### Basic NANC System with Online SPM  
By applying [PERSON]'s method [14] to an NANC system, [PERSON] [26] introduced and analyzed a basic NANC system with online SPM. The block diagram of the NANC system equipped with auxiliary white noise is shown in Figure 1. The whole system is composed of two subsystems. One, called the NANC subsystem, synthesizes the secondary source signal, for which a linear combiner (LC) is used as the controller and an adaptive notch filter is used as a noise canceler to mitigate the primary noise \(p(n)\). The other, called the SPM subsystem, estimates the secondary path, for which fixed-power auxiliary noise is used as the training signal. The primary noise can be represented as  
\[p(n)=\sum_{i=1}^{q}\left\{a_{i}\cos\omega_{i}n+b_{i}\sin\omega_{i}n\right\}+v_{p }(n) \tag{1}\]  
where \(q\) is the number of frequency components of the primary noise, \(\omega_{i}\) is the frequency of the \(i\)th component, and \(v_{p}(n)\) is a zero-mean additive white noise with variance \(\sigma_{p}^{2}\). \(\{a_{i}(n),b_{i}(n)\}\) are the discrete Fourier coefficients (DFC) of the \(i\)th frequency components. The reference sine and cosine waves, which are generated by a synchronization signal from a nonacoustic sensor, such as a tachometer [29], are defined as  
\[x_{a_{i}}(n)=\cos\left(\omega_{i}n\right),x_{b_{i}}(n)=\sin\left(\omega_{i}n\right) \tag{2}\]  
where \(\omega_{i}\) is identified from the synchronization signal. The output of the secondary source signal generated by the NANC subsystem is expressed as  
\[y_{p}(n)=\sum_{i=1}^{q}y_{i}(n)=\sum_{i=1}^{q}\left\{\hat{a}_{i}(n)x_{a_{i}}(n )+\hat{b}_{i}(n)x_{b_{i}}(n)\right\} \tag{3}\]  
where \(\{\hat{a}_{i}(n),\hat{b}_{i}(n)\}_{i=1}^{q}\) are DFC estimations of the secondary source, updated using some adaptive algorithms. The true secondary path is modeled by a moving average (MA) system  
\[S(z)=\sum_{j=0}^{M-1}s_{j}z^{-j} \tag{4}\]  
where \(M\) is the model order and \(\{s_{j}\}_{j=0}^{M-1}\) are the model coefficients. The error signal is measured using an error microphone, given by  
\[e(n)=p(n)-y_{s}(n)=p(n)-\sum_{j=0}^{M-1}s_{j}y(n-j) \tag{5}\]  
where  
\[y(n)=y_{p}(n)-v(n) \tag{6}\]  
Figure 1: Narrowband active noise control (NANC) system with online secondary path modeling (SPM, \(i\)th channel). where \(v(n)\) is the auxiliary white Gaussian noise with zero mean and variance \(\sigma_{v}^{2}\). In the NANC subsystem, the FXLMS algorithm is used to update the DFC estimation of the secondary source, given by  
\[\hat{a}_{i}(n+1) =\hat{a}_{i}(n)+\mu_{i}e(n)\hat{x}_{a_{i}}(n) \tag{7}\] \[\hat{b}_{i}(n+1) =\hat{b}_{i}(n)+\mu_{i}e(n)\hat{x}_{b_{i}}(n) \tag{8}\]  
where \(\mu_{i}\) is the step size, which directly determines the convergence speed, and \(\hat{x}_{a_{i}}(n)\) and \(\hat{x}_{b_{i}}(n)\) are the input filtered reference signals, which can be calculated using  
\[p(n)=\sum_{i=1}^{q}\left\{a_{i}\cos\omega_{i}n+b_{i}\sin\omega_{i}n\right\}+v_{p }(n) \tag{1}\]0 \[p(n)=\sum_{i=1}^{q}\left\{a_{i}\cos\omega_{i}n+b_{i}\sin\omega_{i}n\right\}+v_{p }(n) \tag{1}\]1 (10) \[p(n)=\sum_{i=1}^{q}\left\{a_{i}\cos\omega_{i}n+b_{i}\sin\omega_{i}n\right\}+v_{p }(n) \tag{1}\]2 (11) \[p(n)=\sum_{i=1}^{q}\left\{a_{i}\cos\omega_{i}n+b_{i}\sin\omega_{i}n\right\}+v_{p }(n) \tag{1}\]3  
where \(\hat{S}(z)\) is the estimate of the secondary path, which is identified and utilized simultaneously by the SPM subsystem while the whole system is in operation. Parameters \(\hat{M}\) and \(\hat{s}_{j}\) are the order and finite pulse response (FIR) coefficients of the secondary path estimate \(\hat{S}(z)\), respectively. In the SPM subsystem, \(v(n)\) is the training signal of the subsystem.