# Analysis of Dual-Band Direction of Arrival Estimation in Multipath Scenarios
## 3 Localization System
### DoA Estimation Method  
The azimuthal DoA estimation method implemented in the proposed simulation environment is based on the procedure described in [16; 38]. In both papers, this estimation is obtained as a result of a Maximum-Likelihood (MLH) criterion driven by comparing the actual real-time RSSI measurement with the expected signal magnitude, where the latter one is derived from the antenna beam-set. To better clarify all the issues involved in the DoA estimation method, the following paragraph briefly summarize the key elements of the approach implemented in the present paper. In accordance with Figure 1, the target, located in the position \((x_{T},y_{T})\), acts as a transmitter of message sequences. The power of the packets sequentially received from the anchor throughout the beam-set, are stored as a vector of RSSI readings. Since the target position is unknown, the angle of view \(\theta\) relative to the anchor node is unknown as well. The RSSI signal through the \(n\)-th beam at the \(i\)-th frequency of operation, labelled as \(S_{n}^{(i)}\), is the result of the following observation model:  
\[S_{n}^{(i)}=G_{n}^{(i)}(\theta)+P_{inc}^{(i)}(D,\theta^{\prime})+n_{n}^{(i)} \tag{2}\]  
where fixed the \(i\)-th frequency, \(G_{n}^{(i)}(\theta)\) is the \(n\)-th gain pattern through the nominal direction (\(\theta\)) of the target with respect to the anchor node reference (Figure 1), and \(P_{inc}^{(i)}(D,\theta^{\prime})\) is the power of the wavefront impinging at the anchor section from the distance \(D\), through direction described by the \(\theta^{\prime}\) angle with respect to the target system reference (Figure 1). Finally, \(n_{n}^{(i)}\) is a noise term, which also include all those multi-path effects that usually occur but cannot be explicitly stated mathematically. With reference to (2) the above-mentioned incident power can be expressed as:  
\[P_{inc}^{(i)}(D,\theta)=P_{T}^{(i)}+\alpha_{(i)}\,10\,\log\left(\frac{4\pi D} {\lambda_{(i)}}\right)+G_{T}^{(i)}(\theta^{\prime}) \tag{3}\]  
where the additive term \(P_{T}^{(i)}\) is the power transmitted by the target node, \(\alpha_{(i)}\) is the pathloss exponent, and \(G_{T}^{(i)}(\theta^{\prime})\) is the gain of the target antenna, defined in its own reference. In principle, the mathematical solution of the DoA estimation problem may be obtained reverting the relationship in (2), which in turn is a non-linear system in \(\theta\). However, being unknown the noise term introduced in the model, the latter cannot be solved in closed form. Figure 3: Radiation patterns of the SBA antennas implemented in the simulation environment. This problem can be easily solved by means of the aforementioned MLH criteria which makes use of the _residual term_**R**. As already introduced in [38], at the \(i\)-th frequency this one is defined as: where \(\textbf{R}^{(i)}\) and \(\textbf{S}^{(i)}\) are the vector form of the gain and the signal in (2). One of the most popular MLH approaches is the Least Square Estimator (LSE). At the \(i\)-th frequency, the cost of LSE is defined as:  
\[\mathrm{C}^{(i)}_{\mathrm{LSE}}(\theta)=||\textbf{R}^{(i)}(\theta)||=\sqrt{ \sum_{n=1}^{N}\left(S^{(i)}_{n}-G^{(i)}_{n}(\theta)\right)^{2}} \tag{4}\]  
thus, the estimated azimuthal DoA \(\widehat{\theta}\) is identified as the one which satisfies:  
\[\widehat{\theta}^{(i)}=\underset{\theta}{\mathrm{argmin}}\,\mathrm{C}^{(i)}_ {\mathrm{LSE}}(\theta) \tag{5}\]  
As an evolution of the former, in [39] was introduced another more suited estimator for dual-band operation,-based on the variance of the residual.