline{->}(4,3)\uput[135](2,1.5){$\|\mathbf{y}\|$}}
\only<3->{
\psline[linecolor=blue]{->}(4,3)(6,1)\uput[45](5,2){$\|\mathbf{x-y}\|$}}
\end{pspicture}
\end{center}
\end{frame}
\begin{frame} \frametitle{Distance in $L_2[-1,1]$: the Mean Square Error}
\begin{center}
$\|\mathbf{x} - \mathbf{y}\|^2 = \int_{-1}^{1} |x(t)-y(t)|^2 \, dt \only<3>{= 2}$
\begin{figure}
\begin{dspPlot}[xticks=1,xout=true,yticks=2,sidegap=0]{-1, 1}{-2.2, 2.2}
\moocStyle
\only<1-2>{\dspText(1.3,-.5){{\color{darkred} $\mathbf{x} = \sin(4\pi t)$}}}
\only<2>{\dspText(1.3,.5){{\color{orange} $\mathbf{y} = \sin(5\pi t)$}}}
\only<1-2>{\dspFunc{x 180 mul 4 mul sin}}
\only<2>{\dspFunc[linecolor=orange]{x 180 mul 5 mul sin}}
\only<3>{
\dspFunc[linecolor=lightgray]{x 180 mul 4 mul sin}
\dspFunc[linecolor=lightgray]{x 180 mul 5 mul sin}
\dspFunc[linecolor=blue]{x 180 mul 4 mul sin x 180 mul 5 mul sin sub}}
\end{dspPlot}
\end{figure}
\end{center}
\end{frame}
\begin{frame} \frametitle{A familiar result}
\begin{columns}
\begin{column}{.6\paperwidth}
Pythagorean theorem:
\[
\| \mathbf{x} + \mathbf{y} \|^2 = \| \mathbf{x}\|^2 + \| \mathbf{y} \|^2 \mbox{ for } \mathbf{x} \perp \mathbf{y}
\]
\end{column}
\begin{column}{.3\paperwidth}
\centering
\includegraphics[height=6.5cm]{pythagoras.eps}\\
\footnotesize From Euclid's elements by Oliver Byrne (1810 -– 1880)\\
\end{column}
\end{columns}
\end{frame}
\begin{frame} \frametitle{Inner product for signals}
\begin{center}
\only<1>{
\[
\langle \mathbf{x} , \mathbf{y} \rangle = \sum_{n = 0}^{N-1} x^*[n] y[n]
\]
well defined for all finite-length vectors (i.e. vectors in $\mathbb{C}^{N}$)}
\only<2>{
\[
\langle \mathbf{x} , \mathbf{y} \rangle = \sum_{n = {\color{red} -\infty}}^{{\color{red}\infty}} x^*[n] y[n]
\]
careful: sum may explode!}
\only<3>{
\[
\langle \mathbf{x} , \mathbf{y} \rangle = \sum_{n = -\infty}^{\infty} x^*[n] y[n]
\]
We require sequences to be {\em square-summable}: $\sum |x[n]|^2 < \infty$ \\
\vspace{1em}
Space of square-summable sequences: $\ell_2(\mathbb{Z})$}
\end{center}
\end{frame}
\end{document}
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