## 2 Adaptive Digital Filtering System  
Commonly, digital filters are used for modeling the response properties of given inputs on acoustic and structural systems. The FIR filter is widely used for modeling system responses on account of its intrinsic stability, easy application, and simplicity. Adaptive digital filters also contain a notable capability for analysis and control of adaptive structures. Adaptive digital signal processing is used to adjust the coefficients of digital filters so as to generate a wanted signal to a given system. Adaptive filters update their filter coefficients with a reference signal and error, based on a repetitive algorithm, and are commonly used in active control applications such as noise and vibration attenuation, inverse modeling, and system identification.  
Among a number of adaptive filtering algorithms, the LMS method is the most prevalent feedforward control algorithm because of its relative simplicity [15]. It adjusts a digital filter by the reference input \(u_{k}\), with frequency components to be dealt with; the measured error \(e_{k}\) at the current moment; the gap of the adaptive filter output \(y_{k}\) and the desired value \(d_{k}\). This algorithm makes the output of the filter close to the plant output, which is the desired signal. Updating the equation for the filter coefficients of the LMS can be acquired with the minimization of the mean squared error chosen by a cost function [16].  
\[\mathbf{w}_{k+1}=\mathbf{w}_{k}+\mu\mathbf{u}_{k}\cdot e_{k} \tag{1}\]  
Here, \(\mathbf{w}_{k}\) stands for the weight vector of the filter and \(\mu\) adjusts the stability and the speed of the process. Figure 1 represents the diagram of a common digital filter system. It can be noticed that the filter coefficients cannot track the actual surface of the steepest descent, since the weight changes are calculated by gradient estimates with simplification assumptions. This factor would induce a rather crude tracking performance.  
The LMS algorithm is suitable for cases where there are no dynamics in the secondary path of the control system. Nevertheless, in most practical situations, some influences could exist from the sensors, actuators, amplifiers, filters, digital converters, and so on. The algorithm would not function appropriately with such effects. The most usual type of the secondary path dynamics is a time delay or a phase shift, and the overall system seems to converge to a trivial value or become unstable, because the signal \(y_{k}\)' and the filter output \(y_{k}\) are distinct.  
The FX-LMS is a modification of the above LMS method that incorporates the dynamics of the secondary path, which have been obtained beforehand [17]. This algorithm consists of a filtered reference and it is broadly known that FX-LMS is more stable when the dynamics of the secondary path cannot be ignored. The below equation defines the FX-LMS.  
\[\hat{\mathbf{w}}_{k+1}=\hat{\mathbf{w}}_{k}+\mu\mathbf{u}_{k}\mathbf{{}^{*}} \cdot e_{k} \tag{2}\]  
The significant property of the \(P_{c}\)'(z) filter is that its response with impulse contains the identical quantity of time delay with the dynamics of the secondary path \(P_{c}\)(z). However, there are noticeable defects of the FX-LMS, which a system identification (ID) procedure is needed so as to obtain the \(P_{c}\)'(z) offline in advance. An online system ID might be possible also, but it causes the overall control system complex and it is difficult to implement.