## 3 Modified Structures for Hybrid Active Sound Quality Control System
### The Modified Structure for a Hybrid Active Sound Quality Control Structure  
Considering that the use of the leaky FxLMS algorithm in the simplified structure for the hybrid sound quality control system weakens the ability of noise attenuation in the feedback control structure, a modified structure for the hybrid sound quality control system is proposed to solve this problem. A schematic diagram of the modified structure for the hybrid active sound quality control system is shown in Figure 3. Unlike the traditional structure, the reference signal \(x_{b}(n)\) in the feedback control structure is now synthesized by the residual error signal \(e(n)\), the output signal \(y_{c}(n)\) in the line-spectrum noise cancellation control structure, and the secondary signal \(y_{b}(n)\) of the feedback control structure filtered by the estimated model of the secondary path \(\hat{S}(z)\). By using the \(Z\)-transforms, the reference signal in the feedback control structure is described as  
\[X_{b}(z)=E(z)+\hat{S}(z)Y_{b}(z)-Y_{c}(z). \tag{18}\]  
Figure 3: Schematic diagram of the modified structure for the hybrid active sound quality control system.  
When \(S(z)=\hat{S}(z)\) and \(D(z)=X(z)P(z)\) is established, the residual error signal is derived as follows:  
\[E(z)=[1-S(z)W_{b}(z)]V(z)+\big{[}[1-S(z)W_{b}(z)][P(z)-(1-\beta)S(z)W_{f}(z)]+S( z)W_{c}(z)W_{b}(z)\big{]}X(z). \tag{19}\]  
Because the residual error signal varies with the sound quality control goals, the coupling is discussed based on the residual error signal in Equation (19). It is shown clearly in Equation (19) that the feedback and feedforward control filters in this modified structure are decoupled if \(W_{c}(z)\) is small enough. Furthermore, this modified structure has less of a coupling effect because of the use of the reference signal in Equation (18) when this modified structure is compared with the traditional structure for hybrid active sound quality control system. The term \(1-S(z)W_{b}(z)\) approaches zero to offset the Gaussian random signal \(v(n)\). When the gain factor \(\beta\) is equal to zero, the feedforward control filter \(W_{f}(z)\) is equal to \(P(z)/S(z)\), and the line-spectrum noise control filter \(W_{c}(z)\) is approximately zero, which can eliminate the desired signal. It can be seen that the desired signal \(d(n)\) is reduced by the output signals of the feedforward control filter \(W_{f}(z)\) and feedback control filter \(W_{b}(z)\).  
Moreover, it can be seen that the synthesis of the pseudo-error signal in the feedforward control structure is different by comparing Figures 1 and 3. The pseudo-error signal in the feedforward control structure is composed of the residual error signal \(e(n)\), the secondary signal \(y_{b}(n)\) filtered by the estimated model of the secondary path \(\hat{S}(z)\), and \(\beta\) times the output signal \(y_{f}(n)\) filtered by the estimated model of the secondary path \(\hat{S}(z)\). The pseudo-error signal can be described as  
\[\begin{array}{l}E^{\prime}(z)=E(z)+\hat{S}(z)Y_{b}(z)-\beta\hat{S}(z)Y_{f}( z)=X_{b}(z)+Y_{c}(z)-\beta\hat{S}(z)Y_{f}(z)\\ =D(z)+V(z)+[\hat{S}(z)-S(z)]Y_{b}(z)-S(z)Y_{f}(z)+\beta[S(z)-\hat{S}(z)]Y_{f}( z).\end{array} \tag{20}\]  
If \(S(z)=\hat{S}(z)\) is applied to Equation (20), one can obtain  
\[E^{\prime}(z)=D(z)+V(z)-S(z)Y_{f}(z). \tag{21}\]  
The connection between the pseudo-error signal and the reference input signal in the feedback structure is shown in Equation (20). It is obvious in Equation (21) that the production of the pseudo-error signal does not introduce an extra term compared to the traditional structure.