Commit a0a092d4 by Antonin Dudermel

### 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}} ... ...
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!