The direction-of-arrival problem: coming at you

If you break your leg on a mountain but have a cellular telephone or other transmitter with you, you would hope a rescuer could determine the direction in which to travel to reach you. Similarly, if a navy detects a transmission from a submarine, it would want to determine the signal´s direction of arrival (DOA) to locate that submarine. The problem is complicated if more than one signal appears - especially if the number of signals is unknown - and even more complicated if you or the submarine is moving. Surprisingly, we see that your rescuer can solve an eigenvalue problem (involving the product of some unknown matrices) and use that information to find you. The DOA-finding algorithm we use is called Esprit. To understand the process, we also use several matrix decompositions and illustrate the necessity of using update-techniques for real-time computations.