Network Synchronization and Localization Based on Stolen Signals

We consider an anchor-free, relative localization and synchronization problem where a set of

n

receiver nodes and

m

wireless signal sources are independently, uniformly, and randomly distributed in a disk in the plane. The signals can be distinguished and their capture times can be measured. At the beginning neither the positions of the signal sources and receivers are known nor the sending moments of the signals. Now each receiver captures each signal after its constant speed journey over the unknown distance between signal source and receiver position. Given these

n

m

capture times the task is to compute the relative distances between all synchronized receivers. In a more generalized setting the receiver nodes have no synchronized clocks and need to be synchronized from the capture times of the stolen signals.

For unsynchronized receivers we can compute in time

${\ensuremath \cal O}(n m)$

an approximation of the positions and the clock offset within an absolute error of

${\ensuremath \cal O}\left(\sqrt{\frac{\log m}{m}}\right)$

with probability 1 − 

m

− 

c

 − 

e

− 

c

′

n

(for any

$c\in {\ensuremath \cal O}(1)$

and some

c

′ > 0).

For synchronized receivers we can compute in time

O

(

n

m

) an approximation of the correct relative positions within an absolute error margin of

$ {\cal O}\left(\frac{\log^2 m}{m^2}\right)$

with probability 1 − 

m

− 

c

 − 

e

− 

c

′

n

. This error bound holds also for unsynchronized receivers if we consider a normal distribution of the sound signals, or if the sound signals are randomly distributed in a surrounding larger disk.

If the receiver nodes are connected via an ad hoc network we present a distributed algorithm which needs at most

O

(

n

m

log

n

) messages in total to compute the approximate positions and clock offsets for the network within an absolute error of

${\ensuremath \cal O}\left(\sqrt{\frac{\log m}{m}}\right)$

with probability 1 − 

n

− 

c

if

m

 > 

n

.