# Notes on a New Structure of Active Noise Control Systems
## 3 A New Structure of Active Noise Control Systems  
Values of the discrete-time signal \(e_{0}(i)\) are not technically measurable but they can be estimated with a signal processing algorithm. The technical realization of this idea results in a new structure of an active noise control system presented in Figure 4. To estimate values of the discrete-time signal \(e_{0}(i)\) a discrete-time filter FIR with a transfer function \(F_{est}(z^{-1})\) is included into the secondary path (and into the disturbance path as well) at the output of error signal measurement path. This filter should also be taken into account as a part of the secondary path model. The resulting active noise control system structure is known in the literature as a FeFxLMS control system structure [13; 14; 15; 16; 17; 18; 19; 20; 21; 22]. The literature FeFxLMS is used in another way: To shape properties of the error signal seen by the adaptation algorithm (not perceived by the active noise control system users) as well as to accelerate the rate of convergence of the adaptation algorithm based on the discrete-time signal \(e(i)\). A special case, filtered-error LMS (FeLMS) also uses a filter placed in the error signal measurement path to compensate for the secondary path effects [4; 7; 24; 25; 26; 27]. However, in the literature the existence of the error signal measurement path is not treated in a correct way from the point of view of control theory. The error signal measurement path is included into the primary and secondary paths. It is an optimal solution only for an unrealistic case of the error signal \(e(i)\) equal to 0. In this case the error signal \(e_{0}(i)\) at the error microphone is also equal to 0. In the paper the coefficients of the filter \(F_{est}(z^{-1})\) are calculated offline, before the activation of the active noise control system, using a routine for the equalization of communication transmission channels [28; 29; 30]. It means that the following objective function:  
\[\mathcal{E}\left\{\left(\delta_{0}(i)-e_{0}(i-\Delta)\right)^{2}\right\} \tag{1}\]  
Figure 3: The classical FxLMS control system structure with the featured error signal measurement path embedded in the simulation environment. is minimized with the LMS, LS (least squares), or the RLS (recursive least squares) algorithm and an artificially offline generated discrete-time signal \(e_{0}(i)\). Realizations of persistently exciting signals of a proper order should be used as \(e_{0}(i)\)[31]. In the objective function (1) \(\hat{e}_{0}(i)\) is the output of a discrete-time filter \(F_{est}(z^{-1})\) excited by the output \(e(i)\) of the error signal measurement path and \(\Delta\) is a discrete-time delay being the parameter of equalization algorithm. It is assumed that the dynamical model of the error signal measurement path is known before the equalization is performed and may be identified offline in any way before the start of the active noise control system [32]. Values of \(e(i)\) necessary for equalization are calculated using the known model of the error signal measurement path and artificially generated values of discrete-time signal \(e_{0}(i)\) used as the excitation of the model of the error signal measurement path. It is worth noticing that the identified linear dynamical model of the error signal measurement path may contain unstable zeros that influence the quality of estimates \(\hat{e}_{0}(i)\) obtained. An offline calculation of the filter \(F_{est}(z^{-1})\) coefficients may be considered as an inverse modeling of the error signal measurement path in the minimum variance sense and this filter is called in the paper an equalization filter. The use of identification techniques for inverse modeling overcomes problems associated with inverting dynamic systems with unstable zeros. The filter \(F_{est}(z^{-1})\) may be incorporated into the error signal measurement path as an additional hardware or into the adaptive control algorithm implemented in a signal processor.