# Dynamic Matching of Reconstruction and Anti-Aliasing Filters in Adaptive Active Noise Control
## 2 Background and Limitations: Conventional Methods  
Single-rate systems and multi-rate systems are two commonly used kinds of traditional ANC systems [1; 33]. In a single-rate system, the reconstruction filter and anti-aliasing filter are implemented with analog circuits that are typically composed of capacitors, inductors, resistors, and operational amplifiers. The transfer functions in acoustic regions from the primary source to the reference microphone and error microphone are denoted by \(F_{p}(s)\) and \(P_{p}(s)\), respectively. The principles of a single-channel multi-rate ANC system are shown in Figure 1. The primary sound source \(v_{p}(t)\) reaches the reference microphone and error microphone after passing through \(F_{p}(s)\) and \(P_{p}(s)\), respectively, and the corresponding signals are \(x_{p}(t)\) and \(d_{p}(t)\), respectively. \(d_{p}(t)\) is the noise reduction target of ANC, and \(u_{\text{p}}(t)\) is the control signal given by the speaker. After passing through the secondary path of the sound field \(G_{\text{p}}(s)\), the signal from the error microphone is \(y_{\text{p}}(t)\). The transfer functions of the analog anti-aliasing filters are denoted by \(T_{xa}(s)\) and \(T_{\text{ea}}(s)\), and the transfer function of the analog reconstruction filter is denoted by \(T_{\text{ua}}(s)\).  
Compared with single-rate systems, a multi-rate system has additional digital regions with a high sampling rate. \(T_{\text{rd}}(z)\) and \(T_{\text{ed}}(z)\) correspond to digital anti-aliasing filters, and \(T_{\text{ud}}(z)\) corresponds to a digital reconstruction filter. The reference signal and error signal  
Figure 1: Conventional multi-rate ANC systems consist of four regions.  
in the regions with the high sampling rate are denoted by \(x_{h}(n)\) and \(e_{h}(n)\), respectively. According to the FxLMS algorithm, the output signal of the digital control system is  
\[u(n)=\mathbf{w}^{\text{T}}(n)\mathbf{x}_{c}(n) \tag{1}\]  
where \(\mathbf{w}(n)=\left[\begin{array}{cc}w_{0}(n)&w_{1}(n)&\cdots&w_{I-1}(n)\end{array} \right]^{\text{T}}\) denotes a column vector composed of a number of coefficients at time \(n\), and \(\mathbf{x}_{c}(n)=\left[\begin{array}{cc}x(n)&x(n-1)&\cdots&x(n-I+1)\end{array} \right]^{\text{T}}\). The adaptive filter coefficients are updated with  
\[\mathbf{w}(n+1)=\mathbf{w}(n)-\mu\mathbf{f}^{\text{T}}(n)e(n) \tag{2}\]  
where \(\mu\) denotes the update step size, \(G(z)\) denotes the true transfer function from \(u(n)\) through the loudspeaker to \(e(n)\), and \(\hat{G}(z)\) is the estimate of \(G(z)\). The filtered reference vector \(\mathbf{f}(n)=\left[\begin{array}{cc}\hat{P}(n)&\hat{P}(n-1)&\cdots&\hat{P}( n-I+1)\end{array}\right]^{\text{T}}\), where \(\hat{r}(n)\) donates the digital signal obtained by filtering the reference signal \(x(n)\) with \(\hat{G}(z)\).  
Regardless of whether a single-rate or multi-rate active noise control system is considered, reconstruction and anti-aliasing filters are placed before the ANC block in current systems, and the filter parameters remain fixed during the control process. These preset filters are generally determined based on specific models that only consider their attenuation characteristics, and, so, mismatches may occur when the signal features change. The roll-off rate describes how rapidly the filter attenuates the signal after the cut-off frequency and is typically measured in decibels per octave (dB/octave) or decibels per decade (dB/decade). If the roll-off rate of the designed filter is insufficient, excessive mirror noise or reconstruction noise may be introduced into the system; on the contrary, if the roll-off rate of the designed filter is too fast, excessive group delay may be introduced by the filter, and the control system may not meet the causal requirements, leading to decreased controller performance or even system failure [1].