dsp_files/make_init-mdoc/tex/make_init.tex

   1 %% This documentation was generated with Faust version 0.9.58
   2 %% Thu Jun 13 10:45:16 2013
   3 %% http://faust.grame.fr
   4
   5 \documentclass{article}
   6
   7 \usepackage[utf8]{inputenc}
   8 \usepackage{graphicx}
   9 \usepackage[usenames]{color}
  10 \usepackage{listings}
  11 \usepackage{supertabular}
  12 \usepackage{amsmath}
  13 \usepackage{latexsym, amssymb}
  14 \usepackage{breqn}
  15
  16 % No indent
  17 \setlength{\parindent}{0pt}
  18
  19 % Make LaTeX output a dot when typing an asterisk
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  21
  22 % lstlistings setup
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  25 \definecolor{mylstbg}{rgb}{0.98,0.98,0.98} % a really pale grey.
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  28
  29 \lstset{%
  30   language=C++,
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  69
  70 \newcommand{\faustfilename}{/home/wang/Documents/OcamlProjects/interpreter/interpreter11/dsp_files/make_init.dsp}
  71 \newcommand{\faustdocdir}{make_init-mdoc}
  72 \newcommand{\faustprogname}{make_init}
  73 \newcommand{\faustversion}{0.9.58}
  74 \newcommand{\faustdocdate}{June 13, 2013}
  75
  76 \begin{document}
  77 \title{make_init}
  78 \date{\today}
  79 \maketitle
  80
  81 \bigskip
  82 This document provides a mathematical description of the Faust program text stored in the \texttt{\faustfilename} file. See the notice in Section\,\ref{notice} (page\,\pageref{notice}) for details.
  83
  84
  85 \section{Mathematical definition of \texttt{process}}
  86 \label{equation}
  87
  88 The \emph{\faustprogname} program evaluates the signal transformer denoted by \texttt{process}, which is mathematically defined as follows:
  89
  90 % Set of Faust formulas (corresponding to an <equation> tag).
  91 \begin{enumerate}
  92
  93 \item Output signal $y$ such that
  94         \begin{dgroup*}
  95                 \begin{dmath*}
  96                                 y(t) = r_{1}(t)
  97                 \end{dmath*}
  98         \end{dgroup*}
  99
 100 \item Input signal (none)
 101
 102 \item Intermediate signals  $r_i$ for $i \in [1,2]$,  $q_1$ and  $m_1$ such that
 103         \begin{dgroup*}
 104                 \begin{dmath*}
 105                                 r_{2}(t) = 1
 106                 \end{dmath*}
 107                 \begin{dmath*}
 108                                 r_{1}(t) = 2.000001 + q_{1}(t)
 109                 \end{dmath*}
 110         \end{dgroup*}
 111
 112 \begin{displaymath}
 113 q_{1}(t) =
 114 \left\{\begin{array}{ll}
 115 2 & \mbox{if \,} \mathrm{int}\left(m_{1}(t)\right) = 0\\
 116 r_{1}(t\!-\!1) & \mbox{if \,} \mathrm{int}\left(m_{1}(t)\right) = 1
 117 \end{array}\right.
 118 \end{displaymath}
 119
 120 \begin{displaymath}
 121 m_{1}(t) =
 122 \left\{\begin{array}{ll}
 123 0 & \mbox{, when \,} t = 0\\
 124 r_{2}(t\!-\!1) & \mbox{, when \,} t > 0
 125 \end{array}\right.
 126 \end{displaymath}
 127
 128 \end{enumerate}
 129
 130
 131 \section{Block diagram of \texttt{process}}
 132 \label{diagram}
 133
 134 The block diagram of \texttt{process} is shown on Figure\,\ref{figure1} (page\,\pageref{figure1}).
 135 \begin{figure}[ht!]
 136         \centering
 137         \includegraphics[width=\textwidth]{../svg/svg-01/process}
 138         \caption{Block diagram of \texttt{process}}
 139         \label{figure1}
 140 \end{figure}
 141
 142
 143 \section{Notice}
 144 \label{notice}
 145
 146
 147 \begin{itemize}
 148         \item This document was generated using Faust version \faustversion\ on \faustdocdate.
 149         \item The value of a Faust program is the result of applying the signal transformer denoted by the expression to which the \texttt{process} identifier is bound to input signals, running at the $f_S$ sampling frequency.
 150         \item Faust (\emph{Functional Audio Stream}) is a functional programming language designed for synchronous real-time signal processing and synthesis applications. A Faust program is a set of bindings of identifiers to expressions that denote signal transformers. A signal $s$ in $S$ is a function mapping\footnote{Faust assumes that $\forall \, s \in S, \forall \, t \in \mathbb{Z}, s(t) = 0 \mathrm{\ when\ } t < 0$.} times $t \in \mathbb{Z}$ to values $s(t) \in \mathbb{R}$, while a signal transformer is a function from $S^n$ to $S^m$, where $n,m\in \mathbb{N}$. See the Faust manual for additional information (\textsf{http://faust.grame.fr}).
 151         \item Every mathematical formula derived from a Faust expression is assumed, in this document, to having been normalized (in an implementation-depen\-dent manner) by the Faust compiler.
 152         \item A block diagram is a graphical representation of the Faust binding of an identifier I to an expression E; each graph is put in a box labeled by I. Subexpressions of E are recursively displayed as long as the whole picture fits in one page.
 153         \item $\forall \, x \in \mathbb{R}$,
 154         \begin{displaymath}
 155                 \mathrm{int}(x) =
 156                 \left\{\begin{array}{cl}
 157                         \lfloor x \rfloor & \mbox{if \,} x > 0 \\
 158                         \lceil x \rceil & \mbox{if \,} x < 0 \\
 159                         0 & \mbox{if \,} x = 0 \\
 160                 \end{array}\right..
 161         \end{displaymath}
 162         \item The \texttt{\faustdocdir/} directory may also include the following subdirectories:
 163 \begin{itemize}
 164         \item   \texttt{cpp/} for Faust compiled code;
 165         \item   \texttt{pdf/} which contains this document;
 166         \item   \texttt{src/} for all Faust sources used (even libraries);
 167         \item   \texttt{svg/} for block diagrams, encoded using the Scalable Vector Graphics format (\textsf{http://www.w3.org/Graphics/SVG/});
 168         \item   \texttt{tex/} for the \LaTeX\ source of this document.
 169 \end{itemize}
 170 \end{itemize}
 171
 172
 173 \section{Faust code listing}
 174 \label{listing}
 175
 176 This section provides the listing of the Faust code used to generate this document.
 177
 178 \bigskip\bigskip
 179 \begin{lstlisting}[caption=\texttt{make_init.dsp}]
 180 rec(a, b, i) = a ~ (i : b);
 181
 182 vect0 = case {
 183         ((y, z)) => vect0(y) : vect0(z);
 184         (y) => _ , y : +;
 185 };
 186
 187 init_value = case {
 188         ((x,y)) => init_value(x) : vect0(y);
 189         (0) => 0;
 190         (x) => 0 : vect0(x);
 191 };
 192
 193 make_init(x) = (0, 1 : prefix), init_value(x), _ : select2;
 194
 195 A = +;
 196 B = _;
 197 //B = serialize : _, 2 : vectorize;
 198 I = make_init((2));
 199
 200 //process = _,_ : > ;
 201
 202 //process = (0.000001, 2 : +) : rec(A,B,I);
 203
 204 process = 0.0000010000000000,2:+:+~((0,1:prefix),(0:_,2:+),_:select2:_);
 205 \end{lstlisting}
 206
 207
 208 \end{document}
 209