## 4 Proposed Method for Physical Layer Performance Estimation
### Proposed Power Estimation Algorithm  
The method we propose herein is straightforward and is based on estimating the position of blind nodes from the measurements performed by the anchors and using these estimates to infer the channel power for the entire network links. We use the passive device to store the packets exchanged on the network. We analyze the MAC layer to evaluate the RSSI field, which is shown in Figure 3. The RSSI represents information about the power of the signal received by the end-device. Hence, knowing the environment and the model that describes the propagation of the signal, we can easily extract information about the geometric relationships among the blind nodes and the anchors. We use this information to resolve an OLS problem and obtain an initial estimation of blind node positions. This step is necessary to have the input to apply the algorithm proposed in Section 3.2. Now we compute all the coefficients and linearize the function, with the objective to write the bound problem, which is described in Equation (13). Making use of the solution Equation (14), we obtain the estimation of the corrections to be applied to the initial position of each blind node. We are now able to perform an iteration of the method using the new position (the initial position obtained by algorithm OLS with the computed corrections) as the  
Figure 2: Reference scenario with 5 anchors and 6 blind nodes. new input. The criterion is iterated until the correction is smaller than a threshold selected a priori in accordance to the desired accuracy. When the algorithm terminates for every blind node, we obtain all the position estimations for the devices. Hence, we easily evaluate the relative distance among the blind nodes, and in accordance to the environment, we apply the appropriate propagation model to estimate the transmission power among all the blind nodes. For more details, see Figure 4.  
Figure 4: Flow diagram for the proposed algorithm. Figure 3: IEEE 802.15.4 packet. * _Step 1_: each anchor \(i\) measures and stores the received signal for the packets transmitted during a timeframe \(T\) and related to \(S_{ik}\) packets per blind node \(k\). From these measures the anchors compute the received signal power \(T_{ik}^{A}\). * _Step 2_: on the basis of information about the ambient environment, each anchor makes use of a channel model for each of the blind nodes \(k\), namely \(F_{ik}^{A}\), to estimate the distances \(D_{ik}^{A}\). * _Step 3_: for any \(k\), we set a vector \(r=D_{ik}^{A}\), for \(k=1,...,N_{B}\). We then write the system Equation (3) and apply the OLS method described in Section 3.1 to obtain the estimation of the position of each blind node. These estimations are then referred to as \(\tilde{P}_{k}^{B}\). * _Step 4_: for any \(k\), we also compute the estimated residual by applying Equation (5). From this step on, we enter an interactive WLS algorithm applied to the indirect observations described in Section 3.2. * _Step 5_: we compute the coefficients of matrix \(A_{\partial}\), which represents the partial derivates of \(Ap\) in \(dx\) and \(dy\). This is performed by a linearization of each function in the system \(Ap\) by a Taylor expansion around the point estimate in Step 3. The general equation for the linearization of a multivariable function \(f(X)\) at a point \(Q\) is: \(f(X)\approx f(Q)+\bigtriangledown f\mid_{Q}(X-Q)\)
* _Step 6_: we then apply Equation (14) to obtain the estimation of the correction to be applied to the initial position of each blind node \(k\) at each iteration \(s\): \(\delta P_{k}^{B,s}\). We compute again the new estimated position: \(\tilde{P}_{k}^{B,s}\). * _Step 7_: for any blind node \(k\) we go back for another iteration if \(\delta P_{k}^{B,s}<\epsilon\), where \(\epsilon\) is a threshold set as the stop criterion. Otherwise we go ahead with the next step.