<div>Positioning is a fundamental issue for mobile robots. Therefore, a
performance analysis is suitable to determine the behavior of a system,
and to optimize its working. Unfortunately, some systems are only
evaluated experimentally, which makes the performance analysis and
design decisions very unclear.</div>


<div>In [4], we have proposed a new angle measurement system, named
BeAMS, that is the key element of an algorithm for mobile robot
positioning. BeAMS introduces a new mechanism to measure angles: it
detects a beacon when it enters and leaves an angular window. A
theoretical framework for a thorough performance analysis of BeAMS has
been provided to establish the upper bound of the variance, and to
validate this bound through experiments and simulations. It has been
shown that the estimator derived from the center of this angular window
provides an unbiased estimate of the beacon angle.</div>


<div>This document complements our paper by going into further details
related to the code statistics of modulated signals in general, with an
emphasis on BeAMS. In particular, the probability density function of
the measured angle has been previously established with the assumption
that there is no correlation between the times a beacon enters the
angular window or leaves it. This assumption is questionable and, in
this document, we reconsider this assumption and establish the exact
probability density function of the angle estimated by BeAMS (without
this assumption).</div>


<div>The conclusion of this study is that the real variance of the estimator
provided by BeAMS was slightly underestimated in our previous work. In
addition to this specific result, we also provide a new and extensive
theoretical approach that can be used to analyze the statistics of any
angle measurement method with beacons whose signal has been modulated.
To summarize, this technical document has four purposes:</div>


<div>(1) to establish the exact probability density function of the angle
estimator of BeAMS,</div>


<div>(2) to calculate a practical upper bound of the variance of this
estimator, which is of practical interest for calibration and tracking
(see Table 1, on page 13, for a summary),</div>


<div>(3) to present a new theoretical approach to evaluate the performance of
systems that use modulated (coded) signals, and</div>


<div>(4) to show how the variance evolves exactly as a function of the
angular window (while remaining below the upper bound).</div>