Performance Limits and Geometric Properties of Narrowband Array Localization

Location-aware networks are of great importance and interest in both civil and military applications. This paper determines the localization accuracy of an agent in a wireless network, where the agent is equipped with an antenna array making measurements with several anchor nodes in a far-field environment. In view of the Cram\'er-Rao bound, we derive that the localization information is a weighed sum of Fisher information matrices from each anchor-antenna measurement pair that employs narrowband transmission. Each matrix can be further decomposed into a distance part with intensity proportional to the squared baseband effective bandwidth, and a direction part with intensity associated with the normalized anchor-antenna visual angle. Moreover, we show that in the dynamic case, the Doppler shift contributes additional direction information, with intensity determined by the speed and the root mean squared time duration of the transmitted signal. Furthermore, two measures are proposed to evaluate the performance of wireless localization network with different anchor-agent and antenna-array geometries. Both formulae and simulations are provided to illustrate the superior anchor and/or antenna array geometries for achievable localization performances.