## 3 Design and Analysis for WPTDDM-FxLMS Algorithm
### Algorithm Design for WPTDDM-FxLMS  
In Section 3.2, complex signals will be decomposed into multiple single frequency line spectrum signals by the wavelet method mentioned in Equations (19)-(24), and then the MIMO system will be transformed into a parallel SISO subsystem. It is assumed that the input signal \(\mathbf{X}(n)\) can be decomposed into time-domain signals \(f_{1}(n)\),\(f_{2}(n)\),\(\cdots\)\(f_{l}(n)\) by passing the wavelet packet function \(P_{j}(n)\), and all of them do not overlap with each other in different frequency bands. If each frequency band is in parallel and can operate independently, \(y_{h}^{j}(n)e_{h}^{j}(n)\), and \(u_{h}^{j}(n)\) are the controller output signal, error signal, and iteration step size for the h-th path of the \(j\)-th frequency band, respectively. Then, the \(h\)-th path total output will be elucidated as  
\[y_{h}(n)=\sum_{j=1}^{J}y_{h}^{j}(n)(h=1,2,\cdot\cdot\cdot,H) \tag{25}\]  
Then, the \(h\)-th error signal also can be expressed as  
\[e_{h}(n)=d_{h}(n)+\sum_{k=1}^{H}y_{k}(n)S_{hk}(n)=d_{h}(n)+\sum_{k=1}^{H} \left(\sum_{j=1}^{J}y_{k}^{j}(n)\right)S_{hk}(n) \tag{26}\]  
According to the steepest descent principle, the minimum response of the error sensor is regarded as the objective function. Thus, Equation (18) will be upgraded, and then the update rule of each path control filter in the \(j\)-th frequency band can be obtained as follows:  
\[\mathbf{w}_{h}^{j}(n+1)=\mathbf{w}_{h}^{j}(n)+\mu_{h}^{j}(n)\mathbf{f}_{mh} ^{j}(n)e_{h}^{j}(n)\ (h=1,2,\cdot\cdot\cdot,H) \tag{27}\]  
\[\mathbf{f}_{mh}^{j}(n)=\sum_{i=0}^{M-1}\hat{S}_{mh}(n)f_{j}(n-i)\ (h=1,2,\cdot\cdot,H) \tag{28}\]where \(\mu_{h}^{j}(n)\) is an iterative step-size, the filtered reference signal \(\mathbf{f}_{mh}^{j}(n)\) is filtered by the diagonalization identification matrix \(f_{j}(n)\), and \(e_{h}^{j}(n)\) is also transformed from the error signal \(e_{h}(n)\) by the wavelet packet, which can be expressed as  
\[f_{j}(n)=P_{j}(n)X(n)\;j=1,2,\ldots\;J \tag{29}\]  
\[e_{h}^{j}(n)=P_{j}(n)e_{h}(n)\;j=1,2,\ldots\;J \tag{30}\]  
In addition, in order to obtain the optimal control effect, the reference and the error signals can be introduced to correct the step-size, which can achieve a faster convergence speed with a large step-size in the early stage and a smaller steady-state error with a small step-size later. Then, the corrected iteration step-size can be rewritten as  
\[\mu_{h}^{j}(n+1)=\alpha\mu_{h}^{j}(n)+(1-\alpha)\frac{e_{h}^{j}(n)*e_{h}^{j}(n )}{f_{j}^{T}(n)f_{j}(n)+eps}\;(h=1,2,\cdots,H) \tag{31}\]  
For the determined transfer function in the secondary path and the reference input signal, the controller convergence speed can be guaranteed by adjusting the iteration step-size; if the step-size is too small, the convergence performance will also be reduced. Thus, the control signal for each path on the \(j\)-th frequency band subsystems will be expressed as  
\[y_{h}^{j}(n)=\mathbf{w}_{h}^{j}(z)f_{j}(n)\;(h=1,2,\cdots,H) \tag{32}\]  
Based on the above, Equations (27)-(32) present the controller updating law, the filtered reference signal, and the iterative step-size, and the output can update for each path on the \(j\)-th frequency band; the specific process and structure are shown in Figure 3 (\(H\) is equal to four), a novel improved control strategy, referred to as the WPTDDM-FxLMS algorithm. The line spectrum signal controller of the \(j\)-th frequency band can be called the control subsystem of the \(j\)-th frequency band. This algorithm will quickly acquire the frequency characteristics of the complicated signal by the wavelet packet transform, effectively suppressing the path coupling by the feedback compensation and decomposing it into multiple narrowband control signals at the same time. Therefore, it will have a higher computational efficiency, complex signal processing, and coupled vibration control capabilities and can be applied to active vibration control in MIMO systems.