## 2 Problem Formulation  
In this section, we present the general aspects of the distributed blind calibration problem by generalizing the approach based on affine sensor models presented in the previously published group of papers [34; 35; 36; 44; 45; 46]. Consider \(n\)-distributed sensors to be measuring an _unknown signal_, \(x_{t}\), at discrete time instants, \(t\), \(t\in\mathfrak{I}^{+}\), where \(\mathfrak{I}^{+}\) is a set of non-negative integers. The local _output_ of the \(i\)-th sensor is defined by the following _general model:_  
\[y^{i}_{t}(x_{t})=\beta^{i}+a^{i}x_{t}+v^{i}(x_{t})+\xi^{i}_{t}, \tag{1}\]  
where the _offset_, \(\beta^{i}\), and the _gain (drift)_, \(a^{i}>0\), are _unknown_ constants; \(v^{i}(x_{t})\) is an unknown bounded function, modeling the part of the sensor output that is not explained by the linear model, \(\beta^{i}+a^{i}x_{t}\); \(\xi^{i}_{t}\) is the local _additive measurement noise_, \(i=1,\ldots,n\). The _calibration_ of each sensor is performed by applying the following _affine calibration transformation_, producing the _corrected sensor output_:  
\[z^{i}_{t}=b^{i}+a^{i}y^{i}_{t}=f^{i}+g^{i}x_{t}+a^{i}v^{i}(x_{t})+a^{i}\xi^{i} _{t}, \tag{2}\]  
where \(a^{i}\) and \(b^{i}\) are the _gain (drift) correction_ and the _offset correction (calibration) parameters_, respectively; \(g^{i}=a^{i}a^{i}\) in (2) represents the _corrected gain (drift)_; \(f^{i}=a^{i}\beta^{i}+b^{i}\) represents the _corrected offset_. **Remark 1**. : _A simple idea behind the calibration transformation (2) is to achieve \(g^{i}\approx 1\) and \(f^{i}\approx 0\) through a proper choice of the calibration parameters \(a^{i}\), \(b^{i}\), and \(i=1,\ldots,n\). The simplified case, corresponding to \(v^{i}(x_{t})=0,\forall t\), has been treated in the vast majority of papers dealing with sensor calibration, proposing different methodologies for determining \(a^{i}\) and \(b^{i}\) (see the Introduction and the references therein). However, the fact that the sensor parameters \(a^{i}\) and \(\beta^{i}\) are unknown and the fact that there is no information about the measured signal, \(x_{t}\), make the blind calibration problem highly nontrivial._  
**Remark 2**. : _In this paper, we assume that \(x_{t}\) is an unknown bounded deterministic function of time, implying a deterministic character of \(v^{i}(x_{t})\). Therefore, the analysis presented below will be based on deterministic concepts and the stability properties of deterministic time-varying systems. Notice that an alternative approach is possible, based on the assumption that \(x_{t}\) is a random sequence (see, e.g., [33; 44; 45; 46])._  
We assume that the observed sensors form a _network_ with specific structural properties, which can be formally represented by a _directed graph_ (digraph), \(\mathcal{G}(\mathcal{N},\mathcal{E})\), where \(\mathcal{N}\) is the set of nodes and \(\mathcal{E}\) is the set of directed edges (arcs) [33; 44; 48]. Let \(A_{ad}=[a^{ij}_{ad}]\) be the \(n\times n\)_adjacency matrix_ of \(\mathcal{G}\), such that \(a^{ij}_{ad}=1\) when the \(j\)-th sensor sends its messages to the \(i\)-sensor; otherwise, \(a^{ij}_{ad}=0\). Let \(N^{in}_{i}\) represent the set of the in-neighboring nodes of node \(i\), i.e., the nodes satisfying \(a^{ij}_{ad}=1\) for a given \(i\) and all \(j\); \(\mathcal{N}^{out}_{i}\) represents the out-neighborhood of node \(i\). In the following, we start from the above sensor and calibration models and require that the corrected sensor outputs, \(z^{i}_{t}\) (and/or their increments \(\Delta z^{i}_{t}\)), are _asymptotically equal_, without any information of the measured signal, \(x_{t}\). A simple interpretation of this approach in practice is to exploit the dominant influence of the well-calibrated sensors, which are in the majority in typical exploitation conditions.