Time Delay of Arrival (TDOA) is a method used for determining the location of a signal source using the received signal at multiple locations. In this method we determine how much longer it takes for the signal to reach one location as compared to another. From this information and the known distance between the receivers, we can determine a set of possible locations for the source. This is complicated by noise in the received signal. Particle Swarm Optimization (PSO) can be used to overcome this complication.
PSO is an algorithm first introduced in 1995 that was an outgrowth of a study of the flocking of birds. In PSO, we have “particles” which move in a semirandom manner in search of the optimum value of a function ƒ. The algorithm requires evaluation of ƒ at the position of each particle and then each particle’s position needs to be updated based on its own history and the history of the entire group. The particles end up swarming around the optimum value of ƒ.
For TDOA, the function to be optimized is

where L is the number of receivers, x̄_{k} is the location of the k^{th} receiver, c is the speed of propagation of the signal, D_{k} is the time difference between receiver 1 and receiver k, and x̄ is the estimated location of the source. In order to increase the speed of convergence, and decrease to probability of converging to a local minimum, we seed the particles in a random manner around the estimate of x̄ given by the least squares method.
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