# Estimating estimate the distance of a node from

algorithms estimate the distance of a node from a given node, which transmits a
signal, by measuring the power of the signal received by the receiving node.
The node which transmits the signal are static and are aware of their location.
These nodes are called anchor nodes or beacon nodes. If the power received by a
node which is at a given distance from an anchor node is known, then the power
at a distance d is given as follows
3:

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Where Pr is the power received at a
distance d from the anchor node, A is the power received at a unit
distance from the anchor node and np
is the transmission factor whose value describes the environment and
is different from place to place. The parameters A and np are
constant over a given area for outdoor environment, but they vary considerably
over short distances in an indoor environment. Hence, for indoor localization,
the environment has to divided into smaller areas over which the parameters A and np can be considered constant. Each area would have a
different value for the parameters and can be estimated experimentally 4.
Once the parameters A and np are known, the distance of
a node from the anchor node by the following equation:

Predicting the location with the measured distances.

To predict the
location of a node, in a two-dimensional area, distances has to be measures
from at least three anchor nodes which do not fall in the same line. With the
locations of these anchor nodes, location of the node is the point of
intersection of the three circles, drawn with the anchor nodes as centre and
radius equal to the distances of the node from the anchor nodes. If (xa, ya), (xb,
yb) and (xc, yc) are the coordinates of three anchor nodes and da, db, dc
are the distances of a node from these anchor nodes respectively, then the
coordinates of the node (x, y) is estimated by solving the equations:

In case of a
three-dimensional localization, distances of the node from 4 anchor nodes,
which do not fall on the same plane has to be measured. Location can be
estimated by finding the intersection of the four spheres formed from the
anchor nodes

Considering effects of noise.

Due to the presence
of noise in the signals, randomness in the indoor environment and other factors
introduce error in the distances measured by the nodes and the location data
sent by the anchor nodes. Due to these errors, the constant distance circle
from the anchor nodes a, b and c may not meet at a common point as shown in
fig1. Which means the equations may not have a single point solution. Hence, we
estimate the location to the point of intersection of lines l, m
and n, which are lines obtained by
connecting the points of intersection of a pair of circles. Equations of l, m
and n can be obtained by taking the
difference of equations 4 3, 5 4 and 3 5 respectively. Solving the so obtained
equations would give the estimated location of the node 10. 