# Blockwise Frequency Domain Active Noise Controller Over Distributed Networks
## 2 Collaborative Distributed Algorithm for an \(N\)-Nodes ASN Based on an Incremental Strategy
From Equation (3) and applying the gradient operator in Equation (1), we obtain  
\[\mathbf{w}(t)=\mathbf{w}(t-1)\ -\mu\mathbf{x}_{f}(t)\mathbf{e}(t), \tag{4}\]  
that can be rewritten as  
\[\mathbf{w}(t)=\mathbf{w}(t-1)\ -\mu\sum_{k=1}^{N}\ \mathbf{x}_{f,k}(t)\ e_{k}(t). \tag{5}\]  
Note that the computation of \(\mathbf{x}_{f,k}(t)\) involves the real acoustic path \(\mathbf{h}_{jk}\) which, in practical environments, can be estimated as \(\mathbf{s}_{jk}\). Thus, the vector of size \(LN\times 1\) that contains the last \(L\) samples of reference signal \(x(t)\) filtered through \(\mathbf{s}_{jk}\) is named as \(\mathbf{v}_{k}(t)\) and substituting in Equation (5) leads to  
\[\mathbf{w}(t)=\mathbf{w}(t-1)\ -\mu\sum_{k=1}^{N}\ \mathbf{v}_{k}(t)\ e_{k}(t). \tag{6}\]  
It can be seen in Equation (6) that all the error signals are necessary to calculate the coefficients of each filter, so that a central unit that receives and transmits all the information through the network is required (see Figure 3a). The problem is that, if the number of nodes increases or if multichannel nodes are used, it is not straightforward to transmit information between the central unit and each node due to an increase in the bandwidth required for communication. Moreover, any failure in the central unit will cause no information to be processed. Therefore, a distributed network (see Figure 3b) with the computational burden shared among the nodes becomes necessary to solve these problems. Since the number of signals processed at every node is low, this type of processing provides a more efficient computational performance. In addition, depending on the strategy used to exchange information among the nodes, the bandwidth in data transmission can be reduced. It should be noted that a distributed ASN in the context of this paper means that, not only are the nodes physically distributed in the area of interest, but also the processing (or computation) is divided among the nodes. Previous works [16; 21] showed that the implementation of the LMS algorithm over distributed networks using collaborative strategies achieves good results. However, those works do not consider acoustic nodes that acoustically interact with the environment both controlling and modifying it, as it was introduced in [20]. A distributed ANC system based on the Multiple Error Filtered-x Least Mean Square (MEFxLMS) algorithm [20] and using incremental communication strategies with sample-by-sample data acquisition was presented in [22]. In the case of a ring network based on an incremental strategy, the coefficients of the adaptive filters are calculated by distributing the calculation among different nodes by transmitting information to an adjacent node in a consecutive order. Thus, every node can calculate a portion of the sum of the filter updating equation and supply to the next node the partial result to update the coefficients with its respective information. If this step is performed with an incremental strategy, the  
Figure 3: A centralized ASN (**a**) and distributed ASN (**b**). CU is the central unit in the centralized case. last node will have the complete updated coefficients. Finally, these coefficients are disseminated to the rest of the nodes to allow the system to generate the appropriate cancelation signals before the next iteration begins. This means \(2(N-1)\) interchanges of the filter coefficients among the nodes (see Figure 4). Every \(k\)th node can share information with its adjacent \(k+1\)th node. Each node should calculate the adaptive filters of all the nodes of the network so, in a network of \(N\) nodes, the network state would be defined by \(N\) adaptive filters one of each node.