# Blockwise Frequency Domain Active Noise Controller Over Distributed Networks
## 2 Collaborative Distributed Algorithm for an \(N\)-Nodes ASN Based on an Incremental Strategy
Notice that only the \(F\) partitions of \(2B\) coefficients of their adaptive filter are needed to generate the \(k\)th node output signal:  
\[\textbf{W}_{k}[n-1]=\hat{\textbf{W}}_{N}[n-1]_{(;1+F(k-1):Fk)}. \tag{9}\]  
Moreover, we define  
\[\textbf{V}_{k}[n]=\left[ \begin{array}{c}\textbf{V}_{1k}[n],\textbf{V}_{2k}[ n],\ldots\textbf{V}_{Nk}[n],\end{array} \right] \tag{10}\]  
being \(\textbf{V}_{jk}[n]=\left[ \begin{array}{c}\textbf{V}_{jk}^{1}[n],\textbf{V}_ {jk}^{2}[n],\ldots,\textbf{V}_{jk}^{F}[n]\end{array} \right]\) a matrix of size \([2B\times F]\) where vector \(\textbf{V}_{jk}^{f}[n]\) contains the \(2B\)-FFT of the reference signal filtered with the \(f\)th partition of the estimated acoustic channel \(\textbf{s}_{jk}\). Each node estimates the coefficients of the rest of the nodes to achieve a global solution using the information of the previous node and the adaptation matrix calculated using the signals that each node own. Derived from the FPBFxLMS algorithm in [26], we can rewrite Equation (7) in the frequency domain with blocks of B samples. Thus, the update equation of the adaptive filter coefficients of the \(k\)th node at the \(n\)th block iteration is given by  
\[\hat{\textbf{W}}_{k}[n]=\hat{\textbf{W}}_{k-1}[n]\ -\mu\ \text{FFT}\{\{\left[ \begin{array}{c}\text{IFFT}\{\textbf{E}_{k}[n]\circ\textbf{V}_{k}[n]^{*}\} \end{array} \right]_{[1:B,z]}\quad\textbf{0}_{[B\times FN]}\ \}\}, \tag{11}\]  
where \(\textbf{E}_{k}[n]\) is the multiplication of vector \(\textbf{E}_{k}[n]\) by \(\textbf{1}_{[1\times FN]}\), a row vector of ones of size \(FN\). Constant \(\mu\) is the step-size parameter and the operators FFT and IFFT perform the direct and inverse fast Fourier transform of size \(2B\) of each column of the matrixes involved. \(\circ\) denotes the element-wise product of two matrices, \(*\) denotes complex conjugation and \(\textbf{0}_{[B\times FN]}\) is a matrix of zeros of size \([B\times FN]\). Note that we only consider the first \(B\) samples of the \(2B\)-IFFT operation. Once all the nodes have finished the filter coefficient updates, the global vector \(\hat{\textbf{W}}_{N}[n]\) is disseminated to the rest of the nodes for the (\(n\)+1)th iteration. Note that in (11), \(\hat{\textbf{W}}_{0}[n]=\hat{\textbf{W}}_{N}[n-1]\), as we stated in Equation (7). **Algorithm 1** illustrates the summary of the algorithm instructions, which are executed per block iteration at each node. \begin{table}
\begin{tabular}{c l} \hline \(B\) & Block size \\ \(L\) & Length of the adaptive filters \\ \(M\) & Length of the Finite Impulse Response (FIR) filters that model the estimated secondary paths \\ \(F\) & \(L/B\), number of partitions of the adaptive filters \\ \(P\) & \(M/B\), Number of partitions of the estimated secondary paths \\ \(n\) & index that denotes block iteration \\ \(f,p\) & super-indexes that denote partition number. \\ \(\textbf{s}_{jk}\) & M-length estimation of the acoustic path that links the actuator at the \(j\)th node with the sensor at the \(k\)th node. \\ \(\textbf{S}_{jk}^{p}\) & Fast Fourier Transform (FFT) of size \(2B\) of the \(p\)th partition of the acoustic path \(s_{jk}\). \\ \hline \end{tabular}
\end{table} 
Table 1: Notation of the description of the FPBFxLMS algorithm.