## 2 The Simplified Frequency-Domain Feedback Active Noise Control Algorithm
### The Derivation of the WSMANC Algorithm  
Figure 1 shows a physical representation of feedback ANC system. Both the desired signal and error signal can be detected simultaneously using an error microphone. The desired signal comes from the primary noise. Then the error signal is superposed by the desired signal, and the signal is produced by the secondary loudspeaker. Then the error signal is fed back to the adaptive filter to produce the anti-noise using feedback ANC algorithms.  
Figure 2 displays the principal flow chart of the WSMANC algorithm. \(x(n)\) denotes the internal synthesis input signal, \(d(n)\) denotes the original interference, \(y(n)\) is the output of the adaptive filter, \(s(n)\) is detected by the error sensor from the secondary loudspeaker, the error signal is represented as \(e(n)\), \(x_{f}(n)\) is the time-delay filtering reference signal, \(W(z)\) represents the adaptive controller, \(C(z)\) is the true secondary path, and \(\Delta\) denotes the total acoustic delay.  
Based on the SAIMC algorithm, the error signal includes primary noise information; therefore, it can be used as the internal input signal for simplicity The internal synthesized signal \(x(n)\) is expressed as follows:  
\[x(n)=e(n) \tag{2}\]  
The input signal \(y(n)\) of the secondary loudspeaker is the internal synthesized reference signal \(x(n)\) filtered by the \(N\)-order adaptive filter \(W(z)\) as follows:  
\[y(n)=\sum_{i=0}^{N-1}x(n-i)w(i) \tag{3}\]  
where \(\{w(0)\ldots w(N-1)\}\) are the coefficients of \(W(z)\).  
The signal \(s(n)\) is detected using an error microphone and produced using a filtered output signal \(y(n)\) with the secondary path transfer function \(C(z)\) as follows:  
\[s(n)=\sum_{i=0}^{L_{p}-1}y(n-i)c(i) \tag{4}\]  
where \(c(i)\) is the coefficients of the secondary path impulse response, \(I=0\), \(1\), \(\ldots\), \(L_{p}-1\), and \(L_{p}\) denotes the length of the modeling secondary path. For the sake of calculation simplicity, the secondary path length \(L_{p}\) is equal to the adaptive filter length \(N\).  
\(\Delta\) is the sampling number, and it can be calculated using the total delay \(\tau\) and the sampling frequency. The time-delay filtering signal \(x(n)\) is obtained as follows:  
\[x_{f}(n)=x(n-\Delta) \tag{5}\]  
Figure 1: Physical block diagram of the feedback ANC system.  
Figure 2: Principal flowchart of the WSMANC algorithm.  
The signal \(s(n)\) is added to the original interference signal \(d(n)\), which produces the error signal \(e(n)\) as follows:  
\[e(n)=d(n)+\sum_{i=0}^{L_{p}-1}y(n-i)c(i) \tag{6}\]  
The objective function \(J(n)\) is written as follows:  
\[J(n)=e^{2}(n) \tag{7}\]  
Based on the gradient descent algorithm, the adaptive filter coefficient \(w(n)\) is updated in the time domain as follows:  
\[w(n)=w(n-1)-\mu_{0}x_{f}^{T}(n)e(n) \tag{8}\]  
where the variable factor \(\mu_{0}\) is a positive value and denotes the step size.  
\[w(n)=\left[w(N-1)\quad w(N-2)\quad\ldots\quad w(0)\right]^{T} \tag{9}\]  
\[x_{f}(n)=\left[x_{f}(0)\quad x_{f}(1)\quad\ldots\quad x_{f}(N-1)\right]^{T} \tag{10}\]  
The process of converting Equation (2) to (10) above is called the narrowband feedback active noise control algorithm without secondary path modeling algorithm (WSMANC for short).