# Blockwise Frequency Domain Active Noise Controller Over Distributed Networks
## 2 Collaborative Distributed Algorithm for an \(N\)-Nodes ASN Based on an Incremental Strategy  
Consider a network of \(N\) nodes that supports an ANC system composed by \(N\) sensors and \(N\) actuators, as shown in Figure 2. For the sake of simplicity, we consider one disturbance noise and a distributed network of single-channel acoustic nodes in a homogeneous network. This implies that all the nodes have the same computation and communication capabilities, execute the same algorithm and are composed of a single sensor and a single actuator. The signals recorded at the sensors are called error signals and denoted by \(\epsilon_{k}(t)\) (where \(k=1,2,\ \ldots\,N\)), the actuators emit the filter output signals \(y_{j}(t)\) (where \(j=1,2,\ \ldots\,N\)), the acoustic channel impulse response between actuator \(j\) and sensor \(k\) is estimated as \(\mathbf{s}_{jk}\), which is defined as the Finite Impulse Response (FIR) filter that models the estimation of the real acoustic channel \(\mathbf{h}_{jk}\). In feedforward systems, the reference signal \(x(t)\), is captured by a reference sensor employed to detect the acoustic noise far away from the area of interest. There exits only one noise source and all the nodes share the same reference signal \(x(t)\). We aim at canceling the acoustic noise signal in the sensors location, \(d_{k}(t)\), designing an adaptive filter \(\mathbf{w}_{k}(t)\) at every node. Assuming that \(\mathbf{w}_{k}(t)\) varies slowly, it can be written:  
\[\mathbf{e}(t)=\mathbf{d}(t)\ +\ \mathbf{w}^{\mathrm{T}}(t)\mathbf{x}_{f}(t), \tag{1}\]  
where the vectors \(\mathbf{e}(t)=[e_{1}(t)\ e_{2}(t)\cdots\ e_{N}(t)]^{T}\) and \(\mathbf{d}(t)=[d_{1}(t)\ d_{2}(t)\cdots\ d_{N}(t)]^{T}\) contain the error signals and desired signals, respectively, of the \(N\) nodes of the network, vector \(\mathbf{w}(t)=[\mathbf{w}_{1}(t)\ \mathbf{w}_{2}(t)\cdots\ \mathbf{w}_{N}(t)]^{T}\) of size \(LN\times 1\), concatenates the \(N\) adaptive filters \(\mathbf{w}_{k}(t)\) that contain the \(L\) filter coefficients of the \(k\)th node at the time instant \(t\). Matrix \(\mathbf{x}_{f}(t)=[\mathbf{x}_{f,1}(t)\ \mathbf{x}_{f,2}(t)\cdots\ \mathbf{x}_{f,N}(t)]\) is the concatenation of N vectors of size \(LN\times 1\) defined as \(\mathbf{x}_{f,k}(t)=[x_{f,1k}(t)\ x_{f,2k}(t)\cdots\ x_{f,Nk}(t)]^{T}\) that contain the last \(L\) samples of the reference signal \(x(t)\) filtered through the acoustic channel \(\mathbf{h}_{jk}\) that links the actuator at the \(j\)th node with the sensor at the \(k\)th node. The objective is to estimate the coefficients vector \(\mathbf{w}(t)\) that minimizes a cost function \(J(t)\) that depends on the error signals \(e_{k}(t)\). To do this, we use a gradient-descent method to estimate the coefficients in an iterative manner,  
\[\mathbf{w}(t)=\mathbf{w}(t-1)\ -\mu\
abla_{\mathbf{w}}\ E\{J(t)\}, \tag{2}\]  
where \(\mu\) is the step-size parameter, \(E\{.\}\) is the expectation operator and \(\
abla_{\mathbf{w}}=[\
abla_{\mathbf{w}_{0}}\ \
abla_{\mathbf{w}_{1}}\ \cdots\ \ \
abla_{\mathbf{w}_{N}}]\) being \(\
abla_{\mathbf{w}_{k}}\) the gradient operator defined as the partial derivatives with respect to the coefficients vector \(\mathbf{w}(t)\). The cost function is approximated by its instantaneous value by using the Least Mean Square (LMS) method [19], \((\
abla_{\mathbf{w}}E\{J(t)\}\approx\
abla_{\mathbf{w}}\ (J(t)))\). Moreover and as stated in [20], we consider the sum of the power of the \(N\) instantaneous error signals as cost function,  
\[J(t)=\sum_{k=1}^{N}\ e_{k}^{2}(t)\ =\mathbf{e}(t)\mathbf{e}^{T}(t). \tag{3}\]  
Figure 2: Acoustic Sensor Network (ASN) of \(N\) nodes for an ANC system.