How to make noisy signal in TikZ

Question

Any idea how to do this, please?

Answer 1

I will give you an idea with pgfplots and the declare function facility of TikZ.

I define a function (using a piecewise approach with the ternary function test ? true value : false value) and its more-or-less derivative, and then a "fake" noise function; fake in the sense that it's just a sinusoidal with a bit of randomness added to it, to simulate noise visually.

I also use styles to minimize repetitions and to concentrate common definitions in one site.

\documentclass[border=10pt]{standalone}
\usepackage[T1]{fontenc}
\usepackage{tikz}
\usepackage{pgfplots}\pgfplotsset{compat=newest}% change to the current release
\begin{document}
\pgfplotsset{
    base plot/.style={% common style for the plots
        width=10cm, height=5cm,
        xmin=0, xmax=10,
        domain=0:10,
        samples=100,
        axis x line = center,
        axis y line = center,
        enlarge x limits,
        enlarge y limits,
        xlabel = {$x$},
        % not used, but...
        legend style = {nodes={right, font=\scriptsize}, at={(0.05,0.6)}, anchor=west},
        clip mode = individual,
}}
\begin{tikzpicture}[declare function={
        % constant-ramp-constant function 
        xrampz(\x)=\x<2 ? 0 : (\x<6 ? \x-2 : 4);
        % generic derivate of the above
        Dxrampz(\x)=\x<2 ? 0 : (\x<6 ? 1 : 0);
        % pseudo-noise function, created "by feel"
        fakenoisesin(\A,\f,\x)=\A(0.6rand+sin(\f*\x));
    }]
    \begin{axis}[
        base plot,
        ymin=-1, ymax=5,
        ylabel = {$y$},
        ]
        \addplot[thick] {xrampz(x)+fakenoisesin(.1,400,x)};
        \addplot[red, thin] {xrampz(x)};
    \end{axis}
    \begin{axis}[
        base plot, yshift=-4cm,% move it down 
        ymin=-1, ymax=2,
        ylabel = {$y'$},
        ]
        \addplot[thick] {Dxrampz(x)+fakenoisesin(.2,400,x)};
        \addplot[red, thin] {Dxrampz(x)};
    \end{axis}
\end{tikzpicture}
\end{document}

Answer 2

Just for fun: The blue line is random white noise, and the red line has been processed by a 5 point weighted moving average filter.

One should really do this using C++ or matlab for both speed and accuracy.

\documentclass[border=10pt]{standalone}
%\usepackage[T1]{fontenc}
\usepackage{tikz}
\usepackage{pgfplots}\pgfplotsset{compat=newest}% change to the current release
\usepackage{pgfplotstable}
\begin{document}
\pgfmathsetmacro{\noise}{rand}
\foreach \x in {1,...,50} {\pgfmathsetmacro{\y}{1.0*rand}%
  \xdef\noise{\noise,\y}}%
\edef\noise{{\noise}}% pgfmath array
\pgfmathsetmacro{\ma}{0.375\noise[0]+0.25\noise[1]+0.0625\noise[2]}%
\pgfmathsetmacro{\y}{0.25\noise[0]+0.375\noise[1]+0.25\noise[2]+0.0625\noise[3]}%
\xdef\ma{\ma,\y}%
\foreach \x in {2,...,48} {\pgfmathsetmacro{\y}{0.0625\noise[\x-2]+0.25\noise[\x-1]+0.375\noise[\x]+0.25\noise[\x+1]+0.0625\noise[\x+2]}%
  \xdef\ma{\ma,\y}}%
\pgfmathsetmacro{\y}{0.0625\noise[47]+0.25\noise[48]+0.375\noise[49]+0.25\noise[50]}%
\edef\ma{\ma,\y}
\pgfmathsetmacro{\y}{0.0625\noise[48]+0.25\noise[49]+0.375*\noise[50]}%
\edef\ma{{\ma,\y}}
\pgfplotstablenew[columns={x,random,banded},
  create on use/x/.style={create col/set list={0,1,...,50}},
  create on use/random/.style={create col/set list/.expand once=\noise},
  create on use/banded/.style={create col/set list/.expand once=\ma}]
{51}\mytable
%\pgfplotstabletypeset{\mytable}
\begin{tikzpicture}
    \begin{axis}
        \addplot[blue] table[y=random] \mytable;
        \addplot[red] table[y=banded] \mytable;
    \end{axis}
\end{tikzpicture}
\end{document}