--- pubs/books/ucbka/trunk/c_qua0/c_qua0.tex	2016/10/07 01:36:46	6
+++ pubs/books/ucbka/trunk/c_qua0/c_qua0.tex	2017/07/03 01:59:16	140
@@ -1,223 +1,223 @@
-%$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_qua0/c_qua0.tex,v 1.3 2001/06/29 23:40:53 dtashley Exp $
-
-\chapter{\cquazerolongtitle{}}
-
-\label{cqua0}
-
-\section{Introduction}
-%Section tag INT
-
-A microcontroller can inherently manipulate only integers:  usually 8-bit
-integers, 16-bit integers, or 32-bit integers.  Typically, the less
-expensive a microcontroller is, the smaller the maximum data sizes that
-it can accomodate; the least expensive devices can easily manipulate
-only integers no larger than 8 bits.
-
-This chapter deals with the error analysis of \emph{quantization}.  By
-\emph{quantization}, we mean three [distinct] mechanisms of error introduction
-in microcontroller software.
-
-\begin{itemize}
-\item \textbf{Input Quantization:} the unavoidable conversion from
-      $\vworkrealset$ to $\vworkintset$ performed by interface hardware.
-      For example, interface hardware may convert a voltage which is
-      conceptually continuous to an integer (which can assume only discrete
-      values), or may convert a time period which is conceptually
-      continuous to an integer.
-\item \textbf{Arithmetic Quantization:} microcontroller arithmetic algorithms
-      must often discard precision in order to restrict intermediate results of
-      calculations to data sizes which the microcontroller can economically
-      manipulate.  This type of quantization error is most often injected
-      by discarding the remainder of an integer quotient.
-\item \textbf{Output Quantization:} microcontroller output hardware can
-      produce continuous outputs (such as voltages or pulse widths)
-      only in discrete steps.  Often, this lack of ability to control
-      outputs precisely introduces additional uncertainty into the
-      system which must be analyzed.
-\end{itemize}
-
-Note that the error-injection mechanism that we call \emph{quantization}
-(in some sense, the creation of a discrete-\emph{data} system)
-is not related to and is orthogonal to the notion of a
-discrete-\emph{time} (or sampled) system.
-
-
-\section{Modeling Of Quantization}
-%Section tag: MOQ
-
-For the analytical treatment of quantization, the \emph{floor($\cdot{}$)}
-function, denoted $\lfloor \cdot \rfloor$, is used, often preceded by a
-scaling factor.  For example, in the case of an A/D converter which
-converts a voltage $\in [0, 5]$ volts into an integer
-$\in [0,255]_{\vworkintsetnonneg{}}$, we may model the function which
-maps from voltage to A/D count as
-
-\begin{equation}
-\label{eq:cqua0:smoq:001}
-f(x) = \left\lfloor {\frac{255 x }{5}} \right\rfloor .
-\end{equation}
-
-Inherent in (\ref{eq:cqua0:smoq:001})
-is the assumption that quantization will
-choose an integer by rounding \emph{down}.  Other
-assumptions are possible
-(\ref{eq:cqua0:smoq:002}, \ref{eq:cqua0:smoq:003}).
-
-\begin{equation}
-\label{eq:cqua0:smoq:002}
-f(x) = \left\lceil {\frac{255 x }{5}} \right\rceil
-\end{equation}
-
-\begin{equation}
-\label{eq:cqua0:smoq:003}
-f(x) = \left\lfloor {\frac{255 x }{5} + \frac{1}{2}} \right\rfloor
-\end{equation}
-
-At first glance, it may seem intuitively likely
-that (\ref{eq:cqua0:smoq:003}) leads
-to smaller error terms than (\ref{eq:cqua0:stqn:001})
-or (\ref{eq:cqua0:smoq:002})---that rounding to the
-nearest integer is a better strategy than rounding
-down or rounding up.  In this case, intuition may be misleading.
-(\ref{eq:cqua0:smoq:003}) more precisely \emph{centers the
-expected value} of the error than (\ref{eq:cqua0:smoq:001})
-or (\ref{eq:cqua0:smoq:002}), but the \emph{span} of the
-error---the largest error minus the smallest error---remains
-one.  In a practical system, the \emph{span} of the error
-is the dominant effect.  In practice,
-(\ref{eq:cqua0:smoq:001}), (\ref{eq:cqua0:smoq:002}),
-and (\ref{eq:cqua0:smoq:003}) lead to near-identical
-error terms.  For algebraic convenience,
-(\ref{eq:cqua0:smoq:001}) is used preferentially.
-
-Error terms are denoted by the Greek
-letter \emph{epsilon} ($\varepsilon$) and
-are viewed as the perturbation to the
-``ideal'' to yield the ``actual''; so that a negative error
-term leads to a result less than than it
-``should'' be, and a positive
-error term leads to a result greater than
-it ``should'' be.  If the \emph{floor($\cdot{}$)}
-is used to model quantization, the relationship
-in (\ref{eq:cqua0:smoq:004}) holds.
-
-\begin{equation}
-\label{eq:cqua0:smoq:004}
-\lfloor x \rfloor = x - \varepsilon{}; \; \varepsilon \in [0,1)
-\end{equation}
-
-
-\section{Error Analysis Of Addition Of Quantized Inputs}
-%Section tag: eaqi
-
-If we add two quantized values $\lfloor a \rfloor$ and
-$\lfloor b \rfloor$, both $a$ and $b$ contain quantization
-error, and a question of interest is how much
-error the sum $\lfloor a \rfloor + \lfloor b \rfloor$ may
-contain; that is, how different it may be from $a+b$.\footnote{For
-addition and subtraction, this question is nearly trivial; but for
-multiplication and division the relationships are more complex; and for
-an arbitrary network of addition, subtraction, multiplication, and
-division we are not sure how to answer this question easily.
-Please see \ldots{}.}
-We seek an inequality which bounds
-
-\begin{equation}
-\label{eq:cqua0:eaqi:001}
-\varepsilon{} = \left( {\lfloor a \rfloor + \lfloor b \rfloor} \right)
-                      - \left( {a + b} \right)  .
-\end{equation}
-
-Noting that quantization introduces an error $\varepsilon \in [0,1)$
-(Eq. \ref{eq:cqua0:stqn:004}) leads to
-(\ref{eq:cqua0:eaqi:002}) and (\ref{eq:cqua0:eaqi:003}), which are equivalent statements.
-
-\begin{equation}
-\label{eq:cqua0:eaqi:002}
-a + b - 2 < \lfloor a \rfloor + \lfloor b \rfloor \leq a + b
-\end{equation}
-
-\begin{equation}
-\label{eq:cqua0:eaqi:003}
-\varepsilon \in (-2,0]
-\end{equation}
-
-Extending (\ref{eq:cqua0:eaqi:002}) and (\ref{eq:cqua0:eaqi:003})
-to an arbitrary number $N \in \vworkintsetpos{}$ of quantized inputs leads to
-(\ref{eq:cqua0:eaqi:004}) and (\ref{eq:cqua0:eaqi:005}),
-which are equivalent statements.
-
-\begin{equation}
-\label{eq:cqua0:eaqi:004}
-\sum_{i=1}^{N} x_i - N < \sum_{i=1}^{N} \lfloor x_i \rfloor \leq \sum_{i=1}^{N} x_i
-\end{equation}
-
-\begin{equation}
-\label{eq:cqua0:eaqi:005}
-\varepsilon \in (-N,0]
-\end{equation}
-
-\section{Error Analysis Of Subtraction Of Quantized Inputs}
-
-\section{Error Analysis Of Multiplication Of Quantized Inputs}
-
-\section{Error Analysis Of Division Of Quantized Inputs}
-
-\begin{equation}
-\frac{p-1}{q}
-<
-\frac{\lfloor p \rfloor}{\lfloor q \rfloor}
-<
-\frac{p}{q-1}; \; p,q > 1
-\end{equation}
-
-\section{Error Analysis Of Arbitrary Algebraic Functions}
-
-\section{Error Analysis Of Rational Sweeps}
-
-\section{Exercises}
-
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\noindent\begin{figure}[!b]
-\noindent\rule[-0.25in]{\textwidth}{1pt}
-\begin{tiny}
-\begin{verbatim}
-$RCSfile: c_qua0.tex,v $
-$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_qua0/c_qua0.tex,v $
-$Revision: 1.3 $
-$Author: dtashley $
-$Date: 2001/06/29 23:40:53 $
-\end{verbatim}
-\end{tiny}
-\noindent\rule[0.25in]{\textwidth}{1pt}
-\end{figure}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% $Log: c_qua0.tex,v $
-% Revision 1.3  2001/06/29 23:40:53  dtashley
-% Spelling mistake corrected.
-%
-% Revision 1.2  2001/06/29 23:39:56  dtashley
-% Conversion from binary to CVS archives.
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% $History: c_qua0.tex $
-% 
-% *****************  Version 3  *****************
-% User: Dashley1     Date: 12/22/00   Time: 12:56a
-% Updated in $/uC Software Multi-Volume Book (A)/Chapter, QUA0, Quantization
-% Tcl automated method of build refined.
-% 
-% *****************  Version 2  *****************
-% User: David T. Ashley Date: 7/11/00    Time: 8:30p
-% Updated in $/uC Software Multi-Volume Book (A)/Chapter, QUA0, Quantization
-% Separation and enhancement of quantization chapter.
-%
-% *****************  Version 1  *****************
-% User: David T. Ashley Date: 7/11/00    Time: 6:07p
-% Created in $/uC Software Multi-Volume Book (A)/Chapter, QUA0, Quantization
-% Initial check-in.
-%
-%End of file C_QUA0.TEX
+%$Header$
+
+\chapter{\cquazerolongtitle{}}
+
+\label{cqua0}
+
+\section{Introduction}
+%Section tag INT
+
+A microcontroller can inherently manipulate only integers:  usually 8-bit
+integers, 16-bit integers, or 32-bit integers.  Typically, the less
+expensive a microcontroller is, the smaller the maximum data sizes that
+it can accomodate; the least expensive devices can easily manipulate
+only integers no larger than 8 bits.
+
+This chapter deals with the error analysis of \emph{quantization}.  By
+\emph{quantization}, we mean three [distinct] mechanisms of error introduction
+in microcontroller software.
+
+\begin{itemize}
+\item \textbf{Input Quantization:} the unavoidable conversion from
+      $\vworkrealset$ to $\vworkintset$ performed by interface hardware.
+      For example, interface hardware may convert a voltage which is
+      conceptually continuous to an integer (which can assume only discrete
+      values), or may convert a time period which is conceptually
+      continuous to an integer.
+\item \textbf{Arithmetic Quantization:} microcontroller arithmetic algorithms
+      must often discard precision in order to restrict intermediate results of
+      calculations to data sizes which the microcontroller can economically
+      manipulate.  This type of quantization error is most often injected
+      by discarding the remainder of an integer quotient.
+\item \textbf{Output Quantization:} microcontroller output hardware can
+      produce continuous outputs (such as voltages or pulse widths)
+      only in discrete steps.  Often, this lack of ability to control
+      outputs precisely introduces additional uncertainty into the
+      system which must be analyzed.
+\end{itemize}
+
+Note that the error-injection mechanism that we call \emph{quantization}
+(in some sense, the creation of a discrete-\emph{data} system)
+is not related to and is orthogonal to the notion of a
+discrete-\emph{time} (or sampled) system.
+
+
+\section{Modeling Of Quantization}
+%Section tag: MOQ
+
+For the analytical treatment of quantization, the \emph{floor($\cdot{}$)}
+function, denoted $\lfloor \cdot \rfloor$, is used, often preceded by a
+scaling factor.  For example, in the case of an A/D converter which
+converts a voltage $\in [0, 5]$ volts into an integer
+$\in [0,255]_{\vworkintsetnonneg{}}$, we may model the function which
+maps from voltage to A/D count as
+
+\begin{equation}
+\label{eq:cqua0:smoq:001}
+f(x) = \left\lfloor {\frac{255 x }{5}} \right\rfloor .
+\end{equation}
+
+Inherent in (\ref{eq:cqua0:smoq:001})
+is the assumption that quantization will
+choose an integer by rounding \emph{down}.  Other
+assumptions are possible
+(\ref{eq:cqua0:smoq:002}, \ref{eq:cqua0:smoq:003}).
+
+\begin{equation}
+\label{eq:cqua0:smoq:002}
+f(x) = \left\lceil {\frac{255 x }{5}} \right\rceil
+\end{equation}
+
+\begin{equation}
+\label{eq:cqua0:smoq:003}
+f(x) = \left\lfloor {\frac{255 x }{5} + \frac{1}{2}} \right\rfloor
+\end{equation}
+
+At first glance, it may seem intuitively likely
+that (\ref{eq:cqua0:smoq:003}) leads
+to smaller error terms than (\ref{eq:cqua0:stqn:001})
+or (\ref{eq:cqua0:smoq:002})---that rounding to the
+nearest integer is a better strategy than rounding
+down or rounding up.  In this case, intuition may be misleading.
+(\ref{eq:cqua0:smoq:003}) more precisely \emph{centers the
+expected value} of the error than (\ref{eq:cqua0:smoq:001})
+or (\ref{eq:cqua0:smoq:002}), but the \emph{span} of the
+error---the largest error minus the smallest error---remains
+one.  In a practical system, the \emph{span} of the error
+is the dominant effect.  In practice,
+(\ref{eq:cqua0:smoq:001}), (\ref{eq:cqua0:smoq:002}),
+and (\ref{eq:cqua0:smoq:003}) lead to near-identical
+error terms.  For algebraic convenience,
+(\ref{eq:cqua0:smoq:001}) is used preferentially.
+
+Error terms are denoted by the Greek
+letter \emph{epsilon} ($\varepsilon$) and
+are viewed as the perturbation to the
+``ideal'' to yield the ``actual''; so that a negative error
+term leads to a result less than than it
+``should'' be, and a positive
+error term leads to a result greater than
+it ``should'' be.  If the \emph{floor($\cdot{}$)}
+is used to model quantization, the relationship
+in (\ref{eq:cqua0:smoq:004}) holds.
+
+\begin{equation}
+\label{eq:cqua0:smoq:004}
+\lfloor x \rfloor = x - \varepsilon{}; \; \varepsilon \in [0,1)
+\end{equation}
+
+
+\section{Error Analysis Of Addition Of Quantized Inputs}
+%Section tag: eaqi
+
+If we add two quantized values $\lfloor a \rfloor$ and
+$\lfloor b \rfloor$, both $a$ and $b$ contain quantization
+error, and a question of interest is how much
+error the sum $\lfloor a \rfloor + \lfloor b \rfloor$ may
+contain; that is, how different it may be from $a+b$.\footnote{For
+addition and subtraction, this question is nearly trivial; but for
+multiplication and division the relationships are more complex; and for
+an arbitrary network of addition, subtraction, multiplication, and
+division we are not sure how to answer this question easily.
+Please see \ldots{}.}
+We seek an inequality which bounds
+
+\begin{equation}
+\label{eq:cqua0:eaqi:001}
+\varepsilon{} = \left( {\lfloor a \rfloor + \lfloor b \rfloor} \right)
+                      - \left( {a + b} \right)  .
+\end{equation}
+
+Noting that quantization introduces an error $\varepsilon \in [0,1)$
+(Eq. \ref{eq:cqua0:stqn:004}) leads to
+(\ref{eq:cqua0:eaqi:002}) and (\ref{eq:cqua0:eaqi:003}), which are equivalent statements.
+
+\begin{equation}
+\label{eq:cqua0:eaqi:002}
+a + b - 2 < \lfloor a \rfloor + \lfloor b \rfloor \leq a + b
+\end{equation}
+
+\begin{equation}
+\label{eq:cqua0:eaqi:003}
+\varepsilon \in (-2,0]
+\end{equation}
+
+Extending (\ref{eq:cqua0:eaqi:002}) and (\ref{eq:cqua0:eaqi:003})
+to an arbitrary number $N \in \vworkintsetpos{}$ of quantized inputs leads to
+(\ref{eq:cqua0:eaqi:004}) and (\ref{eq:cqua0:eaqi:005}),
+which are equivalent statements.
+
+\begin{equation}
+\label{eq:cqua0:eaqi:004}
+\sum_{i=1}^{N} x_i - N < \sum_{i=1}^{N} \lfloor x_i \rfloor \leq \sum_{i=1}^{N} x_i
+\end{equation}
+
+\begin{equation}
+\label{eq:cqua0:eaqi:005}
+\varepsilon \in (-N,0]
+\end{equation}
+
+\section{Error Analysis Of Subtraction Of Quantized Inputs}
+
+\section{Error Analysis Of Multiplication Of Quantized Inputs}
+
+\section{Error Analysis Of Division Of Quantized Inputs}
+
+\begin{equation}
+\frac{p-1}{q}
+<
+\frac{\lfloor p \rfloor}{\lfloor q \rfloor}
+<
+\frac{p}{q-1}; \; p,q > 1
+\end{equation}
+
+\section{Error Analysis Of Arbitrary Algebraic Functions}
+
+\section{Error Analysis Of Rational Sweeps}
+
+\section{Exercises}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\noindent\begin{figure}[!b]
+\noindent\rule[-0.25in]{\textwidth}{1pt}
+\begin{tiny}
+\begin{verbatim}
+$RCSfile: c_qua0.tex,v $
+$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_qua0/c_qua0.tex,v $
+$Revision: 1.3 $
+$Author: dtashley $
+$Date: 2001/06/29 23:40:53 $
+\end{verbatim}
+\end{tiny}
+\noindent\rule[0.25in]{\textwidth}{1pt}
+\end{figure}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% $Log: c_qua0.tex,v $
+% Revision 1.3  2001/06/29 23:40:53  dtashley
+% Spelling mistake corrected.
+%
+% Revision 1.2  2001/06/29 23:39:56  dtashley
+% Conversion from binary to CVS archives.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% $History: c_qua0.tex $
+% 
+% *****************  Version 3  *****************
+% User: Dashley1     Date: 12/22/00   Time: 12:56a
+% Updated in $/uC Software Multi-Volume Book (A)/Chapter, QUA0, Quantization
+% Tcl automated method of build refined.
+% 
+% *****************  Version 2  *****************
+% User: David T. Ashley Date: 7/11/00    Time: 8:30p
+% Updated in $/uC Software Multi-Volume Book (A)/Chapter, QUA0, Quantization
+% Separation and enhancement of quantization chapter.
+%
+% *****************  Version 1  *****************
+% User: David T. Ashley Date: 7/11/00    Time: 6:07p
+% Created in $/uC Software Multi-Volume Book (A)/Chapter, QUA0, Quantization
+% Initial check-in.
+%
+%End of file C_QUA0.TEX