## 2 Theoretical Considerations
### Proposed Additive ASP Control Algorithm with a Feedforward Enhancement  
Figure 3 shows the block diagram of the proposed additive ASP algorithm that consists of the pure ASP algorithm and the feedforward enhancement. The inside of the dashed box indicates the added feedforward enhancement. The subscripts noted as 1 and 2 in the symbols in Figure 3 mean the signals relevant to the pure ASP algorithm and the feedforward enhancement, respectively. Figure 2: Pure and additive ASP control schemes. (**a**) Noise canceling for a lower order and (**b**) sound enhancing for a higher order using the pure ASP algorithm. (**c**) Sound enhancing for the higher order using the proposed additive ASP algorithm with the feedforward enhancement. As shown in Figure 3, these two sub-algorithms included in the additive ASP algorithm are operated in parallel, and they make their own control signal \(u_{1}(n)\) and \(u_{2}(n)\), respectively. Then, the control signal \(u(n)\), which is the output of the proposed algorithm is selectively switched to either \(u_{1}(n)\) or \(u_{2}(n)\) in accordance with the engine speed. This conversion is conducted at the switch block depicted in Figure 3. Thus, when the ASP control logic is \"on\", the control signal \(u(n)\) is given by Equation (6a), but when the added feedforward enhancement is turned \"on\", it is given by Equation (6b). \[u(n)=u_{1}(n)\text{ for ASP}, \tag{6a}\] \[u(n)=u_{2}(n)\text{ for feedforward enhancement} \tag{6b}\]  
By the way, the feedforward enhancement in Figure 3 has a straightforward structure with a fixed control filter \(A(z)\) to compensate for the effect of the secondary path. This structure can be derived by simplifying a feedback ASP algorithm with the internal model control (IMC). The block diagram of the feedback algorithm is illustrated in Figure 4a, where \(\hat{y}(n)\) is the estimate of the output signal of the secondary path model \(\hat{S}(z)\), which represents the internal model. The complementary sensitivity function \(T(z)\) of this feedback ASP algorithm can be given as [1; 12]  
\[T(z)=\frac{Y(z)}{C_{2}(z)-D(z)}=\frac{A(z)S(z)}{1+A(z)\big{[}S(z)-\hat{S}(z) \big{]}} \tag{7}\]  
where \(Y(z)\), \(C_{2}(z)\) and \(D(z)\) are the z-transforms of \(y(n)\), \(c_{2}(n)\) and \(d(n)\), respectively. If \(\hat{S}(z)=S(z)\) and \(d(n)\) is ignorable as \(|D(z)|\ll|C_{2}(z)|\), then, Equation (7) can be rewritten by  
\[T(z)=\frac{Y(z)}{C_{2}(z)}=A(z)S(z). \tag{8}\]  
Thus, from Equation (8), the feedback ASP algorithm illustrated in Figure 4a can be simplified as the purely feedforward form depicted in Figure 4b, and this is the feedforward enhancement in the proposed additive ASP algorithm in Figure 3.  
Figure 3: Block diagram of the proposed additive ASP control algorithm with the feedforward enhancement. If the control filter \(A(z)\) of the feedforward enhancement in Figure 4b is the perfect inverse of \(S(z)\), \(e(n)\) will exactly follow \(c_{2}(n)\). However, in practice, since \(S(z)\) using the ESG-cowl pair is a non-minimum phase, its stable perfect inverse cannot be obtained. One possible approach to obtaining a realizable inverse filter of \(S(z)\) for \(A(z)\) is to use the Wiener filter design process [12] with a modeling delay \(\Delta\)[15]. This approach can offer a stable and causal FIR filter of \(A(z)\) by designing this as the delayed inverse of the secondary path \(S_{\Delta}^{-1}(z)\), although \(A(z)\) makes \(e(n)\) track the target signal \(c_{2}(n)\) delayed by \(\Delta\) samples. The proper tracking performance of \(A(z)\) can be obtained with a suitable choice of \(\Delta\). By the way, from a practical point of view, it is required for sound enhancing to just augment the amplitude of \(e(n)\) to that of \(c_{2}(n)\) with regardless of the phase. This can also be achieved by compensating for just the magnitude response of \(S(z)\) instead of both magnitude and phase responses of \(S(z)\).