# Dynamic Matching of Reconstruction and Anti-Aliasing Filters in Adaptive Active Noise Control
## 3 An Active Noise Control System with Dynamically Adapted Reconstruction and Anti-Aliasing Filters
### Construction of a Digital Filter Library in Regions with a High Sampling Rate  
The adaptation of digital reconstruction and anti-aliasing filters can be implemented with online design methods or library-based methods. Online design methods dynamically calculate the filter parameters during the control process, while library-based methods set a library composed of a large number of filters before the control starts. In order to simplify the filter-matching process and ensure the stable operation of the filters, this study adopts a library-based method. Before the control starts, a large number of filters are arranged to form a filter library. In the control process, only the most suitable filters need to be selected from the filter library, with no need for online filter design. First, \(T_{xd}(z)\), \(T_{ud}(z)\), and \(T_{cd}(z)\) are designed. These transfer functions have the following general form:  
\[H(z)=\frac{Y(z)}{X(z)}=\frac{\sum\limits_{k=0}^{q}b_{k}z^{-k}}{1+\sum\limits _{k=1}^{p}a_{k}z^{-k}} \tag{5}\]  
where \(\{b_{k}\}\) and \(\{a_{k}\}\) are filter coefficients that determine the locations of the zeros and poles of \(H(z)\), respectively, \(X(z)\) is the z-transform of the input signal, and \(Y(z)\) is the z-transform of the output signal. Different FIR filters or IIR filters can be implemented based on different values of \(\{a_{k}\}\) and \(\{b_{k}\}\). Regardless of whether one is using an FIR filter or IIR filter, one can refer to mature filter models such as the Butterworth, Chebyshev, Elliptics, and Bessel models when constructing the filter library. Chebyshev-I low-pass filters exhibit equal ripples in the passband, fast attenuation in the transition band, and a monotonic decrease attenuation with the frequency in the stopband. The squared amplitude response of the Nth-order analog Chebyshev filter is  
\[\left|H_{a}(\Omega)\right|^{2}=\frac{1}{1+\varepsilon^{2}T_{N}^{2}\Big{(} \Omega/\Omega_{p}\Big{)}} \tag{6}\]where \(\Omega_{p}\) denotes the passband cut-off frequency of the analog filter, \(\varepsilon\) is a parameter that determines the size of the ripples within the passband, and \(T_{N}(x)\) is the first type of Chebyshev polynomial [33]:  
\[T_{N}(x)=\left\{ \begin{array}{cc}\cos(N\arccos x)&|x|\leq 1\\ \cosh(N\arccos x)&|x|>1\end{array} \right. \tag{7}\]  
Selecting different values of \(\Omega_{p}\), \(\varepsilon\), and \(N\), we obtain the constant \(H_{0}\) and poles \(p_{k}\). Then, the following transfer function of the analog filter can be obtained:  
\[H_{a}(s)=\frac{H_{0}}{\prod\limits_{k}(s-p_{k})} \tag{8}\]  
After the completion of the analog filter design, the corresponding digital Chebyshev-I-type low-pass filter can be constructed with such methods as bilinear transformation:  
\[H(z)=H_{a}(s)\Big{|}_{s=\frac{z-1}{z+1}}=\frac{H_{0}}{\prod\limits_{k}\left( \frac{z-1}{z+1}-p_{k}\right)} \tag{9}\]  
After sorting, Equation (9) is transformed into the form shown in Equation (5), thus yielding the filter output  
\[y(n)=\sum\limits_{k=0}^{q}b_{k}x(n-k)-\sum\limits_{k=1}^{p}a_{k}y(n-k) \tag{10}\]  
and the actual frequency response of the digital filter  
\[H(e^{i\omega})=H(z)\big{|}_{z=e^{i\omega}}=\frac{\sum\limits_{k=0}^{q}b_{k}e^{ -i\omega k}}{1+\sum\limits_{k=1}^{p}a_{k}e^{-i\omega k}}=\Big{|}H(e^{i\omega} )\Big{|}e^{i\varphi(\omega)} \tag{11}\]  
where \(\omega\) is the radian frequency, \(\big{|}H(e^{i\omega})\big{|}\) is the amplitude response of digital filter \(H(z)\), and \(\varphi(\omega)\) is the phase response of \(H(z)\). Then, the group delay response of the filter is  
\[\tau(\omega)=-\frac{d\varphi(\omega)}{d\omega} \tag{12}\]  
For the first, second, third, and fourth orders of Chebyshev-I-type filters, assuming that the high sampling rate is 40 kHz and the cut-off frequency of the passband is 2200 Hz, we can derive the amplitude response and group delay curves, as shown in Figure 3.  
Figure 3: Frequency response and group delay curves of Chebyshev-I-type filters with different orders: (**a**) amplitude response; (**b**) group delay.