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‎lecture_notes/figures/block_diagram_w_feedback.tikz

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\begin{tikzpicture}
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%draw matrix of nodes
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\matrix[row sep=5mm,column sep=10mm]{
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& & & \node (desired) {$d(n)$}; & & &\\
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\node (input) {$u(n)$}; & \node [dotnode] (dotin) {}; & \node [blocknode] (filter) {$w_m(n)$}; & \node [combnode] (sumyd) {$\Sigma$}; & \node [dotnode] (doterr) {}; & \node (error) {$e(n)$};\\
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& & \node [blocknode] (weight control) {Weight Control}; & & &\\
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};
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%draw arrows
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\begin{scope}[->,>=latex]
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%column 1
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\draw (desired) -- (sumyd);
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%column 2
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\draw (input) -- (filter);
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\draw (filter) -- node[above,midway]{$y(n)$} (sumyd);
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\path (sumyd) [late options={label=195:$-$}];
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\draw (sumyd) -- (error);
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%column 3
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\draw (dotin) |- (weight control);
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\draw (doterr) |- (weight control);
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\draw ([xshift=-5mm] weight control.north) -- ([shift={(-5mm,-2.5mm)}] filter.south) -- ([xshift=-3.75mm] filter.south) ([xshift=3.75mm] filter.north) -- ([shift={(5mm,2.5mm)}] filter.north);
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\end{scope}
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\end{tikzpicture}

‎lecture_notes/figures/const_opt.tikz

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\begin{tikzpicture}
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%axes
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\begin{scope}[->,>=latex]
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\draw (-1,0) -- (6,0) node[below] {$x_1$};
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\draw (0,-0.5) -- (0,4) node[left] {$x_2$};
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\end{scope}
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\coordinate (xu) at (2.5,2); %unconstrained optimum
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\coordinate (xc) at (2.837,1.0427); %constrained optimum
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\coordinate (g) at (2.837+0.3527,1.0427-1);
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\begin{scope}[shift={(xu)},rotate=-45]
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%unconstrained optimum
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\filldraw (xu) circle (1pt) node[above] {$\vect{x}_u$};
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%constrained optimum
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\filldraw (xc) circle (1pt) node[above] {$\vect{x}_c$};
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%contours of the cost function
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\draw (0,0) circle (2) node[shift={(-1.75,1.75)}] {$f(\vect{x})$};
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\draw (0,0) circle (1.0148);
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\end{scope}
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%constraint
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\draw[domain=0.5:6,samples=100] plot (\x,{ln(\x)}) node[below] {$h(\vect{x})$};
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%gradients
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\begin{scope}[->,>=stealth,every node/.style={fill=white,inner sep=1pt}]
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\draw (xc) -- (g) node[below,yshift=-1mm] {$\vect{\nabla}f(\vect{x}_c)$};
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\draw (xc) -- ($(xc)+0.6*(g)-0.6*(xc)$) node[right,xshift=1mm] {$\vect{\nabla}h(\vect{x}_c)$};
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\end{scope}
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% filter values
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\end{tikzpicture}
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%matlab
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%find yi so that f=0
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%y=linspace(1,5,10000);x1=2.5;x2=2;f=y.^2-x1*y-x2+log(y);plot(y,f)
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%then find the length from constraint to xo
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%norm([x1,x2]-[yi,log(yi)])
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%the end coordinate of the gradient is
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% xc + [1/yi,-1]

‎lecture_notes/figures/exponential_window.tikz

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\begin{tikzpicture}[y=2cm]
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%axes
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\draw[->,>=latex] (-0.5,0) -- (8,0) node [below] {$i$};
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\draw[->,>=latex] (0,-0.2) -- (0,1.2) node [left] {$\beta(n,i)$};
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\draw (1pt,1) -- (-1pt,1) node[left] {1};
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%window
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\draw plot[id=ew,domain=0:7,samples=100] function{0.70**(7-x)} node[above] {$\lambda^{n-i}$};
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\draw[dashed] (7,1) -- (7,0) node[below] {$n$};
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\draw[fill=white] (0,0.08235) circle (1pt);
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\end{tikzpicture}

‎lecture_notes/figures/growing_window.tikz

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\begin{tikzpicture}[y=2cm]
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%axes
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\draw[->,>=latex] (-0.5,0) -- (8,0) node [below] {$i$};
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\draw[->,>=latex] (0,-0.2) -- (0,1.2) node [left] {$\beta(n,i)$};
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%window
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\draw (0,1) -- (7,1);
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\draw[fill=white] (0,1) circle (1pt) node [left] {1};
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\draw[dashed] (7,1) -- (7,0) node[below] {$n$};
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\end{tikzpicture}

‎lecture_notes/figures/learning_curve_sd.tikz

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\begin{tikzpicture}
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%axes
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\begin{scope}[->,>=latex]
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\draw (-0.5,0) -- (6,0) node[below] {$n$};
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\draw (0,-0.2) -- (0,2.2) node[left] {$J_1(\vect{w}(n))$};
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\end{scope}
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%cost function
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\draw[dashed] (0,0.25) node[left] {$J_1(\vect{w}_o)$}-- (6,0.25);
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\draw plot[id=J,domain=0:6,samples=100] function{0.25+3*2**(-2*x-1)};
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\end{tikzpicture}

‎lecture_notes/figures/learning_curve_sgm.tikz

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\begin{tikzpicture}
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%axes
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\begin{scope}[->,>=latex]
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\draw (-0.5,0) -- (6,0) node[below] {$n$};
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\draw (0,-0.2) -- (0,3.2) node[left] {$J_1(\vect{w}(n))$};
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\end{scope}
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%cost function
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\draw[dashed] (0,0.2) node[left] {$J_\textup{min}$}-- (6,0.2);
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\draw[dashed] (0,0.5) node[left] {$J_\textup{min}+J_\textup{ex}$}-- (6,0.5);
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\draw plot[id=J,domain=0:6,samples=100] function{0.5+5*2**(-2*x-1)};
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\draw [<->,>=stealth] (0.5,0) -- (0.5,7/4) node[right] {MSE};
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\draw [<->,>=stealth] (6,0.2) -- node[midway,right] {EMSE} (6,0.5);
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\end{tikzpicture}

‎lecture_notes/figures/ls_windowing.tikz

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\begin{tikzpicture}
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\matrix (phi) [
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matrix of nodes,
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left delimiter=\lbrack,
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right delimiter=\rbrack,
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column sep=5mm,
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row sep=5mm
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]
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{ & [4cm] & 0\\
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0 & & \\
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};
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\node[left,xshift=-2mm] at (phi.west) {$\vect{A}^T = $};
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%create two coordinates
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\coordinate (phisw) at ([xshift=1cm]phi.south west);
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\coordinate (phise) at ([xshift=-1cm]phi.south east);
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%draw diamond
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\draw[fill=gray!20] (phi.north west) -- (phisw) -- (phi.south east) -- (phise |- phi.north west) -- cycle;
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\draw[dashed] (phisw) -- (phisw |- phi.north west);
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\draw[dashed] (phise) -- (phise |- phi.north west);
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%braces
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\begin{scope}[decoration={brace,mirror}]
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\draw[decorate] ([yshift=-1mm]phisw) -- node[midway,below] {$\vect{A}_1^T$} ([yshift=-1mm]phise);
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\draw[decorate] ([yshift=-7mm]phi.south west) -- node[midway,below] {$\vect{A}_2^T$} ([yshift=-7mm]phi.south east);
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\draw[decorate] ([yshift=-13mm]phi.south west) -- node[midway,below] {$\vect{A}_3^T$} ([yshift=-13mm]phise);
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\draw[decorate] ([yshift=-19mm]phisw) -- node[midway,below] {$\vect{A}_4^T$} ([yshift=-19mm]phi.south east);
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\end{scope}
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\end{tikzpicture}

‎lecture_notes/figures/nonwss_block_diagram.tikz

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\begin{tikzpicture}
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%draw matrix of nodes
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\matrix[row sep=5mm,column sep=10mm]{
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& & \node (desired) {$d(n)$}; & &\\
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\node (input) {$u(n)$}; & \node [blocknode] (filter) {$w_m(n)$}; & \node [combnode] (sumyd) {$\Sigma$}; & \node (error) {$e(n)$};\\
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};
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%draw arrows
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\begin{scope}[->,>=latex]
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%column 1
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\draw (desired) -- (sumyd);
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%column 2
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\draw (input) -- (filter);
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\draw (filter) -- node[above,midway]{$y(n)$} (sumyd);
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\path (sumyd) [late options={label=195:$-$}];
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\draw (sumyd) -- (error);
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\end{scope}
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\end{tikzpicture}

‎lecture_notes/figures/optimal_step_sd.tikz

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\begin{tikzpicture}[y=2cm]
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%axes
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\draw[->,>=latex] (-0.5,0) -- (8,0) node [below] {$\mu$};
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\draw[->,>=latex] (0,-0.2) -- (0,1.2) node [left] {$|1-\mu\lambda_m|$};
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\draw (-2pt,1) node [left] {$1$} -- (2pt,1);
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%eigenvalues
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\coordinate (lam1) at (1,0);
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\coordinate (lam2) at (2.5,0);
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\coordinate (lam3) at (4,0);
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\coordinate (muo) at (1.6,0.6);
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%draw modes
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\draw (0,1) -- (lam1) -- ($2*(lam1)+(0,1)$);
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\draw (0,1) -- (lam2) -- ($2*(lam2)+(0,1)$);
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\draw (0,1) -- (lam3) -- ($2*(lam3)+(0,1)$);
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%write text
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\begin{scope}[below]
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\node at (lam1) {$\frac{1}{\lambda_\textup{max}}$};
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\node at (lam2) {$\frac{1}{\lambda_2}$};
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\node at (lam3) {$\frac{1}{\lambda_\textup{min}}$};
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\draw[dashed] (muo) -- (muo |- 0,0) node {$\mu_o$};
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\filldraw (muo) circle (1pt);
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\end{scope}
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\end{tikzpicture}

‎lecture_notes/figures/orthogonality_principle.tikz

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\begin{tikzpicture}[->,>=latex]
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%axes
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\draw (-1,0) -- (7,0);
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\draw (0,-0.5) -- (0,3.5);
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%desired signal
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\draw[thick,>=stealth] (0,0) -- (1.5,3) node[near end,left,yshift=1mm] {$\vect{d}$};
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%column range of A
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\draw[>=] (-1,-1/3) -- (7,7/3) node[very near end,below,yshift=-2mm] {$\mathcal{R}(\vect{A})$};
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%optimal projector
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\draw[thick,>=stealth] (0,0) -- (9/4,3/4) node[below,yshift=-2.5mm] {$\vect{P}_u\vect{d}=\vect{A w}_o$};
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%error signal
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\draw[thick,>=stealth,dashed] (9/4,3/4) -- (1.5,3) node[midway,right,yshift=2mm] {$\vect{e}_o=\vect{P}_u^{\perp}\vect{d}$};
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%90 degree symbol
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\draw[shift={(9/4,3/4)},rotate=18.43] (0,0) rectangle +(-0.2,0.2);
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\end{tikzpicture}

‎lecture_notes/figures/overview.tikz

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\begin{tikzpicture}
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%draw matrix of nodes
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\matrix[row sep=5mm,column sep={40mm,between origins},nodes=blocknode]{
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& \node[left,xshift=15mm,fill=white,minimum width=7cm,text width=6.5cm] (bd) {\includegraphics[width=7cm]{figures/wss_block_diagram}};\\
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\node (wh) {Wiener-Hopf Equations}; & \node (ls) {Normal Equations}; \\[5mm]
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\node (sd) {Steepest Descent}; & \node (lms) {LMS}; \\
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& \node (nlms) {NLMS}; \\
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& \node (apa) {Affine Projections}; \\
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& \node (rls) {Recursive Least-Squares}; \\
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};
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%draw grid
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\draw [dashed] (bd.south) -- (bd.south |- rls.south);
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\draw [dashed] ([yshift=-5mm] wh.south west) -- ([yshift=-5mm] ls.south east);
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%draw arrows between blocks
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\begin{scope}[->,>=latex]
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%column 1
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\draw (bd) -- (wh);
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\draw (bd) -- (ls);
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%column 2
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\draw (wh) -- (sd);
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\draw (ls) -- ([xshift=10mm] ls.east) |- (rls);
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%column 3
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\draw (sd) -- (lms);
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\draw (sd) |- (nlms);
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\draw (lms) -- (nlms);
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%column 5
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\draw (apa) -- (nlms);
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\end{scope}
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%text
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\node[yshift=2mm,above] at (wh.north |- bd.north) {Known Statistics};
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\node[yshift=2mm,above] at (ls.north |- bd.north) {Unknown Statistics};
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\node[shift={(-2mm,2mm)},above,rotate=90] at (wh.north west) {WSS Signals};
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\node[shift={(-2mm,2mm)},above,rotate=90] at (sd.west |- nlms.south) {Non-WSS Signals};
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\draw[decorate,decoration={brace}] ([xshift=1mm]lms.north east) -- node[midway,above,rotate=-90,yshift=1mm] {Stochastic Gradient Methods} ([xshift=1mm]apa.south east);
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\draw[->,>=stealth] ([xshift=15mm] lms.north -| rls.east) -- node[midway,above,rotate=-90,yshift=1mm,text width=6cm,text centered] {Increasing Computational Complexity and Performance} ([xshift=15mm] rls.south east);
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\end{tikzpicture}

‎lecture_notes/figures/rand_proc_relation.tikz

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\begin{tikzpicture}
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\draw (0,0) ellipse (4 and 1.5);
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\draw (-0.75,0) ellipse (2.25 and 1.3);
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\node[draw,circle,text width=1.5cm,text centered] at (-1.75,0) {stationary processes};
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\node[circle,text width=1.5cm,text centered] at (0.25,0) {WSS processes};
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\node[circle,text width=2cm,text centered] at (2.75,0) {all stochastic processes};
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\end{tikzpicture}

‎lecture_notes/figures/sliding_window.tikz

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\begin{tikzpicture}[y=2cm]
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%axes
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\draw[->,>=latex] (-0.5,0) -- (8,0) node [below] {$i$};
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\draw[->,>=latex] (0,-0.2) -- (0,1.2) node [left] {$\beta(n,i)$};
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\draw (1pt,1) -- (-1pt,1) node[left] {1};
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%window
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\draw (3,1) -- (7,1);
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\draw[dashed] (3,1) -- (3,0) node[below] {$n-L$};
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\draw[dashed] (7,1) -- (7,0) node[below] {$n$};
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\draw[fill=white] (3,1) circle (1pt);
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\end{tikzpicture}

‎lecture_notes/figures/steepest_descent_idea_2d.tikz

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\begin{tikzpicture}
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%axes
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\begin{scope}[->,>=latex]
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\draw (-1,0) -- (6,0) node[below] {$w$};
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\draw (0,-0.5) -- (0,3.5);
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\end{scope}
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%cost function
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\draw[domain=-1:6,samples=100] plot (\x,{0.2*\x^2-1*\x+2}) node[right] {$J_1(w)$};
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%optimal filter
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\draw[dashed] (2.5,0) node[below] {$w_o$} -- (2.5,0.75);
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\filldraw (2.5,0.75) circle (1pt);
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%gradients
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\begin{scope}[->,>=stealth]
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%w(0)
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\draw (-0.5,51/20) -- (-1.7,51/20) node[below] {$g(w(0))$};
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\draw (-0.5,51/20) -- (4,51/20) node[above] {$-\frac{\mu}{2}g(w(0))$};
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%w(1)
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\draw (4,1.2) -- (4.6,1.2) node[below] {$g(w(1))$};
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\draw (4,1.2) -- (1.75,1.2) node[above] {$-\frac{\mu}{2}g(w(1))$};
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\end{scope}
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% filter values
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%w(0)
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\draw[dashed] (-0.5,0) node[below] {$w(0)$} -- (-0.5,51/20);
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\filldraw (-0.5,51/20) circle (1pt);
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%w(1)
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\draw[dashed] (4,0) node[below] {$w(1)$} -- (4,51/20);
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\filldraw (4,1.2) circle (1pt);
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%w(2)
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\draw[dashed] (1.75,0) node[below] {$w(2)$} -- (1.75,1.2);
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\filldraw (1.75,0.8625) circle (1pt);
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\end{tikzpicture}

‎lecture_notes/figures/steepest_descent_idea_3d.tikz

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\begin{tikzpicture}
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%axes
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\begin{scope}[->,>=latex]
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\draw (-1,0) -- (6,0) node[below] {$w_0$};
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\draw (0,-0.5) -- (0,4) node[left] {$w_1$};
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\end{scope}
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\coordinate (wo) at (2.3,1.5); %R = [5,3;3,5]/8; r = [2;1.8];
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\begin{scope}[shift={(wo)},rotate=-45]
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%optimal filter
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\filldraw (wo) ellipse (1pt and 0.5pt) node[left] {$\vect{w}_o$};
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%contours of the cost function
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\draw (0,0) ellipse (2.9967 and 1.4983);
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\draw (0,0) ellipse (1.7980 and 0.8990);
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\end{scope}
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%gradients
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\begin{scope}[->,>=stealth,every node/.style={fill=white,inner sep=1pt}]
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%w(0)
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\draw (4,1.8) -- (6.350,3.450) node[above left] {$\vect{g}(\vect{w}(0))$};
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\draw (4,1.8) -- (2.120,0.480) node[below right] {$-\frac{\mu}{2}\vect{g}(\vect{w}(0))$};
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%w(1)
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\draw (2.120,0.480) -- (1.13,-0.93) node[left] {$\vect{g}(\vect{w}(1))$};
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\draw (2.120,0.480) -- (2.912,1.608) node[above,yshift=2mm] {$-\frac{\mu}{2}\vect{g}(\vect{w}(1))$};
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\end{scope}
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% filter values
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%w(0)
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\filldraw (4,1.8) circle (1pt) node[right] {$\vect{w}(0)$};
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%w(1)
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\filldraw (2.120,0.480) circle (1pt) node[left] {$\vect{w}(1)$};
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\end{tikzpicture}
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%matlab
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%w
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%R=A;r=[2;1.8];wo=R\r;[X,L,dum]=svd(R);w = [4;1.8]; sqrt([1,0.25]*(X'*(w-wo)).^2)*[2,1],g = 2*R*w-2*r;g+w,u=2/trace(L);w-u*g/2

‎lecture_notes/figures/sys_id_block_diagram.tikz

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\begin{tikzpicture}
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%draw matrix of nodes
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\matrix[row sep=5mm,column sep=10mm]{
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& & & \node (noise) {$v(n)$}; & &\\
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\node (input) {$u(n)$};& \node [dotnode] (dotinp) {}; & \node [blocknode] (system) {$w_{o,m}$}; & \node [combnode] (sumzv) {$\Sigma$}; & &\\
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& & \node [blocknode] (filter) {$w_m(n)$}; & \node [combnode] (sumyd) {$\Sigma$}; & \node (error) {$e(n)$};\\
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};
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%draw arrows
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\begin{scope}[->,>=latex]
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%column 1
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\draw (noise) -- (sumzv);
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%column 2
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\draw (input) -- (dotinp) -- (system);
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\draw (system) -- node[above,midway]{$z(n)$} (sumzv);
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\draw (sumzv) -- node[right,near end]{$d(n)$} (sumyd);
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%column 3
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\draw (dotinp) |- (filter);
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\draw (filter) -- node[above,midway]{$y(n)$} (sumyd);
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\path (sumyd) [late options={label=195:$-$}];
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\draw (sumyd) -- (error);
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\end{scope}
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% draw dashed block around the adaptive filter
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\draw[dashed] ([shift={(-15mm,-2mm)}]filter.south west) rectangle ([shift={(40mm,2mm)}]filter.north east) node [above right,anchor=south east] {Adaptive Filter};
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\end{tikzpicture}

‎lecture_notes/figures/wss_block_diagram.tikz

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\begin{tikzpicture}
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%draw matrix of nodes
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\matrix[row sep=5mm,column sep=10mm]{
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& & \node (desired) {$d(n)$}; & &\\
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\node (input) {$u(n)$}; & \node [blocknode] (filter) {$w_m$}; & \node [combnode] (sumyd) {$\Sigma$}; & \node (error) {$e(n)$};\\
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};
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%draw arrows
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\begin{scope}[->,>=latex]
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%column 1
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\draw (desired) -- (sumyd);
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%column 2
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\draw (input) -- (filter);
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\draw (filter) -- node[above,midway]{$y(n)$} (sumyd);
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\path (sumyd) [late options={label=195:$-$}];
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\draw (sumyd) -- (error);
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\end{scope}
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\end{tikzpicture}

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