## 2 Numerical and Control Methods
### Active Control Method of the Flow-Induced Resonance Noise  
The narrowband ANC system for resonance noise mainly consists of the computer, the controller, the amplifier, the secondary source and the error microphone, as shown in Figure 1. The error microphone is installed on the side-wall of the upstream duct of the cavity to pick up the residual noise signal \(e(n)\), while the secondary source is placedat the bottom of the cavity. The controller calculates the output signal \(y(n)\) using the estimated resonance frequency, which is then amplified and used as the stimulus for the secondary source. As a result, an acoustic wave of the same amplitude and the opposite phase corresponding to the narrowband resonance noise would be generated, with which the noise cancellation could be achieved.  
The narrowband FxLMS algorithm used in this paper is shown in Figure 2. The secondary path \(S(z)\) is the system between the controller output and the error microphone, whose model \(\hat{S}(z)\) could be obtained offline. The sinusoidal reference signal \(x(n)\) is synthesized internally in the controller based on an estimation of the resonance frequency. The output signals of the control filter and the secondary path are denoted as \(y(n)\) and \(s(n)\), respectively. The disturbance signal \(d(n)\) is the external narrowband resonance noise. The control filter is denoted as \(W(z)\).  
The controller coefficient vector and the reference signal vector can be expressed as  
\[\mathbf{w}(n)=\left[w_{0}(n),w_{1}(n),\ldots,w_{L-1}(n)\right]^{T} \tag{9}\]  
\[\mathbf{x}(n)=\left[x(n),x(n-1),\ldots,x(n-L+1)\right]^{T} \tag{10}\]  
where \(L\) is the order of the controller and \(n\) is the time index. The controller coefficients are adjusted adaptively according to the FxLMS algorithm, which could be described by the following set of equations  
\[y(n)=\mathbf{x}^{T}(n)\mathbf{w}(n) \tag{11}\]  
\[e(n)=d(n)+s(n)=d(n)+\sum_{i=0}^{L_{q}-1}h_{i}y(n-i) \tag{12}\]  
\[r(n)=\sum_{i=0}^{\hat{L}_{q}-1}\hat{h}_{i}x(n-i) \tag{13}\]  
\[\mathbf{r}(n)=\left[r(n),r(n-1),\ldots,r(n-L+1)\right]^{T} \tag{14}\]  
Figure 1: Schematic diagram of the ANC system for acoustic resonance noise.  
Figure 2: The narrowband FxLMS algorithm.  
\[\mathbf{w}(n+1)=\mathbf{w}(n)-\mu e(n)\mathbf{r}(n) \tag{15}\]  
where \(r(n)\) is the filtered-x signal,\(\mu\) is the stepsize, \(h_{i}\) (\(i\) = 0,..., \(L_{s}\)) and \(\hat{h}_{i}\) (\(i\) = 0,..., \(\hat{L}_{s}\)) are the coefficients of the impulse responses of the secondary path and its model, respectively.  
How to generate the reference signal is one of the key issues for narrowband ANC systems. In this paper, the reference signal is internally generated with an estimate of the resonance frequency [23; 24; 25; 26; 27]. Considering that the frequency of the narrowband resonance noise depends on the physical structure, the frequency is fixed and does not vary with time. Thus, it is tested offline and used directly in the algorithm, as shown in Figure 2. It should be noted that for ordinary linear systems, the control performance of such a narrowband system is extremely sensitive to the FM, which is the frequency difference between the reference signal and the disturbance signal. In principle, the offline estimation of the resonance frequency should be made as accurate as possible.