Fourier series & synthesis

Some basic examples of Fourier series, synthesis and series of sines, cosines, rectangle, sawtooth, triangle functions, as well as a 3D figure to breakdown synthesis into the time and frequency domain.
For more Fourier analysis figures, please see the “fourier analysis” tag. These figures are used in Ben Kilminster’s lecture notes for PHY111.

fourier_series-001.pngfourier_series-002.pngfourier_series-003.pngfourier_series-004.pngfourier_series-005.pngfourier_series-006.pngfourier_series-007.pngfourier_series-008.pngfourier_series-009.pngfourier_series-011.png

Edit and compile if you like:

% Author: Izaak Neutelings (January 2021)
% http://pgfplots.net/tikz/examples/fourier-transform/
% https://tex.stackexchange.com/questions/127375/replicate-the-fourier-transform-time-frequency-domains-correspondence-illustrati
% https://www.dspguide.com/ch13/4.htm
\documentclass[border=3pt,tikz]{standalone}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{physics}
\usepackage[outline]{contour} % glow around text
\usepackage{xcolor}
\usetikzlibrary{intersections}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{angles,quotes} % for pic
\usetikzlibrary{calc}
\usetikzlibrary{3d}
\contourlength{1.3pt}

\tikzset{>=latex} % for LaTeX arrow head
\colorlet{myred}{red!85!black}
\colorlet{myblue}{blue!80!black}
\colorlet{mycyan}{cyan!80!black}
\colorlet{mygreen}{green!70!black}
\colorlet{myorange}{orange!90!black!80}
\colorlet{mypurple}{red!50!blue!90!black!80}
\colorlet{mydarkred}{myred!80!black}
\colorlet{mydarkblue}{myblue!80!black}
\tikzstyle{xline}=[myblue,thick]
\def\tick#1#2{\draw[thick] (#1) ++ (#2:0.1) --++ (#2-180:0.2)}
\tikzstyle{myarr}=[myblue!50,-{Latex[length=3,width=2]}]
\def\N{90}

\begin{document}


% SQUARE WAVE 
\def\xmin{-0.7*\T}   % min x axis
\def\xmax{6.0}       % max x axis
\def\ymin{-1.04}     % min y axis
\def\ymax{1.3}       % max y axis
\def\A{0.67*\ymax}   % amplitude
\def\T{(0.35*\xmax)} % period
\def\f#1{\A*4/pi/(#1)*sin(360/\T*#1*Mod(\t,\T))} %Mod(360*#1*\t/\T,360)
\begin{tikzpicture}
  \message{^^JSquare wave}
  
  % AXIS
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] ({\xmin},0) -- (\xmax,0) node[below=1,right=1] {$t$ [s]};
  
  % PLOT
  \begin{scope}
    \clip ({0.9*\xmin},-1.1*\A) rectangle (0.95*\xmax,1.1*\A);
    \foreach \i [evaluate={\x=\i*\T/2;}] in {-2,...,5}{
      %\coordinate (A\i) at (\x,{-\A});
      %\coordinate (B\i) at (\x,{-\A});
      \ifodd\i
        \draw[xline,very thick,line cap=round] (\x,{-\A}) --++ ({\T/2},0);
        \draw[xline,dashed,thin,line cap=round]
          ({\x+\T/2},{-\A}) --++ (0,2*\A);
      \else
        \draw[xline,very thick,line cap=round] (\x,{\A}) --++ ({\T/2},0);
        \draw[xline,dashed,thin,line cap=round]
          ({\x+\T/2},{\A}) --++ (0,-2*\A);
      \fi
    }
  \end{scope}
  
  % TICKS
  \tick{{ -\T/2},0}{90} node[below=-1,scale=0.8] {\contour{white}{$-T/2$}};
  \tick{{  \T  },0}{90} node[below=-1,scale=0.8] {\contour{white}{$T$}};
  \tick{{  \T/2},0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$T/2$}};
  \tick{{3*\T/2},0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$3T/2$}};
  \tick{{2*\T  },0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$2T$}};
  \tick{{5*\T/2},0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$5T/2$}};
  \tick{0,{ \A}}{  0} node[left=-1,scale=0.9] {$A$};
  \tick{0,{-\A}}{180} node[right=-2,scale=0.9] {$-A$};
  
\end{tikzpicture}


% SQUARE WAVE - offset
\begin{tikzpicture}
  \message{^^JSquare wave}
  \def\ymin{-0.6} % min y axis
  \def\ymax{ 1.8} % max y axis
  
  % AXIS
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] ({\xmin},0) -- (\xmax,0) node[below=1,right=1] {$t$ [s]};
  
  % PLOT
  \begin{scope}
    \clip ({0.9*\xmin},1.1*\ymin) rectangle (0.95*\xmax,1.1*\A);
    \foreach \i [evaluate={\x=\i*\T/2;}] in {-2,...,5}{
      \ifodd\i
        \draw[xline,very thick,line cap=round] (\x,{0}) --++ ({\T/2},0);
        \draw[xline,dashed,thin,line cap=round]
          ({\x+\T/2},{0}) --++ (0,\A);
      \else
        \draw[xline,very thick,line cap=round] (\x,{\A}) --++ ({\T/2},0);
        \draw[xline,dashed,thin,line cap=round]
          ({\x+\T/2},{\A}) --++ (0,-\A);
      \fi
    }
  \end{scope}
  
  % TICKS
  \tick{{ -\T/2},0}{90} node[below=-1,scale=0.8] {\contour{white}{$-T/2$}};
  \tick{{  \T  },0}{90} node[below=-1,scale=0.8] {\contour{white}{$T$}};
  \tick{{  \T/2},0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$T/2$}};
  \tick{{3*\T/2},0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$3T/2$}};
  \tick{{2*\T  },0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$2T$}};
  \tick{{5*\T/2},0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$5T/2$}};
  \tick{0,{ \A}}{  0} node[left=-1,scale=0.9] {$A$};
  
\end{tikzpicture}


% SAW-TOOTH WAVE
\begin{tikzpicture}
  \message{^^JSaw-tooth wave}
  
  % AXIS
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] ({\xmin},0) -- (\xmax,0) node[below=1,right=1] {$t$ [s]};
  
  % PLOT
  \begin{scope}
    \clip ({0.9*\xmin},-1.1*\A) rectangle (0.95*\xmax,1.1*\A);
    \foreach \i [evaluate={\x=\i*\T-\T/2;}] in {-2,...,4}{
      \draw[xline,dashed,thin,line cap=round]
        ({\x+\T},\A) --++ (0,-2*\A);
      \draw[xline,very thick,line cap=round]
        (\x,-\A) --++ ({\T},2*\A);
    }
  \end{scope}
  
  % TICKS
  \tick{{ -\T/2},0}{90} node[below=-1,scale=0.8] {\contour{white}{$-T/2$}};
  \tick{{  \T  },0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$T$}};
  \tick{{  \T/2},0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$T/2$}};
  \tick{{3*\T/2},0}{90} node[right=0,below=-1,scale=0.8] {\contour{white}{$3T/2$}};
  \tick{{2*\T  },0}{90} node[right=0,below=-1,scale=0.8] {\contour{white}{$2T$}};
  \tick{{5*\T/2},0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$5T/2$}};
  \tick{0,{ \A}}{0} node[left=-1,scale=0.9] {$A$};
  \tick{0,{-\A}}{0} node[below=1,left=-1,scale=0.9] {$-A$};
  
\end{tikzpicture}


% TRIANGLE WAVE
\begin{tikzpicture}
  \message{^^JSaw-tooth wave}
  \def\T{(0.355*\xmax)} % period
  
  % AXIS
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] ({\xmin},0) -- (\xmax,0) node[below=1,right=1] {$t$ [s]};
  
  % PLOT
  \begin{scope}
    \clip ({0.9*\xmin},-1.1*\A) rectangle (0.95*\xmax,1.1*\A);
    \draw[xline,very thick,line cap=round]
      \foreach \i [evaluate={\x=\i*\T;}] in {-2,...,4}{
        (\x,0) --++ ({0.25*\T},\A) --++ ({\T/2},-2*\A) --++ ({0.25*\T},\A)};
  \end{scope}
  
  % TICKS
  \tick{{ -\T/2},0}{90} node[below=-1,scale=0.8] {\contour{white}{$-T/2$}};
  \tick{{  \T  },0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$T$}};
  \tick{{  \T/2},0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$T/2$}};
  \tick{{3*\T/2},0}{90} node[right=0,below=-1,scale=0.8] {\contour{white}{$3T/2$}};
  \tick{{2*\T  },0}{90} node[right=0,below=-1,scale=0.8] {\contour{white}{$2T$}};
  \tick{{5*\T/2},0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$5T/2$}};
  \tick{0,{ \A}}{0} node[left=-1,scale=0.8] {$A$};
  \tick{0,{-\A}}{180} node[below=0,right=-1,scale=0.9] {$-A$};
  %\tick{0,{-\A}}{0} node[below=2,left=-1,scale=0.9] {$-A$};
  
\end{tikzpicture}


% SQUARE WAVE SYNTHESIS - time domain
\begin{tikzpicture}
   \message{^^JSquare wave synthesis - time}
   \def\xmin{-0.65*\T}  % max x axis
   \def\T{(0.465*\xmax)} % period
  
  % SQUARE WAVE
  \begin{scope}
    \clip ({-0.54*\T},-1.1*\A) rectangle (0.97*\xmax,1.1*\A);
    \foreach \i [evaluate={\x=\i*\T/2;}] in {-2,...,4}{
      \ifodd\i
        \draw[myblue!80!black!30,line cap=round] (\x,{-\A}) --++ ({\T/2},0);
        \draw[myblue!80!black!30,dashed,thin,line cap=round]
          ({\x+\T/2},{-\A}) --++ (0,2*\A);
      \else
        \draw[myblue!80!black!30,line cap=round] (\x,{\A}) --++ ({\T/2},0);
        \draw[myblue!80!black!30,dashed,thin,line cap=round]
          ({\x+\T/2},{\A}) --++ (0,-2*\A);
      \fi
    }
  \end{scope}
  
  % AXIS
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] ({\xmin},0) -- (\xmax,0) node[below=1,right=1] {$t$ [s]};
  
  % PLOT
  \draw[xline,samples=\N,smooth,variable=\t,domain=-0.55*\T:0.94*\xmax]
    plot(\t,{\f{1}});% node[pos=0.3,above] {$n=1$};
  \draw[xline,mygreen,samples=3*\N,smooth,variable=\t,domain=-0.54*\T:0.94*\xmax]
    plot(\t,{\f{1}+\f{3}});
  \draw[xline,myred,samples=5*\N,smooth,variable=\t,domain=-0.53*\T:0.94*\xmax]
    plot(\t,{\f{1}+\f{3}+\f{5}});
  \draw[xline,myorange,line width=0.7,samples=7*\N,smooth,variable=\t,domain=-0.52*\T:0.94*\xmax]
    plot(\t,{\f{1}+\f{3}+\f{5}+\f{7}});
  %\draw[xline,mypurple,samples=9*\N,smooth,variable=\t,domain=-0.52*\T:0.95*\xmax]
  %  plot(\t,{\f{1}+\f{3}+\f{5}+\f{7}+\f{9}});
  %\node[xcol,above=2,right=4] at ({720/\om},\A) {$x(t)=A\cos(\omega t)$};
  
  % NUMBERS
  \node[myblue,  above,scale=0.9] at ({0.16*\T},1.20*\A) {$s_1$};
  \node[mygreen, below,scale=0.9] at ({0.25*\T},0.88*\A) {$s_3$};
  \node[myred,   above,scale=0.9] at ({0.41*\T},1.17*\A) {$s_5$};
  \node[myorange,right,scale=0.9] at ({0.48*\T},0.50*\A) {$s_7$};
  
  % TICKS
  \tick{{  \T},0}{90} node[below right=-2,scale=0.8] {$T$};
  \tick{{2*\T},0}{90} node[below right=-2,scale=0.8] {$2T$};
  %\tick{{2*\T  },0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$2T$}};
  \tick{0,{ \A}}{  0} node[left=-1,scale=0.9] {$A$};
  \tick{0,{-\A}}{180} node[right=-2,scale=0.9] {$-A$};
  
\end{tikzpicture}


% SAWTOOTH WAVE SYNTHESIS - time domain
\begin{tikzpicture}
   \message{^^JSawtooth wave synthesis - time}
   \def\xmin{-0.65*\T}  % max x axis
   \def\T{(0.465*\xmax)} % period
   \def\f#1{\A*2/pi/(#1)*(-1)^(#1-1)*sin(360/\T*#1*Mod(\t,\T))} %Mod(360*#1*\t/\T,360)
  
  % PLOT
  \begin{scope}
    \clip ({0.9*\xmin},-1.1*\A) rectangle (0.98*\xmax,1.1*\A);
    \foreach \i [evaluate={\x=\i*\T-\T/2;}] in {-2,...,4}{
      \draw[myblue!80!black!30,line cap=round]
        (\x,-\A) --++ ({\T},2*\A);
      \draw[myblue!80!black!30,dashed,thin,line cap=round]
        ({\x+\T},\A) --++ (0,-2*\A);
    }
  \end{scope}
  
  % AXIS
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] ({\xmin},0) -- (\xmax,0) node[below=1,right=1] {$t$ [s]};
  
  % PLOT
  \draw[xline,samples=\N,smooth,variable=\t,domain=-0.55*\T:0.95*\xmax]
    plot(\t,{\f{1}});% node[pos=0.3,above] {$n=1$};
  \draw[xline,mygreen,samples=2*\N,smooth,variable=\t,domain=-0.54*\T:0.95*\xmax]
    plot(\t,{\f{1}+\f{2}});
  \draw[xline,myred,samples=3*\N,smooth,variable=\t,domain=-0.53*\T:0.95*\xmax]
    plot(\t,{\f{1}+\f{2}+\f{3}});
  \draw[xline,myorange,samples=4*\N,smooth,variable=\t,domain=-0.52*\T:0.95*\xmax]
    plot(\t,{\f{1}+\f{2}+\f{3}+\f{4}});
  \draw[xline,mypurple,line width=0.7,samples=5*\N,smooth,variable=\t,domain=-0.52*\T:0.95*\xmax]
    plot(\t,{\f{1}+\f{2}+\f{3}+\f{4}+\f{5}});
  
  % NUMBERS
  \node[myblue,  above,scale=0.9] at ({0.09*\T}, 0.47*\A) {$s_1$};
  \node[mygreen, above,scale=0.9] at ({0.21*\T}, 0.68*\A) {$s_2$};
  \node[myred,   above,scale=0.9] at ({0.32*\T}, 0.93*\A) {$s_3$};
  \node[myorange,below,scale=0.9] at ({0.68*\T},-0.88*\A) {$s_4$};
  \node[mypurple,below,scale=0.9] at ({0.53*\T},-0.92*\A) {$s_5$};
  
  % TICKS
  \tick{{  \T},0}{90} node[below right=-2,scale=0.9] {$T$};
  \tick{{2*\T},0}{90} node[below right=-2,scale=0.9] {$2T$};
  \tick{0,{ \A}}{  0} node[left=-1,scale=0.9] {$A$};
  \tick{0,{-\A}}{180} node[right=-2,scale=0.9] {$-A$};
  
\end{tikzpicture}


% TRIANGLE WAVE SYNTHESIS - time domain
\begin{tikzpicture}
  \message{^^JTriangle wave synthesis - time}
   \def\xmin{-0.65*\T}  % max x axis
   \def\T{(0.465*\xmax)} % period
   \def\f#1{\A*8/pi^2/(#1)^2*(-1)^((#1-1)/2)*sin(360/\T*#1*Mod(\t,\T))} %Mod(360*#1*\t/\T,360)
  
  % SQUARE WAVE
  \begin{scope}
    \clip ({-0.59*\T},-1.1*\A) rectangle (0.96*\xmax,1.1*\A);
    \draw[myblue!80!black!30,line cap=round]
      \foreach \i [evaluate={\x=\i*\T;}] in {-2,...,4}{
        (\x,0) --++ ({0.25*\T},\A) --++ ({\T/2},-2*\A) --++ ({0.25*\T},\A)};
  \end{scope}
  
  % AXIS
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
  \draw[->,thick] ({\xmin},0) -- (\xmax,0) node[below=1,right=1] {$t$ [s]};
  
  % PLOT
  \draw[xline,samples=\N,smooth,variable=\t,domain=-0.55*\T:0.96*\xmax]
    plot(\t,{\f{1}});% node[pos=0.3,above] {$n=1$};
  \draw[xline,mygreen,line width=1.2,samples=2*\N,smooth,variable=\t,domain=-0.54*\T:0.96*\xmax]
    plot(\t,{\f{1}+\f{3}});
  \draw[xline,myred,line width=0.6,samples=3*\N,smooth,variable=\t,domain=-0.53*\T:0.96*\xmax]
    plot(\t,{\f{1}+\f{3}+\f{5}});
  %\draw[xline,myorange,line width=0.6,samples=3*\N,smooth,variable=\t,domain=-0.52*\T:0.96*\xmax]
  %  plot(\t,{\f{1}+\f{3}+\f{5}+\f{7}});
  %\draw[xline,mypurple,samples=9*\N,smooth,variable=\t,domain=-0.52*\T:0.95*\xmax]
  %  plot(\t,{\f{1}+\f{3}+\f{5}+\f{7}+\f{9}});
  %\node[xcol,above=2,right=4] at ({720/\om},\A) {$x(t)=A\cos(\omega t)$};
  
  % NUMBERS
  \node[myblue,  above,scale=0.9] at ({0.08*\T},0.53*\A) {$s_1$};
  \node[mygreen, above,scale=0.9] at ({0.38*\T},0.62*\A) {$s_3$};
  \node[myred,   above,scale=0.9] at ({0.55*\T},0.01*\A) {$s_5$};
  %\node[myorange,right,scale=0.9] at ({0.50*\T},0.55*\A) {$s_7$};
  
  % TICKS
  \tick{{  \T},0}{90} node[below right=-2,scale=0.8] {$T$};
  \tick{{2*\T},0}{90} node[below right=-2,scale=0.8] {$2T$};
  %\tick{{2*\T  },0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$2T$}};
  \tick{0,{ \A}}{  0} node[left=-1,scale=0.9] {$A$};
  \tick{0,{-\A}}{180} node[right=-2,scale=0.9] {$-A$};
  
\end{tikzpicture}


%% SQUARE WAVE SYNTHESIS - relative difference
%\begin{tikzpicture}
%  \message{^^JSquare wave synthesis - relative difference}
%  \def\xmin{-0.60*\T} % min x axis
%  \def\ymin{-1.2}     % min y axis
%  \def\ymax{1.4}      % max y axis
%  \def\T{(0.6*\xmax)} % period
%  
%  % AXIS
%  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[left] {$y$};
%  \draw[->,thick] ({\xmin},0) -- (\xmax,0) node[below=1,right=1] {$t$ [s]};
%  
%  % PLOT
%  \foreach \i [evaluate={\x=\i*\T/2;}] in {-1,...,2}{
%    \draw[xline,samples=\N,smooth,variable=\t,domain=\x:\x+\T/2]
%      plot(\t,{(\f{1}-((-1)^\i)*\A)/abs(\A)});
%    \draw[xline,mygreen,samples=\N,smooth,variable=\t,domain=\x:\x+\T/2]
%      plot(\t,{(\f{1}+\f{3}-((-1)^\i)*\A)/abs(\A)});
%    \draw[xline,myred,samples=\N,smooth,variable=\t,domain=\x:\x+\T/2]
%      plot(\t,{(\f{1}+\f{3}+\f{5}-((-1)^\i)*\A)/abs(\A)});
%    \draw[xline,myorange,samples=\N,smooth,variable=\t,domain=\x:\x+\T/2]
%      plot(\t,{(\f{1}+\f{3}+\f{5}+\f{7}-((-1)^\i)*\A)/abs(\A)});
%    \draw[xline,mypurple,samples=\N,smooth,variable=\t,domain=\x:\x+\T/2]
%      plot(\t,{(\f{1}+\f{3}+\f{5}+\f{7}+\f{9}-((-1)^\i)*\A)/abs(\A)});
%  }
%  
%%  % NUMBERS
%%  \node[myblue,  above,scale=0.7] at ({0.19*\T},1.25*\A) {$n=1$};
%%  \node[mygreen, below,scale=0.7] at ({0.25*\T},0.90*\A) {$n=2$};
%%  \node[myred,   above,scale=0.7] at ({0.51*\T},1.21*\A) {$n=3$};
%%  \node[myorange,right,scale=0.7] at ({0.50*\T},0.55*\A) {$n=4$};
%  
%  % TICKS
%  \tick{{  \T},0}{90} node[below right=-2,scale=0.8] {$T$};
%  \tick{{2*\T},0}{90} node[below right=-2,scale=0.8] {$2T$};
%  %\tick{{2*\T  },0}{90} node[right=1,below=-1,scale=0.8] {\contour{white}{$2T$}};
%  \tick{0,{ \A}}{  0} node[left=-1,scale=0.8] {$A$};
%  %\tick{0,{-\A}}{180} node[right=-2,scale=0.8] {$-A$};
%  
%\end{tikzpicture}



% SQUARE WAVE SYNTHESIS - frequency domain
\begin{tikzpicture}
  \message{^^JSquare wave synthesis - frequency}
  \def\ymax{2.3}     % max y axis
  \def\A{0.6*\ymax}  % amplitude
  \def\w{\xmax/10.4} % component spacing
  
  % AXIS
  \draw[->,thick] (0,-0.2*\ymax) -- (0,\ymax) node[above=1,left] {$b_n$};
  \draw[->,thick] ({-0.2*\ymax},0) -- (\xmax,0)
    node[below=1,right=1] {$f_n$ $\left[\frac{1}{\mathrm{s}}\right]$};
  \draw[myblue!90!black,dash pattern=on 2 off 2] (0,{\A*4/pi}) --++ ({\w},0);
  \draw[mygreen!90!black,dash pattern=on 2 off 2] (0,{\A*4/3/pi}) --++ ({3*\w},0);
  
  % ENVELOPE
  \draw[myblue!30,dashed,samples=\N,smooth,variable=\t,domain=0.08*\xmax:0.96*\xmax]
    plot(\t,{\A*4/pi/\t*\w}) node[right=2,above=0,scale=0.7] {$\dfrac{4A}{\pi n}$};
  
  % COMPONENTS
  \foreach \i/\col [evaluate={\x=\w*\i;}] in {
      1/myblue,3/mygreen,5/myred,7/myorange,9/mypurple}{
    \draw[\col!90!black,dash pattern=on 2 off 2]
      (\x,0) --++ (0,{\A*4/pi/\i});
    \fill[\col] (\x,{\A*4/pi/\i}) circle(0.06);
    \tick{\x,0}{90}
      node[below=-1,scale=0.8]
      {$\i/T$};
      %{$\dfrac{\i}{T}$}; %f_\i=\ifnum\i=1 \else \i \fi T
  }
  \foreach \i/\col [evaluate={\x=\w*\i;}] in {2,4,...,8}{
    \fill[myblue!60!black] (\x,0) circle(0.06);
  }
  \tick{0,\A*4/pi}{0} node[myblue!80!black,below=1,left=-1,scale=0.8] {$\dfrac{4A}{\pi}$};
  \tick{0,\A*4/3/pi}{0} node[mygreen!80!black,below=0,left=-1,scale=0.8] {$\dfrac{4A}{3\pi}$};
  
\end{tikzpicture}


% SAWTOOTH WAVE SYNTHESIS - frequency domain
% https://mathworld.wolfram.com/FourierSeriesSawtoothWave.html
\begin{tikzpicture}
  \message{^^JSquare wave synthesis - frequency}
  \def\ymin{-1.6}    % min y axis
  \def\ymax{2.3}     % max y axis
  \def\A{1.18*\ymax} % amplitude
  \def\w{\xmax/7}    % component spacing
  
  % AXIS
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[above=1,left] {$b_n$};
  \draw[->,thick] ({-0.2*\ymax},0) -- (\xmax,0)
    node[below=1,right=1] {$f_n$ $\left[\frac{1}{\mathrm{s}}\right]$};
  \draw[myblue!90!black,dash pattern=on 2 off 2] (0,{\A*2/pi}) --++ ({\w},0);
  \draw[mygreen!90!black,dash pattern=on 2 off 2] (0,{-\A/pi}) --++ ({2*\w},0);
  
  % ENVELOPE
  \draw[myblue!30,dashed,samples=\N,smooth,variable=\t,domain=0.115*\xmax:0.96*\xmax]
    plot(\t,{\A*2/pi/\t*\w}) node[right=2,above=0,scale=0.7] {$\dfrac{2A}{\pi n}$};
  \draw[myblue!30,dashed,samples=\N,smooth,variable=\t,domain=0.166*\xmax:0.96*\xmax]
    plot(\t,{-\A*2/pi/\t*\w}) node[left=1,below=0,scale=0.7] {$-\dfrac{2A}{\pi n}$};
  
  % COMPONENTS
  \foreach \i/\col [evaluate={\x=\w*\i;}] in {
      1/myblue,2/mygreen,3/myred,4/myorange,5/mypurple,6/mycyan}{
    \ifodd\i
      \draw[\col!90!black,dash pattern=on 2 off 2]
        (\x,0) --++ (0,{\A*2/pi/\i});
      \fill[\col] (\x,{\A*2/pi/\i}) circle(0.06);
      \tick{\x,0}{90} node[below=-1,scale=0.8] {\contour{white}{$\i/T$}};
    \else
      \draw[\col!90!black,dash pattern=on 2 off 2]
        (\x,0) --++ (0,{-\A*2/pi/\i});
      \fill[\col] (\x,{-\A*2/pi/\i}) circle(0.06);
      \tick{\x,0}{-90} node[above=-1,scale=0.8] {\contour{white}{$\i/T$}};
    \fi
  }
  \tick{0,\A*2/pi}{0} node[myblue!80!black,below=1,left=-1,scale=0.8] {$\dfrac{2A}{\pi}$};
  \tick{0,-\A/pi}{0} node[mygreen!80!black,below=0,left=-1,scale=0.8] {$-\dfrac{A}{\pi}$};
  
\end{tikzpicture}


% TRIANGLE WAVE SYNTHESIS - frequency domain
% https://archive.lib.msu.edu/crcmath/math/math/f/f270.htm
% https://mathworld.wolfram.com/FourierSeriesTriangleWave.html
\begin{tikzpicture}
  \message{^^JTriangle wave synthesis - frequency}
  \def\ymin{-1.0}    % min y axis
  \def\ymax{2.9}     % max y axis
  \def\A{1.05*\ymax} % amplitude
  \def\w{\xmax/9.1}  % component spacing
  
  % AXIS
  \draw[->,thick] (0,\ymin) -- (0,\ymax) node[above=1,left] {$b_n$};
  \draw[->,thick] ({-0.2*\ymax},0) -- (\xmax,0)
    node[below=1,right=1] {$f_n$ $\left[\frac{1}{\mathrm{s}}\right]$};
  \draw[myblue!90!black,dash pattern=on 2 off 2] (0,{8*\A/pi^2}) --++ ({\w},0);
  \draw[mygreen!90!black,dash pattern=on 2 off 2] (0,{-8*\A/pi^2/9}) --++ ({3*\w},0);
  
  % ENVELOPE
  \draw[myblue!30,dashed,samples=\N,smooth,variable=\t,domain=0.10*\xmax:0.96*\xmax]
    plot(\t,{ 8*\A*(\w/pi/\t)^2}) node[left=2,above=0,scale=0.7] {$\dfrac{8A}{\pi^2 n^2}$};
  \draw[myblue!30,dashed,samples=\N,smooth,variable=\t,domain=0.18*\xmax:0.96*\xmax]
    plot(\t,{-8*\A*(\w/pi/\t)^2}) node[left=5,below=0,scale=0.7] {$-\dfrac{8A}{\pi^2 n^2}$};
  
  % COMPONENTS
  \foreach \i/\col [evaluate={\x=\w*\i;}] in {
    1/myblue,3/mygreen,5/myred,7/myorange}{ %,9/mypurple,11/mycyan
    \draw[\col!90!black,dash pattern=on 2 off 2]
      (\x,0) --++ (0,{8*\A/(pi^2)/(\i^2)*(-1)^((\i-1)/2)});
    \fill[\col] (\x,{8*\A/(pi^2)/(\i^2)*(-1)^((\i-1)/2)}) circle(0.06);
    \ifnum\i=3
      \tick{\x,0}{90} node[below=4,scale=0.8] {$\i/T$}; %{\contour{white}{$\i/T$}};
    \else
      \tick{\x,0}{90} node[below=-1,scale=0.8] {\contour{white}{$\i/T$}};
    \fi
  }
  \foreach \i/\col [evaluate={\x=\w*\i;}] in {2,4,...,8}{
    \fill[myblue!60!black] (\x,0) circle(0.06);
  }
  \tick{0, 8*\A/pi^2  }{0} node[myblue!80!black,below=5,left=-1,scale=0.8] {$\dfrac{8A}{\pi^2}$};
  \tick{0,-8*\A/pi^2/9}{0} node[mygreen!80!black,below=6,left=-1,scale=0.8] {$-\dfrac{8A}{9\pi^2}$};
  
\end{tikzpicture}


% SYNTHESIS 3D
\begin{tikzpicture}[x=(-20:0.9), y=(90:0.9), z=(42:1.1)]
  \message{^^JSynthesis 3D}
  \def\xmax{6.5}        % max x axis
  \def\ymin{-1.2}       % min y axis
  \def\ymax{1.6}        % max y axis
  \def\zmax{5.8}        % max z axis
  \def\xf{1.17*\xmax}   % x position frequency axis
  \def\A{(0.60*\ymax)}  % amplitude
  \def\T{(0.335*\xmax)} % period
  \def\w{\zmax/11.2}    % spacing components
  
  % COMPONENTS
  \foreach \i/\col [evaluate={\z=\w*\i;}] in {
      11/mycyan,9/mypurple,7/myorange,5/myred,3/mygreen,1/myblue}{
    \draw[black!30] ({\T},0.1,\z) --++ (0,-0.2,0);
    \draw[black!30] ({2*\T},0.1,\z) --++ (0,-0.2,0);
    \draw[->,black!30] (0,0,\z) --++ (0.93*\xmax,0,0);
    \draw[xline,\col,opacity=0.8,thick,
          samples=\i*\N,smooth,variable=\t,domain=-0.05*\T:0.87*\xmax]
      plot(\t,{\f{\i}},\z);
  }
  
  % TIME DOMAIN
  \begin{scope}[shift={(0,0,-0.17*\zmax)}]
    \draw[black,fill=white,opacity=0.3,canvas is xy plane at z=0]
      (-0.1*\xmax,-1.25*\ymax) rectangle (1.13*\xmax,1.25*\ymax);
    \draw[->,thick] (-0.05*\xmax,0,0) -- (\xmax,0,0)
      node[below right=-3,canvas is xy plane at z=0] {$t$ [s]};
    \draw[->,thick] (0,\ymin,0) -- (0,\ymax,0)
      node[left,canvas is xy plane at z=0] {$y$};
    \draw[xline,blue!90!black,very thick,
          samples=9*\N,smooth,variable=\t,domain=-0.05*\T:0.9*\xmax]
      plot(\t,{\f{1}+\f{3}+\f{5}+\f{7}+\f{9}+\f{11}},0); %node[above] {$f$};
    \tick{{\T},0,0}{90}
      node[below=-1,scale=0.9,canvas is xy plane at z=0] {\contour{white}{$T$}};
    \tick{{2*\T},0,0}{90}
      node[below=-1,scale=0.9,canvas is xy plane at z=0] {\contour{white}{$2T$}};
    \node[scale=1,canvas is xy plane at z=0] at (0.4*\xmax,-\ymax,0) {Time domain};
  \end{scope}
  
  % FREQUENCY DOMAIN
  \begin{scope}[shift={(\xf,0,0)}]
    \draw[black,fill=white,opacity=0.3,canvas is zy plane at x=0]
      (-0.13*\zmax,-1.25*\ymax) rectangle (1.26*\zmax,1.25*\ymax);
    %\draw[->,thick] (0,0,0) -- (0,0,\zmax) node[above left=-1] {$z$};
    %\draw[->,thick] (\xmax,0,0) --++ (0,0,\zmax);
    \draw[->,thick] (0,0.8*\ymin,0) -- (0,\ymax,0)
      node[above=2,left=0,canvas is zy plane at x=0] {$b_n$};
      %node[pos=0.84,left=2,fill=white,inner sep=0] {$b_n$};
    \draw[->,thick] (0,0,-0.05*\zmax) --++ (0,0,1.13*\zmax)
      node[below right=-1,canvas is zy plane at x=0] {$f_n$ $\left[\frac{1}{\mathrm{s}}\right]$};
    \node[scale=1,canvas is zy plane at x=0] at (0,-\ymax,0.65*\zmax) {Frequency domain};
    \draw[myblue!30,dashed,samples=3*\N,smooth,variable=\t,domain=0.074*\zmax:1.02*\zmax]
      plot(0,{\A*4/pi/\t*\w},\t); %node[right=2,above=0,scale=0.7] {$\dfrac{4A}{\pi n}$};
    \foreach \i/\col [evaluate={\z=\w*\i;}] in {
        11/mycyan,9/mypurple,7/myorange,5/myred,3/mygreen,1/myblue}{
      \draw[\col,dash pattern=on 2 off 2]
        (0,0,\z) --++ (0,{\A*4/pi/\i},0);
      \fill[\col,canvas is zy plane at x=0]
        %(\xf,{\A*4/pi/\i},\z) circle(0.08);
        (\z,{\A*4/pi/\i}) circle(0.07);
      \tick{0,0,\z}{90}
        node[below=-1,scale=0.85,canvas is zy plane at x=0]
        {$\dfrac{\i}{T}$}; %f_\i=\ifnum\i=1 \else \i \fi T
    }
    \foreach \i [evaluate={\z=\w*\i;}] in {2,4,...,10}{
      \fill[myblue!60!black,canvas is zy plane at x=0] (\z,0) circle(0.07);
    }
  \end{scope}
  
\end{tikzpicture}


\end{document}

Click to download: fourier_series.texfourier_series.pdf
Open in Overleaf: fourier_series.tex

Leave a Reply

Your email address will not be published.