Example: Radix-2 FFT signal flow

Published 2006-12-13 | Author: Kjell Magne Fauske

Radix-2 signal flow graph for a 16 point fast Fourier transform (FFT). This diagram is quite complex. However, the most difficult part is keeping track of all the indexes. The foreach command is used extensively to get compact code.

Source: The diagram was inspired by content on this web page.

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Radix-2 FFT signal flow

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\documentclass{article}

\usepackage{tikz}

\usetikzlibrary{arrows}
\begin{document}
\pagestyle{empty}

\tikzstyle{n}= [circle, fill, minimum size=4pt,inner sep=0pt, outer sep=0pt]
\tikzstyle{mul} = [circle,draw,inner sep=-1pt]

% Define two helper counters
\newcounter{x}\newcounter{y}
\begin{tikzpicture}[yscale=0.5, xscale=1.2, node distance=0.3cm, auto]
    % The strategy is to create nodes with names: N-column-row
    % Input nodes are named N-0-0 ... N-0-15
    % Output nodes are named N-10-0 ... N-10-15

    % Draw inputs
    \foreach \y in {0,...,15}
        \node[n, pin={[pin edge={latex'-,black}]left:$x(\y)$}]
              (N-0-\y) at (0,-\y) {};
    % Draw outputs
    \foreach \y / \idx in {0/0,1/8,2/4,3/12,4/2,5/10,6,7/14,
                           8/1,9,10/5,11/13,12/3,13/11,14/7,15}
        \node[n, pin={[pin edge={-latex',black}]right:$X(\idx)$}]
              (N-10-\y) at (7,-\y) {};
   % draw connector nodes
    \foreach \y in {0,...,15}
        \foreach \x / \c in {1/1,2/3,3/4,4/6,5/7,6/9}
            \node[n, name=N-\c-\y] at (\x,-\y) {};
    % draw x nodes
    \foreach \y in {0,...,15}
        \foreach \x / \c  in {1/2,4/5,7/8}
            \node[mul, right of=N-\x-\y] (N-\c-\y) {${\times}$};

    % horizontal connections
    % Note the use of simple counter arithmetics to get correct
    % indexes.
    \foreach \y in {0,...,15}
        \foreach \x in {0,1,3,4,6,7,9}
        {
            \setcounter{x}{\x}\stepcounter{x}
            \path (N-\x-\y) edge[-] (N-\arabic{x}-\y);
       }
    % Draw the W_16 coefficients
    \setcounter{y}{0}
    \foreach \i / \j in {0/0,1/0,2/0,3/0,4/0,5/0,6/0,7/0,
                            0/1,1/1,2/1,3/1,4/1,5/1,6/1,7/1}
    {
        \path (N-2-\arabic{y}) edge[-] node {\tiny $W^{\i\cdot\j}_{16}$}
                (N-3-\arabic{y});
        \stepcounter{y}
    }
    % Draw the W_8 coefficients
    \setcounter{y}{0}
    \foreach \i / \j in {0/0,1/0,2/0,3/0,0/1,1/1,2/1,3/1,
                         0/0,1/0,2/0,3/0,0/1,1/1,2/1,3/1}
    {
        \path (N-5-\arabic{y}) edge[-] node {\tiny $W^{\i\cdot\j}_{8}$}
              (N-6-\arabic{y});
        \addtocounter{y}{1}
    }

    % Draw the W_4 coefficients
    \setcounter{y}{0}
    \foreach \i / \j in {0/0,1/0,0/1,1/1,0/0,1/0,0/1,1/1,
                            0/0,1/0,0/1,1/1,0/0,1/0,0/1,1/1}
    {
        \path (N-8-\arabic{y}) edge[-] node {\tiny $W^{\i\cdot\j}_{4}$}
              (N-9-\arabic{y});
        \stepcounter{y}
    }
    % Connect nodes
    \foreach \sourcey / \desty in {0/8,1/9,2/10,3/11,
                                   4/12,5/13,6/14,7/15,
                                   8/0,9/1,10/2,11/3,
                                   12/4,13/5,14/6,15/7}
       \path (N-0-\sourcey.east) edge[-] (N-1-\desty.west);
    \foreach \sourcey / \desty in {0/4,1/5,2/6,3/7,
                                   4/0,5/1,6/2,7/3,
                                   8/12,9/13,10/14,11/15,
                                   12/8,13/9,14/10,15/11}
        \path (N-3-\sourcey.east) edge[-] (N-4-\desty.west);
    \foreach \sourcey / \desty in {0/2,1/3,2/0,3/1,
                                   4/6,5/7,6/4,7/5,
                                   8/10,9/11,10/8,11/9,
                                   12/14,13/15,14/12,15/13}
        \path (N-6-\sourcey.east) edge[-] (N-7-\desty.west);
    \foreach \sourcey / \desty in {0/1,1/0,2/3,3/2,
                                   4/5,5/4,6/7,7/6,
                                   8/9,9/8,10/11,11/10,
                                   12/13,13/12,14/15,15/14}
        \path (N-9-\sourcey.east) edge[-] (N-10-\desty.west);

\end{tikzpicture}

\end{document}

Comments

  • #1 mario, February 25, 2010 at 6:14 a.m.

    Hola, esta excelente para Filtros digitales gracias

  • #2 Matt, October 5, 2010 at 4:38 p.m.

    This is a beautiful depiction of the flow of a fast Fourier transform algorithm.

    The community is indebted to wonderful examples like this. Keep up the good work!

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