Commit a0a092d4 authored by Antonin Dudermel's avatar Antonin Dudermel
Browse files

presentation continuation

parent cc72fd62
#THX SOF https://tex.stackexchange.com/questions/40738/how-to-properly-make-a-latex-project#40759
FILENAME=finalpres
include ../Makefile.am
......@@ -211,6 +211,73 @@ Biggest (range): \(2^{\msb}\) \hspace{\stretch{1}} Smallest (accuracy): \(2^{\ls
\section{Bounding Signals to Avoid Overflow (MSB)}
\begin{frame}[t]{Interval Arithmetic}
Compute for each signal an interval containing the signal range by translating
elementary operations on intervals:
\[x\in X, y\in Y \Rightarrow x\diamond y \in X \diamond Y\]
\begin{eqnarray*}
y(t) &=& 1/2 (x_0(t) + x_1(t))\\
% Y &=& [1/2,1/2]\times(X_0 + X_1)\\
Y &=& [1/2,1/2]\times([-1,1] + [-1,1])\\
Y &=& [1/2,1/2]\times[-2,2]\\
Y &=& [-1,1]
\end{eqnarray*}
\centering
\only<2>{
\includegraphics[height = 0.15\textheight,valign=c]{../code/plus-sig-old.png}
}
\only<3>{
\includegraphics[height = 0.15\textheight, valign=c]{../code/plus-sig.png}
}
\end{frame}
\begin{frame}{Limitations}
\begin{itemize}
\item \emph{Correlation} between signals is lost:
\begin{itemize}
\item \(x\in [-1,1]\) gives \(x * x \in [-1, 1] * [-1, 1] = [-1, 1]\)
\item \lstinline{frac(n) = n - floor(n)}
\end{itemize}
\item Information may be hidden in the code
\[ x (1 - x) = 1/4 - (x-1/2)^2\]
\end{itemize}
\end{frame}
\begin{frame}{Recursive Signals}
Int phaser : \lstinline{process = \%(10)~ +(1);}
\begin{figure}
\centering
\subfloat[blocks]{\includegraphics[width = 0.4\textwidth, valign=c]{../code/rec-block.png}}
\hspace{\stretch{1}}
\subfloat[graph]{\includegraphics[width = 0.5\textwidth, valign=c]{../code/rec-sig-old.png}}
\end{figure}
Problem: The sample \(s(t)\) of the signal at time \(t\) relies on the
previous sample \(s(t-1)\).
\end{frame}
\begin{frame}{Fixpoint search}
Adapt \emph{Abstract analysis} to signals:
for each time \(t\), find an
interval \(S(t)\) containing \(s(t)\)
\[\small
\recSig{t}{s(t) = 0}{s(t) = f(s(t-1))} \leadsto
\recSig{t}{S(t) = \bot}{S(t) = f(S(t-1)) \bigcup S(t)}
\]
\(\bar{S} = \bigcup_{t\in\Rel} S(t)\) upperbounds the signal \(s(t)\) at each time
\pause
From lattice theory:
\begin{itemize}
\item \(\bar{S}\) upperbounded by a fixpoint of \(F:X \to f(X)\bigcup X\)
\item Such fixpoint can sometimes be found by simply iterating
\(S(N+1) = S(N) \Rightarrow \forall k \geq N, S(k) = S(N)\)
\end{itemize}
\end{frame}
\section{Designing Accuracy for Small but Correct Computations (LSB)}
\section{Conclusion and Future Work}
......
......@@ -228,9 +228,9 @@ To transpose interval arithmetic into signals, we need to attach to a signal
To compute the \(S'(t)\), we could simply transform the recurrence relation
\[
\recSig{i}{s(i) = 0}{s(i+1) = f(s(i)\dots s(i-k))}
\recSig{i}{s(i) = 0}{s(i) = f(s(i-1)\dots s(i-k))}
\text{into}
\recSig{i}{S'(i) = \bot}{S'(i+1) = f(S'(i)\dots S'(i-k))}
\recSig{i}{S'(i) = \bot}{S'(i) = f(S'(i-1)\dots S'(i-k))}
\]
Note that for any function \(f\) on signals, the function \(f\) on interval is
......
......@@ -13,16 +13,6 @@
\usepackage{algorithm}
\usepackage{algpseudocode}
\usepackage{listings}
\lstset{
language=c,
basicstyle=\ttfamily,
literate={~}{{\raisebox{-.25em}{\textasciitilde}}}{1}
% literate={~}{\char `~}{0},
% linewidth=.5\textwidth,
% numbers=left,
% numberstyle=\small
}
\usepackage{amsmath,amsfonts,amssymb, amsthm}
\usepackage[pagebackref]{hyperref}
......
\usepackage{listings}
\lstset{
language=c,
basicstyle=\ttfamily,
literate={~}{{\raisebox{-.25em}{\textasciitilde}}}{1}
% literate={~}{\char `~}{0},
% linewidth=.5\textwidth,
% numbers=left,
% numberstyle=\small
}
\newcommand{\todo}[1]{\textcolor{red}{TODO: #1}}
\newcommand{\figref}[1]{figure \ref{#1}}
......
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