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Top1. Introduction
Wireless sensor networks (WSNs) is the most widely adaptive network comprising of a number of sensor nodes scattered in a topology for specific sensing implementations (Li & Thai, 2008; Panda et. al., 2020). In the network each node tries to improve the estimates of the desired parameters taking into account its own and shared information from each and every sensor node in the network. The distributed estimation can be described mathematically as the optimization of a cost function which depends on all the information in a network (Panigrahi et al., 2016). For this, each sensor node after sensing the environment, processes the data and then sends it to a subset of the neighboring nodes. Therefore, great effort is required to formulate distributed estimation algorithms. The algorithm using the distributed approach improves the estimate of the parameters in each sensor node and minimizing the communication overhead thus providing energy efficiency to the system(Sayed, 2014b).
In case of sparse wireless sensor network or less connected networks the nodes are scattered far- apart in a geographic area. The number of neighbors participating in data aggregation and sharing for each node is less and even less for the nodes that lie at the edge of the network. These nodes face the difficulty in accessing the information from the network required for estimating the parameters(Kong, Lee, Kim, Shin, & Song, 2017; Nayak, Panigrahi, & Sharma, 2015). In this scenario one of the solutions to improve the performance may be the centralized approach. But the major drawback of this approach is that it requires more number of communications between the nodes and the central processor. Thus it is not energy efficient as more energy is consumed in communication in comparison to processing(Bazzi, Rastegarnia, &Khalili, 2014). In order to overcome these problems and to make the sensor syste m energy efficient, distributed parameter estimation approaches are proposed by authors in literature. There are two ways node can cooperate: incremental and diffusion.
Incremental cooperative approach was proposed for estimation of parameter with sharing information (Majhi& Panda, 2013) using particle swarm optimization algorithm. This mode of operation requires a cyclic path among the sensor nodes which is not feasible in large and sparse WSNs (Khalili, Rastegarnia, Bazzi, &Rahmati, 2017). The convergence speed is very slow and the algorithm is not adaptive in real time(Sayed, 2014b). Whereas in the diffusion cooperative strategy, every node updates the estimate and shares it with its one-hop immediate neighbors (Sayed, 2014a). This method in comparison to the incremental approach is adaptive to real time changes in the environment. In most of the literature, the distributed least mean square algorithm or its variants are used to estimate the parameters of the FIR system as it is inherently stable. But FIR systems need a greater number of parameters to have equivalent model of a real system. In fact, a feedback system may not be modeled exactly by using FIR even though with very high order. But the same feedback system can be modeled exactly by using IIR system. In WSNs, there are certain applications where the sensor nodes are modeled as IIR system. In this paper, we have considered an adaptive IIR system at each node. The Diffusion LMS algorithm is used to estimate the parameters of the adaptive IIR system with cooperation.
Further, the diffusion LMS algorithm for IIR system is extended to estimate the parameters in sparse sensor network. In sparse network where the nodes are connected to a smaller number of neighbors, the estimation performance degrades and need a greater number of iterations to achieve the steady state value which leads to the algorithm energy inefficient. Therefore, multi-hop diffusion LMS is proposed here where a sensor node communicate with more than one hop sensor nodes unlikely only to one hop in diffusion LMS algorithm (Hu &Tay, 2015; Kong et al., 2017).