\documentstyle[IEEEtran]{article}
\begin{document}
\tolerance 10000

\title{Computations Based on Delta-Modulation Representation of
Measurement Results}
\author{George Djuro Zrilic}

\pagestyle{myheadings}

\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ }

\maketitle

\auffil{The author is with the Department of Engineering,
New Mexico Highlands University,
Las Vegas, NM 87701,
email djuro@edison.nmhu.edu.}

\section{Direct dynamic measurement with interval uncertainty}

In many real-life situations, we want to 
{\it monitor} the value of a physical quantity $x$
for all moments of time $t$ (e.g., 
to check that the object of measurement is performing in the right
manner). 

We want to make as many measurements as
possible, so we have lots of information to send. 
However, the capacity of the communication channel is limited (and in
many situations, e.g., in space exploration, the cost of adding an
extra communication channel can be enormous). The smaller the interval
between consequent measurements, the more information we need to send.
So, the limit on the capacity of the communication channel restricts
the interval between the measurements. 
Let us denote the smallest interval between
the measurements (that the communication channel can still support) 
by $\Delta t$. If we denote the starting moment for our monitoring by
$t_0$, then, since we want to measure as many values
of the quantity $x(t)$ as possible, we will measure 
the value of $x(t)$ of the quantity $y$ 
in the moments $t_0$, $t_1=t_0+\Delta t$, ..., $t_k=t_0+k\Delta t$,
etc.

Measurements are never absolutely precise \cite{Rabinovich 1993}; 
therefore, the measurement 
result $\tilde x$ can differ from the actual value $x$ of the
corresponding quantity by the {\it measurement error} $\Delta
x=\tilde x-x$. 
For a measuring instrument or a sensor to make sense, the manufacturer
must provide us with the guaranteed error (if there is no guaranteed
error, then we can conclude nothing from the fact that, e.g., the
measured value was $\tilde x=10.0$: the  actual value can be 9.9,
can be 2,000). 
In some cases, we know the probabilities of possible errors $\Delta x$. In many
cases, however, the guaranteed upper bound $\Delta$ 
is the only information
about the errors $\Delta x$ that the manufacturer provides. In this
cases, the only information that we know about the actual value of $x$
is that this actual value belongs to an interval ${\bf x}=
[\tilde x-\Delta,\tilde x+\Delta]$. 

\section{The main idea behind delta-modulation}

A possibility to decrease the number of transmitted bits (and thus, to
make more frequent monitoring measurements)
comes from the fact that the
measured quantities are usually changing continually, and we usually know the
upper estimate $M$ on the rate with which the measured quantity $x(t)$
changes (if we do not have any limits on $M$, 
then we have no information about the
intermediate values $x(t)$, and our monitoring is of limited usage).

In this case, if 
we know the value $x(t_k)$ in the moment of time $t_k$,
then the next value $x(t_{k+1})$ cannot deviate from $x(t_k)$ by more
than $M\cdot \Delta t$. 

Let us give an example of why this idea can
indeed decrease number of bits that is necessary to carry a single
measurement.
\smallskip
 
\noindent{\bf Example.} Let us 
assume that we are measuring the temperature every millisecond, 
with an accuracy of 1 degree: $|\tilde x(t_k)-x(t_k)|\le\Delta=1$.  
Let us also assume that the measured value $\tilde x(t_k)$ 
of the temperature $x$ at some moment $t_k$ 
is equal to $1,826$ degrees, and we know
that during the interval between the two consequent measurement (i.e.,
during 1 millisecond) the temperature can change by no more than 2
degrees, i.e., that $|x(t_{k+1})-x(t_k)|\le 2$. Therefore, the
difference between the {\it measured} values of temperature cannot
exceed 4 degrees: $$|\tilde x(x_{k+1})-\tilde x(t_k)|\le |\tilde x(t_{k+1})-
x(t_{k+1})|+$$ $$|x(t_{k+1}) -x(t_k)|+|x(t_k)-\tilde x(t_k)|\le 1+2+2=4.$$

According to the traditional approach, in the next moment of time
$t_{k+1}$, 
we must send the numerical value of the measured temperature $\tilde
x(t_{k+1})$.
This value is an integer between $1,822=1,826-4$ and $1,830=
1,826+4$. Therefore,
it is an integer between $1,024=2^{10}$ and $2,048=2^{11}$,
so, we need 11 binary digits to describe this measurement
result. 

On the other hand, instead of sending the value $\tilde x(t_{k+1})$, we
can simply send the difference between $\tilde x(t_{k+1})$ and $\tilde
x(t_k)$. This difference is an integer between $-4$ and $4$, so it has
only 9 possible values ($-4, -3, ..., 0, 1, ..., 4$). So, we only
need 4 bits (1 bit for sending a sign, and 3 bits for sending the
absolute value of the difference) as opposed to 11 in the traditional
approach. Because we need fewer bits to send the results of the
measurements, we can hold measurements 11/4 ($>2$) 
times more frequently than before.
\smallskip

We have already mentioned that 
ideally, we should be monitoring the value of $x(t)$ for every moment
$t$, but in reality, we 
only get the values in the moments $t_1,...,t_k,...\ $. So, if
we are interested in the value of $x(t)$ for some intermediate moment
of time $t$, i.e., in a moment of time that lies in between $t_k$ and
$t_{k+1}$ for some $k$, then as an estimate for $x(t)$, we take the latest
available measured value, i.e., $\tilde x(t_k)$. 
Even if we measured $x(t_k)$ precisely, this
difference in times between $t$ and $t_k$ 
would still to an error in this estimate, an error
$x(t)-x(t_k)$ that is limited by $M\cdot(t-t_k)\le M\cdot\Delta
t$.
The additional measurement error $\tilde x(t_k)-x(t_k)$ may increase
the total error $x(t)-\tilde x(t_k)$ of using $\tilde x(t_k)$ as an
estimate for $x(t)$.  

Since we already have an error component of size $M\cdot\Delta t$, 
it makes no big sense to measure the values $x(t_k)$
with accuracy that is much better than $M\cdot\Delta t$
(i.e., there is no sense in trying to achieve
measurement errors that are much smaller than $M\cdot\Delta t$): such
super-accurate measurements would mean using very expensive sensors,
but their usage will not seriously improve the resulting error,
because this error will still be of order $M\cdot\Delta t$.

So, the measurement accuracy $\Delta$ is usually chosen to be
smaller than $M\cdot\Delta t$, but approximately of the same order
($\Delta<M\cdot\Delta t$, $\Delta\approx M\cdot\Delta t$). 
With this choice, the difference 
$x(t_{k+1})-x(t_k)$ (that is $\le M\cdot\Delta t$) is measured with an error
that is close to the value of this difference. 
With such a huge measurement error, we can basically distinguish
between only two cases: 
\begin{itemize}

\item the case when this difference is positive, and

\item the case when this difference is negative.
\end{itemize}

So, the sensor gets the measurement results $\tilde x(t_1), ...,$ but it
sends for processing only one bit per moment of time. This bit
actually represents a sign of the difference 
between the two consequent values of the signal 
(i.e., whether $x$ increased with respect
to the previous moment of time or not), so it is natural to represent
this bit not as 0 or 1, but as a {\it sign}, i.e., as $+1$ or $-1$.
Let us denote the sign bit that comes out of the sensor in the moment
$t_k$ by $s(k)$. Then, at the receiving end of the
communication channel, all we have is a sequence of sign bits
$s(1), ..., s(k), ...\ $. In order to be able to
reconstruct the signal from the this sequence, we must 
know the initial value of the signal $\tilde x(t_0)$. 
How can we reconstruct the
signal from this sequence? There is not much that we can do but follow
the following natural algorithm:  
\begin{itemize}

\item As the initial value $r(0)$ of the reconstructed
signal $r$, we simply take $r(0)=\tilde x(t_0)$.

\item To get further reconstructed values $r(k)$, we 
proceed as follows: 
\begin{itemize}
\item[$\bullet$]if we have already
computed $r(k)$, and the next bit that comes 
out of the communication channel is $+1$, we add 
$\alpha=M\cdot\Delta t$ (i.e., take
$r(k+1)=r(k)+M\cdot\Delta t$); 

\item[$\bullet$]if we have already
computed $r(k)$, and the next bit that comes 
out of the communication channel is $-1$, we subtract $M\cdot\Delta t$
(i.e., take $r(k+1)=r(k)+M\cdot\Delta t$).
\end{itemize}
\end{itemize}
This natural reconstruction algorithm leads to the following
natural idea of 
selecting a signal $s(k)$ that would go through
the communication channel:  
\begin{itemize}
\item At every moment of time, after we generate the
communication bit, we also simulate the reconstruction procedure at the
sensor's end of the communication channel (thus getting $r(0)$,
$r(1)$, ...).

\item After a new measured value $\tilde x(t_k)$ arrives, we
compare it with the previously reconstructed signal $r(k-1)$. Now, we
have two options:
\begin{itemize}
\item[$\bullet$]If we
choose $s=+1$, then the next reconstructed signal $r(k)$ will be
greater than $r(k-1)$. 

\item[$\bullet$]If we
choose $s=-1$, then the next reconstructed signal $r(k)$ will be
greater than $r(k-1)$. 
\end{itemize}
\item[]So, to get $r(k)$ as close to $\tilde x(t_k)$ as possible, we
will choose:
\begin{itemize}
\item[$\bullet$] $s(k)=+1$ if the value of $r(k-1)$ is
smaller than $\tilde x(t_k)$ (and thus needs to be increased), and

\item[$\bullet$] $s(k)=-1$ if the value of $r(k-1)$ is
greater than $\tilde x(t_k)$ (and thus needs to be decreased).
\end{itemize}
\end{itemize}
The resulting algorithm is called {\it delta-modulation}.
\smallskip

\noindent{\it Comment.} For a recent survey of delta-modulation
techniques, see, e.g., \cite{Galton 1994} and references therein. These
methods and results, however, are mainly developed for the statistical
case.

\section{Direct dynamic measurement and its error estimate}

\noindent{\bf Definition 1.}
By a {\it dynamic measuring instrument}, we mean 
a pair $(\Delta, \Delta t)$ of two positive numbers:
\begin{itemize}

\item a number $\Delta>0$ will be called
the {\it measurement accuracy;}

\item a number $\Delta t>0$ will be called a {\it time
quantum}. 
\end{itemize}
\newpage

\noindent{\bf Definition 2.} By a {\it dynamic 
measurement situation}, we mean a tuple $(I, M, t_0, x,
\{t_k\}, \{\tilde x(t_k)\})$, where:
\begin{itemize}
\begin{itemize}
\item[$I$] is a dynamic measuring instrument.

\item[$M$] is a positive real number called the
{\it (prior) bound on the rate of change of the signal.} 
We will assume that $\Delta\le M\cdot\Delta t$.

\item[$t_0$] is a real number called the {\it initial moment of
time}.

\item[$x$] is a function from real numbers into
real numbers that is an $M-$Lipschitz function (i.e.,
$|x(t)-x(s)|\le M|t-s|$ for all $t$ and $s$).

\item[$\{t_k\},$] $0\le k$, is a sequence of real numbers
defined as 
$t_k=t_0+k\cdot\Delta t$. A number 
$t_k$ will be called {\it $k-$th measurement moment.}

\item[$\{\tilde x(t_k)\}$] is a sequence of real numbers 
for which for every $k$, $|\tilde
x(t_k)-x(t_k)|\le\Delta$.
The element $\tilde x(t_k)$ will be called the {\it result of $k-$th
measurement.}
\end{itemize}
\end{itemize}

\noindent{\it Comment.} Let us first consider the case when we do not
use delta-modulation. 
\smallskip

\noindent{\bf Definition 3.} For every dynamic measurement situation,
and for every moment of time $t\ge t_0$, by a {\it monitoring error},
we mean the difference $\tilde x(t_k)-x(t)$, where $k$ is the largest
value for which $t_k\le t$. 
\smallskip

\noindent{\bf PROPOSITION 1.}
\begin{itemize}

\item {\it For every dynamic measurement
situation, and for every moment of time $t$, the absolute value of the
monitoring error
does not exceed $M\cdot\Delta t+\Delta$.}

\item {\it For every $M>0$, $t_0$, $\Delta t>0$, $\Delta>0$, and
$\delta>0$, there exists a dynamic measurement
situation and a moment of time $t\ge t_0$ for which 
the monitoring error is not smaller than 
$M\cdot\Delta t+\Delta-\delta$.}
\end{itemize}

\noindent{\it Comments.} 
\begin{itemize}
\item[1.] These two statements mean that:
\begin{itemize}
\item[$\bullet$]
$M\cdot\Delta t+\Delta$ is the error bound for monitoring error, and

\item[$\bullet$]that no better bound is possible.  
\end{itemize}

\item[2.] The fact that the error bound is $M\cdot\Delta t+\Delta$ can be
easily explained by the fact that we have two sources of error:
\begin{itemize}

\item[$\bullet$]the measurement error, whose bound is $\Delta$, and 

\item[$\bullet$] the error caused by the difference between $t$ and
$t_k$ whose upper bound is $M\cdot\Delta t$.
\end{itemize}
\item[3.]The results from this paper has been proven by 
V. Kreinovich and me.
\end{itemize}

\subsection{Proof of Proposition 1} 
Let us first prove that the
monitoring error is always bounded by $M\cdot\Delta t+\Delta$. Indeed, if
$t_k\le t<t_{k+1}$, then $0\le t-t_k<t_{k+1}-t_k=\Delta t$ and
therefore, by the definition of a measurement situation,
$|x(t)-x(t_k)|\le M\cdot|t-t_k|\le M\cdot\Delta t$. 
From the same definition, it follows that $|\tilde
x(t_k)-x(t_k)|\le\Delta$. Therefore, $|x(t)-\tilde
x(t_k)|\le|x(t)-x(t_k)|+|x(t_k)-\tilde x(t_k)|\le M\cdot\Delta t+\Delta$. 
So, the monitoring error is indeed always bounded by $M\cdot\Delta t+\Delta$.

Let us now show that a smaller bound for a monitoring error is
impossible. Indeed, assume that $\delta>0$, $t_0$, $M$, $\Delta t$ are
fixed. As a measured signal, let us take the function 
$x(t)=M(t-t_0)$. As measurement results, we will take $\tilde
x(t_k)=x(t_k)-\Delta$. Then, for $t=t_1-\delta/M$, the monitoring
error is equal to $$x(t)-\tilde
x(t_0)=M(t_1-\delta/M-t_0)-(M(t_0-t_0)-\Delta)=$$ 
$$M\cdot \Delta t-\delta+\Delta.$$ Q.E.D. 

\section{Delta-modulation: formal definition, and why it
really helps}

Let us now define delta-modulation and show
that its usage (while saving on communication) does not decrease the
resulting monitoring error. Namely, we will show that with
delta-modulation, we can achieve the same monitoring error if we
double the measuring rate. At this rate, we need to transmit twice as
many measurement results. However, since we only need one bit to transmit a
single delta-modulated measurement result, and we need several bits to
transmit the actual measurement result $\tilde x(t_k)$, the resulting total
amount of bits per second that needs to be transmitted is smaller when
we use delta-modulation.
\smallskip

\noindent{\bf Definition 4.} 
\begin{itemize}

\item For every dynamic measurement situation, 
by the {\it result of delta-modulation} (applied to
the sequence $\tilde x(t_k)$), we mean the sequence 
$$s(1), ...,s(k),...$$ whose 
elements are determined by the formula 
$$s(k)={\rm sign}\big[\tilde x(t_k)-\big(\tilde x(t_0)+M\cdot\Delta
t\cdot\sum^{k-1}_{j=1}s(j)\big)\big],\eqno{(1)}$$    
where:
\begin{itemize}
\item[$\bullet$]${\rm sign}(a)=+1$ if $a\ge 0$, and

\item[$\bullet$]${\rm sign}(a)=-1$ if $a<0$.
\end{itemize}

\item For every dynamic measurement situation, 
by the {\it reconstructed} or {\it delta-demodulated} 
signal, we mean a sequence
$$r(k)=\tilde x(t_0)+M\cdot\Delta t\cdot\sum_{j=1}^k s(j).\eqno{(2)}$$
\end{itemize}

\noindent{\bf Definition 5.} For every dynamic measurement situation,
and for every moment of time $t\ge t_0$, by a {\it monitoring
error after (delta-)demodulation},
we mean the difference $r(k)-x(t)$, where $k$ is the largest
value for which $t_k\le t$. 
\smallskip

\noindent{\bf PROPOSITION 2.}
\begin{itemize}

\item {\it For every dynamic measurement
situation, and for every moment of time $t$, the absolute value of the
monitoring error after delta-demodulation
does not exceed $2M\cdot\Delta t+\Delta$.}

\item {\it For every $M>0$, $t_0$, $\Delta t>0$, $\Delta>0$, and
$\delta>0$, there exists a dynamic measurement
situation and a moment of time $t\ge t_0$ for which 
the monitoring error after delta-demodulation is not smaller than  
$2M\cdot\Delta t+\Delta-\delta$.}
\end{itemize}

\noindent{\it Comments.} 
\begin{itemize}
\item[1.] These two statements mean that:
\begin{itemize}
\item[$\bullet$]
$2M\cdot\Delta t+\Delta$ is the error bound for monitoring error after
delta-demodulation, and

\item[$\bullet$]that no better bound is possible.  
\end{itemize}

\item[2.] Because of Proposition 2, 
if for measurements with delta-modulation, 
we take the time quantum $\Delta t$ that is twice smaller than the one
that was used for
regular measurements, we will get exactly the same monitoring error
as for measurements without modulation. Let us give two examples:
\begin{itemize}
\item[$\bullet$]Suppose that 
we measure $x(t_k)$ with an accuracy of 1\%. This means that possible
measurement results run from $-100$ to 100 quanta. The binary representation of
100 takes 7 bits, so, with an extra bit for sign, we need 8 bits to
transmit the result of a single measurement. If we use
delta-modulation, then we only need one bit per measurement, but these
measurements must be two times more frequent. So, when initially we
needed 8 bits, we now need only two. Therefore, if we use
delta-modulation, we can keep the same total error and reduce the
information flow by the factor of four. 

\item[$\bullet$]Suppose now that we measure $x(t_k)$ with an accuracy
of 0.1\%. In this case, possible measurement results run from $-1000$
to 1000. The binary representation of 1000 takes 10 bits, so we need
11 bits to transmit the result of a single measurement. If we use
delta-modulation, and aim at the same accuracy of the final result, we
thus need two bits during the same time quantum $\Delta t$. So, we
decrease the information flow by the factor of 5.5. 
\end{itemize}
\item[]In general, the more accurate the measurements, the more we
save by using delta-modulation.
\end{itemize}

\subsection{Proof of Proposition 2} 
Let us first start with proving
the inequality, and then produce an example that proves the second part of
this proposition. 

To prove the inequality, we will first prove (by induction over $k$)
that $|x(t_k)-r(k)|\le M\cdot\Delta t+\Delta$ for all $k$.
\smallskip

\noindent {\it Induction base.} The initial reconstructed 
value $r(0)$ is defined as $r(0)=\tilde x(t_0)$. But by definition of
a measurement situation, we have $|\tilde x(t_0)-x(t_0)|\le\Delta$.
Therefore, $$|r(0)-x(t_0)|\le \Delta<M\cdot \Delta t+\Delta.$$
\smallskip

\noindent{\it Induction step.} Assume that we have already proved the
desired inequality for $k$, i.e., that
$$|x(t_k)-r(k)|\le M\cdot\Delta t+\Delta.\eqno{(3)}$$
We must prove that a similar inequality holds for $k+1$, i.e., that 
$$|x(t_{k+1})-r(k+1)|\le M\cdot\Delta t+\Delta.\eqno{(4)}$$ 
To prove that, we will consider two cases:
\begin{itemize}

\item the case when $\tilde x(t_{k+1})\ge r(k)$, and therefore,
$s(k+1)=1$.

\item the case when $\tilde x(t_{k+1})<r(k)$, and therefore,
$s(k+1)=-1$.
\end{itemize}

In the first case, $r(k+1)=r(k)+M\cdot\Delta t$. From (3), we
conclude that $$x(t_k)\le r(k)+M\cdot\Delta t+\Delta.\eqno{(5)}$$ 
From the definition of a
measuring situation, we conclude that $x(t_{k+1})-x(t_k)\le
M\cdot(t_{k+1}-t_k)=M\cdot\Delta t$, so $x(t_{k+1})\le
x(t_k)+M\cdot\Delta t$. Replacing $x(t_k)$ by its upper bound taken from (5),
we conclude that $$x(t_{k+1})\le r(k)+M\cdot\Delta
t+\Delta+M\cdot\Delta t.$$ Since
$r(k)+M\cdot\Delta t=r(k+1)$, we conclude that $x(t_{k+1})\le
r(k+1)+\Delta+M\cdot\Delta t$, i.e., that $x(t_{k+1})-r(k+1)\le
M\cdot\Delta t+\Delta$. 

This is a half of the desired inequality (4). 
To complete the proof for this case, it is thus 
necessary to prove the other half of this inequality, i.e., to prove
that $x(t_{k+1})\ge r(k+1)-(M\cdot\Delta t+\Delta)$. Since in the case under
consideration, $r(k+1)=r(k)+M\cdot\Delta t$, this inequality is equivalent to
$x(t_{k+1})\ge r(k)-\Delta$. This follows from the following sequence
of inequalities:
\begin{itemize}

\item in this
case, $\tilde x(t_{k+1})\ge r(k)$;

\item by definition of a measurement situation, 
$$x(t_{k+1})\ge\tilde x(t_{k+1})-\Delta;$$

\item therefore, $x(t_{k+1})\ge\tilde x(t_{k+1})-\Delta\ge
r(k)-\Delta$. 
\end{itemize}

This inequality is thus proved, and so (4) is true in 
the first case. 
\smallskip

The second case is proved similarly. Now, the desired
inequality follows in a manner similar to the proof of Proposition 1:
if $t_k\le t<t_{k+1}$, then $$|x(t)-r(k)|\le |x(t)-x(t_k)|+|x(t_k)-r(k)|\le$$
$$M\cdot |t-t_k|+M\cdot\Delta t+\Delta\le $$ $$M\cdot\Delta t+M\cdot\Delta
t+\Delta.$$ 

Let us now give an example of the measurement situation in which 
the monitoring error is not smaller than $2M\cdot
\Delta t+\Delta-\delta$:
\begin{itemize}
\item[$x$:]we choose the following function $x(t)$:

\item[] 
$\ x(t_2)=-M\Delta t$;

\item[]$\ x(t_k)=0$ for $k\ne 2$:

\item[]$\ x(t)$ is linear on each of the intervals $[t_k,t_{k+1}]$. 

\item[$\tilde x$:]we take 
$\tilde x(t_k)=x(t_k)+\Delta$ for all $k$.

\item[$t$:]we take $t=t_2-\delta/M$.
\end{itemize}
\noindent In this case:
\begin{itemize}
\item $r(0)=\tilde x(t_0)=\Delta$. 

\item Since $\tilde x(t_1)=\Delta\ge r(0)$, we have
$s(1)=+1$ and $r(1)=r(0)+M\cdot\Delta t=\Delta+M\cdot\Delta t$. 

\item We have $t_1\le t<t_2$. 
Since $x$ is linear on $[t_1,t_2]$, we have $$x(t)=x(t_1)\cdot{t-t_1\over
t_2-t_1}+x(t_2)\cdot{t_2-t\over t_2-t_1}=$$
$$(-M\Delta t)\cdot{\Delta t-\delta/M\over\Delta t}=-M\cdot\Delta t+\delta.$$

\end{itemize}
\noindent So, here, $$|r(1)-x(t)|=|(\Delta+M\cdot\Delta
t)-(-M\cdot\Delta t+\delta)|=$$ $$2M\cdot\Delta t+\Delta-\delta.$$
Q.E.D.

\section{Error estimation of the result of
delta-demodulation}
When the signals arrive in a delta-modulated form, the
following problem arises:
modulation and demodulation (reconstruction)
take some extra computation time. So, if we want to process the
measured values, and we simply demodulate and process, then we lose
extra time. Can we somehow modify data processing algorithms so that
they can deal directly with a delta-modulated form and produce the
desired result without spending extra time on demodulation? 
It will be shown that in many real-life cases, we can
process the delta-modulated sequence directly, and this usage will not
only {\it not} increase the computation time, but often drastically
{\it decrease} this time.

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\bibitem{Galton 1994}
I. Galton, {\it Granular quantization noise in a class of delta-sigma
modulators}, IEEE Transactions on Information Theory, 1994, Vol. 40,
No. 3, pp. 848--859. 

\bibitem{Rabinovich 1993}
S. Rabinovich, {\it Measurement errors: theory and practice},
American Institute of Physics, N.Y., 1993. 


\end{thebibliography}
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