## 3 Offset Correction
### Algorithm with Reference  
In the case when one of the sensors (say, w.l.o.g., the first) has been preliminarily micro-calibrated, its influence on the rest of the sensors in a network during calibration can be formally maximized in such a way as to adopt \(\gamma_{kj}=0\), \(\forall j\). The selected sensor can be then considered to be the _leader_ of the whole sensor population (i.e., the network is considered to be _pinned_ to the selected sensor), providing a _reference_ for the whole population of sensors [33; 34; 44; 55]. Theorem 3.2: _Let A1-A3 be satisfied; then, suppose that a fixed reference value \(f_{ref}^{1}\) is selected for the first sensor and that the node \(1\) is a center node of graph \(\mathcal{G}\). Then, there exists such a \(\delta^{\prime\prime}>0\) that \(\forall\delta<\delta^{\prime\prime}\); the algorithm (4), modified in such a way that it consists of_  
\[\hat{b}_{t+1}^{1}=\hat{b}_{t}^{1}, \tag{10}\]  
_with \(b_{0}^{1}=-\beta^{1}\), and_  
\[r_{t+1}=(I+\delta\Gamma^{-})r_{t}+\delta\Gamma^{-}v_{t}^{-}, \tag{11}\]  
_where \(r_{t}=[r_{t}^{2}\cdots r_{t}^{n}]^{T}\), \(r_{t}^{i}=\hat{f}_{t}^{i}-f_{ref}^{1}=\beta^{i}+\hat{b}_{t}^{i}-(\beta^{1}+\hat {b}_{t}^{1})\), \(v_{t}^{-}=[v_{t}^{2}\cdots v_{t}^{n}]^{T}\),_  
\[\Gamma^{-}=\left[ \begin{array}{cccc}-\sum_{j=1,j\
eq 2}^{n}\gamma_{2j}& \gamma_{23}&\cdots&\gamma_{2n}\\ \gamma_{32}&-\sum_{j=1,j\
eq 3}^{n}\gamma_{3j}&\cdots&\gamma_{3n}\\ \vdots&\vdots&\vdots&\vdots\\ \gamma_{n2}&\gamma_{n3}&\cdots&-\sum_{j=1,j\
eq n}^{n}\gamma_{nj}\end{array}  \right],\]satisfies \(\|r_{t}\|<\infty\) and \(\|r_{t}\|\to 0\) when \(t\to\infty\) and \(\|v_{t}\|\to 0\)._  
Proof. : According to the general setting, we have the following \(n-1\) local recursions:  
\[\hat{f}_{t+1}^{i}=\hat{f}_{t}^{i}+\delta\sum_{j=1,j\
eq i}^{n}\gamma_{ij}[(\hat{ f}_{t}^{j}-\hat{f}_{t}^{j})+(v_{t}^{j}-v_{t}^{i})] \tag{12}\]  
\(i=1,\ldots,n\), \(i\
eq 1\), bearing in mind that \(r_{t}^{i}\) is defined by \(\hat{f}_{t}^{i}\) and \(f_{ref}^{1}\). Namely, the global recursion (11) is derived directly from (4) by putting all the underlying recursions together after fixing \(k=1\) and \(\hat{b}_{0}^{1}=-\beta^{1}\), in accordance with the assumption that the reference node has been a priori ideally micro-calibrated. Therefore, \(f_{ref}^{1}=0\). In such a way, it follows that the recursion (11) is a time-invariant dynamic system with an input term depending on \(v_{t}\), which is asymptotically exponentially stable for \(v_{t}=0\), bearing in mind that \(\Gamma^{-}\) is constant and Hurwitz, under the adopted assumptions (see, e.g., [34, 44, 45, 52, 53, 58, 59]). It is also B.I.B.O.-stable (for \(v_{t}\
eq 0\)). The rest of the proof follows directly from [52] and Theorem 1. It is essential for the whole analysis to compare the results provided by (4) and (11) at a certain level of generality, in spite of the fact that the results are dependent on the parameters in \(\Gamma\) and the local sensor characteristics. It is obvious that there is a basic similarity between these algorithms. Both of them provide equal values of corrected offsets when \(v_{t}=0\). However, in the case of (4), the common value of all the corrected offsets is given by \(\pi f_{0}\); meanwhile, in the case of (11), the common value is equal to zero for an ideally micro-calibrated sensor with \(f_{ref}^{1}=0\). In general, it is natural that an additional local micro-calibration should enable us to obtain an initially better-corrected offset estimate. One should bear in mind that a new network quality can be obtained by acquiring the knowledge of one selected sensor output; in large sensor networks, this is not too demanding a requirement.