PN sequences for better reception of the desired signal. In this scenario, however, we
require that the robots do not use different seeded PN sequences, because then, as the
swarm size increases, hardware complexity of the receivers will also increase. Hence, all
the robots must use the same PN sequence with different phases and autocorrelation
property of the PN sequence assumes importance. The autocorrelation function of an m-
sequence [Hay94] is constant at a very small value (-1/M, where M is the sequence
period) for non-zero lags and is 1 at zero lag. Hence, they can be beneficially used in this
application.
Since every robot is at different distances from a particular robot, the received
signal will have the same PN sequence arriving in different phase shifts. Then, when
there are N robots, the received signal r,(t) at the ith robot can be written as
N
ri (t)= Aik g(t Aik) (5.1)
k=l,k i
2
where Ak is the amplitude of the signal coming from robot k. Again, Aik = A / dk where
d,k is the distance between the ith and kth robots. Also, we let g(t) be the assumed PN
sequence, which is common to all the robots. The phase Ak depends on the time-of-
arrival of the signal, which is a function of the distance dk.
If the PN sequence is g(t), then the decoded signal will be at a particular phase shift
L of the PN sequence
i = E[ri (t)g(t- L)] (5.2)
Then, substituting the received signal in the above equation gives
y =E l Aikg(t- Aik)g(t-L) (5.3)
k=1