## 4 NBI Suppression Algorithm
### Linear Cascaded High Order Complex-Coefficient Adaptive Notch Filter  
For the multiple in-band strong NBI case, it is inevitable to utilize high-order ANF. If we directly expand the numerator polynomial into the high-order finite impulse response (FIR) filter and adjust the denominator polynomial at the same time, it is rather hard to guarantee all of the pole points distribute inside the unit circle to satisfy stability. A feasible approach is to cascade the proposed one order complex efficient IIR ANF units into linear cascaded (LC) high order ANF [36]. The implementation of the linear cascaded high order complex-coefficient ANF is depicted in Figure 3. The transfer function of \(K\)-order linear cascaded complex efficient ANF becomes:  
\[H(z)=\prod_{k=1}^{K}H_{k}(z)=\prod_{k=1}^{K}\frac{1-h_{k}z^{-1}}{1-\rho h_{k}z ^{-1}}, \tag{36}\]  
where \(H_{k}(z)\) denotes the transfer function of the \(k\)-th basic IIR notch filter unit and \(h_{k}\) denotes the \(k\)-th adaptive weight coefficient. The frequency response of the above transfer function has a unified gain at all frequencies; except at the notch frequencies in the present of narrow band interferences, their gains approximate to zero. \begin{table}
\begin{tabular}{l l} \hline Step 1 & initialize system variables \(x_{1}(0)=0.1\), \(y_{1}(0)=0.1\), \\  & set initial value \(h_{1}(1)=1\); \\ Step 2 & for \(n=1,2,3...\), calculate the intermediate variables according to \\  & \(T_{1}(n)=x_{1}(n)x_{1}^{*}(n-1)h_{1}^{*}(n)+\rho x_{1}^{*}(n)y_{1}(n-1)h_{1}(n)\) ; \\ Step 3 & for \(n=1,2,3...\), calculate the complex coefficient according to \\  & \(h_{1}(n+1)=h_{1}(n)\{1+j\mu\text{Imag}[T_{1}(n)]-\frac{1}{2}\{\mu\text{Imag}[T_ {1}(n)]\}^{2}\}\) ; \\ Step 4 & calculate the output of the adaptive notch filter unit according to \\  & \(y_{1}(n)=x_{1}(n)-h_{1}(n)x_{1}(n-1)+\rho h_{1}(n)y_{1}(n-1)\), \\  & and then turn back to Step 2. \\ \hline \end{tabular}
\end{table} 
Table 1: The detailed procedure of the one order complex-coefficient notch filter unit with a single constrained zero-pole pair. In the LC high order complex-coefficient ANF framework, the output of the \((k\!-\!1)\)-th one order ANF unit can be considered as the input of the \(k\)-th complex-coefficient one order ANF unit, so the difference equation of the \(k\)-th section is given by:  
\[y_{k}(n)=y_{k-1}(n)-h_{k}(n)y_{k-1}(n-1)+\rho h_{k}(n)y_{k}(n-1). \tag{37}\]  
Thus, the intermediate variable of the \(k\)-th one order complex-coefficient ANF unit becomes:  
\[T_{k}(n)=y_{k-1}(n)y_{k-1}^{*}(n-1)h_{k}^{*}(n)+\rho y_{k-1}^{*}(n)y_{k}(n-1)h_ {k}(n). \tag{38}\]  
The complex coefficients' iteration relationship for the \(k\)-th one order complex-coefficient IIR ANF unit can be expressed as:  
\[h_{k}(n\!+\!1)=h_{k}(n)\{1+j\mu\mathrm{Imag}[T_{k}(n)]-\frac{1}{2}\{\mu\mathrm{ Imag}[T_{k}(n)]\}^{2}\}. \tag{39}\]  
In the LC \(K\)-order complex-coefficient ANF, the adaptive iteration relationship of each one order ANF unit can be updated separately, since the objection frequency of NBI in each ANF unit is different. The processing of the LC high order complex-coefficient ANF algorithm based on one order ANF unit can be found in Table 2. \begin{table}
\begin{tabular}{l l} \hline Step 1 & initialize system variables \(x_{1}(0)=0.1\), \(y_{1}(0)=0.1\), \(y_{k}(0)=0.1\), \\  & set initial value \(h_{k}(1)=1\), for \(k=1,2...,K\); \\ Step 2 & for \(n=1,2,3...\), calculate the intermediate variable according to \\  & \(T_{k}(n)=y_{k-1}(n)y_{k-1}^{*}(n-1)h_{k}^{*}(n)+\rho y_{k-1}^{*}(n)y_{k}(n-1)h_ {k}(n)\); \\ Step 3 & for \(n=1,2,3...\), calculate the adaptive instantaneous complex weight coefficients according to \\  & \(h_{k}(n\!+\!1)\!=\!h_{k}(n)\{1\!+\!j\mu\mathrm{Imag}[T_{k}(n)]\!-\!\frac{1}{2} \{\mu\mathrm{Imag}[T_{k}(n)]\}^{2}\}\) ; \\ Step 4 & calculate the output of the adaptive notch filter unit according to \\  & \(y_{k}(n)=y_{k-1}(n)-h_{k}(n)y_{k-1}(n-1)+\rho h_{k}(n)y_{k}(n-1)\) \\  & and then turn back to Step 2. \\ \hline \end{tabular}
\end{table}