A typical wireless localization process follows three main step: range identification, location estimation, and location refinement. Due to hardware characteristics, ranging is not always available for wireless devices. Rather, it has been extracted from different characteristics of radio signals. Many localization algorithms utilized hybrid RSS-AoA measurements [14
] because they are more accurate than other methods. The tradeoff is that using hybrid system provides more information on sensor location while increasing network complexity and implementation costs. Thus, according to a specific application purpose, we can select an appropriate localization scheme with a given level of complexity and accuracy.
There are some related literature works on hybrid RSS-AoA measurements [14
]. According to the capabilities of diverse hardware, we may classify the conventional hybrid RSS-AoA-based localization into either of two main categories: nonlinear minimization localization or convex approximation localization. In the first category, supposing that the measurements from the reference node are only corrupted by zero-mean Gaussian noise and the reference nodes are identical or much closer to each other than the target, the basic idea is solving (3) with the given information. In particular, this nonlinear minimization problem can be solved by the Newton–Gauss iteration. For instance, ref. [19
] proposed the least squares (LS) and optimization estimators using the distance and the AoA information. In [20
], hybrid RSS-AoA LS and maximum likelihood (ML) estimators were derived for the emitter geolocation. The method of combining ML-LS implicitly assumes that a rough estimate range between anchor-target can be obtained. The cost function given by (3
) weakly depends on this value; thus, the roughness will not significantly affect the solution. It has been reported in [22
] that the accuracy of such a scheme suffers from the environmental dynamics. On the other hand, departing from the first category, most of the works in the second category simplify the location region update due to the target range and the network communication cost. The objective function (3
) can be solved by itself or approximated by an alternative form. The approximation for (3
) often makes use of the convex constraint. One typical example is [22
] where the geometric constraints (distance and angle measurements) between target and anchor node are represented in linear inequalities. All of them are combined to form a single semidefinite programming (SDP) problem. For each iteration, the bounding region for each node is updated accordingly. In [14
], the authors presented a hybrid weighted LS (WLS) approach to intra-cell target localization in non-line-of-sight environments, while another closed-form WLS was developed in [15
]. The common characteristics of those approaches are their concise problem formulation with clear model representation and refined mathematical solutions, while likely precluding themselves in practice due to their complexities. For example, the complexity of solving the SDP in [22
] is at least
is the number of convex constraints needed to describe the network connectivity and measurement information. Other approaches such as SDP and second-order cone programming were proposed in [22
]. However, the aforementioned works were designed either on 2D space [14
], or had a high computational complexity [22
], or only worked well with low noise [14
]. Moreover, the RSS signals mainly depend on the transmit power of the target node
(e.g., its battery) and the PLE
. Only a few studies have considered the transmit power in (1
) as an unknown parameter [15
], while in practice, these two parameters may vary significantly in different times and places. To the best of our knowledge, there is no existing work that derives a solution for hybrid RSS-AoA in 3D space when both transmit power and PLE are unknown variables. Therefore, in the paper, we derive a suboptimal solution for this case.
Moreover, given the combined RSS-AoA measurements, our localization scheme is proposed based on the following assumptions. The first one is that the AoAs can be measured and calibrated from the omnidirectional antennas. The ambiguities in the hybrid receiver architecture and its sensitivity are neglected in this paper. We also do not address the noise uncertainty issue over fading channel factors through the localization problem formulation.