// Default texts for the math documentator of the Faust compiler. // Copyright (C) 2009 GRAME, Centre National de Creation Musicale // Karim Barkati // Remark: // double quotes are useful to visualize spaces and tabulations // at the beginning or at the end of strings. // Metadatas section. :name "name" :author "author" :copyright "copyright" :license "license" :version "version" // Notice section. :faustapply "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." :faustpresentation "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})." :causality "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." :blockdiagrams "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." :faustdocdir "The \texttt{\faustdocdir/} directory may also include the following subdirectories:" "\begin{itemize}" " \item \texttt{cpp/} for Faust compiled code; " " \item \texttt{pdf/} which contains this document; " " \item \texttt{src/} for all Faust sources used (even libraries); " " \item \texttt{svg/} for block diagrams, encoded using the Scalable Vector Graphics format (\textsf{http://www.w3.org/Graphics/SVG/});" " \item \texttt{tex/} for the \LaTeX\ source of this document." "\end{itemize}" :foreignfun ""ff" prefix in some function name stands for C++ "foreign function"." :intcast "$\forall \, x \in \mathbb{R}$, " " \begin{displaymath}" " \mathrm{int}(x) =" " \left\{\begin{array}{cl}" " \lfloor x \rfloor & \mbox{if \,} x > 0 \\" " \lceil x \rceil & \mbox{if \,} x < 0 \\" " 0 & \mbox{if \,} x = 0 \\" " \end{array}\right.." " \end{displaymath}" :operators "This document uses the following integer operations:" :optabtitle "\emph{operation} & \emph{name} & \emph{semantics} \\" :intplus "$i \oplus j$ & integer addition & $\mathrm{normalize}(i+j), \mathrm{~in~} \mathbb{Z}$ \\" :intminus "$i \ominus j$ & integer substraction & $\mathrm{normalize}(i-j), \mathrm{~in~} \mathbb{Z}$ \\" :intmult "$i \odot j$ & integer multiplication & $\mathrm{normalize}(i \cdot j), \mathrm{~in~} \mathbb{Z}$ \\" :intdiv "$i \oslash j$ & integer division & $\mathrm{normalize}(\mathrm{int}(i/j)), \mathrm{~in~} \mathbb{Q}$ \\" :integerops "Integer operations in Faust are inspired by the semantics of operations on the n-bit two's complement representation of integer numbers; they are internal composition laws on the subset $[\,-2^{n-1}, 2^{n-1}\!-\!1\,]$ of $\mathbb{Z}$, with $n = 32$. For any integer binary operation $\times$ on $\mathbb{Z}$, the $\otimes$ operation is defined as: $i \otimes j = \mathrm{normalize}(i \times j)$, with " "$$\mathrm{normalize}(i) = i - N\cdot\mathrm{sign}(i) \cdot \left\lfloor \frac{|i|+N/2+(\mathrm{sign}(i)\!-\!1)/2}{N} \right\rfloor , $$" " where $N = 2^n$ and $\mathrm{sign}(i) = 0 \mathrm{\ if\ } i=0 \mathrm{\ and\ } i / |i| \mathrm{\ otherwise}.$" "Unary integer operations are defined likewise." // Auto-documentation section. :thisdoc "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." :autoeqntitle "\section{Mathematical definition of \texttt{process}}" "\label{equation}" :autoeqntext "The \emph{\faustprogname} program evaluates the signal transformer denoted by \texttt{process}, which is mathematically defined as follows:" :autodgmtitle "\section{Block diagram of \texttt{process}}" "\label{diagram}" :autodgmtext "The block diagram of \texttt{process} is shown on Figure\,\ref{figure1} (page\,\pageref{figure1})." :autontctitle "\section{Notice}" "\label{notice}" :autontctext "This document was generated using Faust version \faustversion\ on \faustdocdate." :autolsttitle1 "\section{Faust code listing}" "\label{listing}" :autolsttext1 "This section provides the listing of the Faust code used to generate this document." :autolsttitle2 "\section{Faust code listings}" "\label{listing}" :autolsttext2 "This section provides the listings of the Faust code used to generate this document, including dependencies." // Titles for each type of formulas. :inputsigtitle1 "Input signal" :inputsigtitle2 "Input signals" :outputsigtitle1 "Output signal" :outputsigtitle2 "Output signals" :constsigtitle1 "Constant" :constsigtitle2 "Constants" :uisigtitle1 "User-interface input signal" :uisigtitle2 "User-interface input signals" :intermedsigtitle1 "Intermediate signal" :intermedsigtitle2 "Intermediate signals" :lateqcomment "% Set of Faust formulas (corresponding to an tag)." :emptyformulafield "none" :defaultvalue "default value" :suchthat "such that" :and "and" :for "for" :rootlevel "(at root level)" :dgmcaption "Block diagram of"