## 4 NBI Suppression Algorithm
### Single Narrow Band Interference Suppression  
To simplify the discussion, without loss generality, we observe the one order complex-coefficient adaptive notch filter unit with a single constrained zero-pole pair first [35]. The transfer function of the one order complex-coefficient infinite impulse response (IIR) adaptive notch filter unit can be expressed as:  
\[H_{1}(z)=\frac{z-z_{1}}{z-p_{1}}, \tag{25}\]  
where \(z_{1}\) and \(p_{1}\) denote the zero point and the pole point of the basic one order complex-coefficient adaptive notch filter unit, respectively. For the one order IIR notch filter, the stability condition requires that \(p_{1}\) must be less than one. We constrain the zero-pole pair as follows: (1) the zero point is strictly located on the unit circle; (2) the pole point is located inside and close to the unit circle; (3) the zero point and the pole point are under the same frequency angle related to the central frequency of narrow band interference. In other words, the relationship between the zero point and the pole point can be described as:  
\[p_{1}=\rho z_{1}=\rho\text{exp}(j\chi_{1}), \tag{26}\]  
where \(\chi_{1}\) denotes the notch angular frequency with \(\chi_{1}=\arg\{z_{1}\}\), \(0\leq\chi_{1}<2\pi\) and \(\rho\) represents the distance between the origin point and the pole point in the complex plane with \(0<\rho<1\). As depicted in Figure 1, the zero point on the unit circle brings about an infinite depth notch around \(\chi_{1}\), and the parameter \(\rho\) controls the bandwidth of the notch. We should note that the parameter \(\rho\) is chosen to be close to, but less than one to obtain the narrow band notch and avoid any filter instability problems. In Figure 1, the mathematic mark \(\arg\{z_{1}\}\) denotes taking the argument of the complex variable \(z_{1}\). From (25) and (26), we can obtain the following relationship in the time domain:  
\[y_{1}(n)-\rho h_{1}(n)y_{1}(n-1)=x_{1}(n)-h_{1}(n)x_{1}(n-1), \tag{27}\]  
where \(x_{1}(n)\) (\(n=1,2,3...\)) denotes the hybrid input of the one order complex-coefficient ANF unit with a single zero-pole pair including the useful ultra-wide band wireless signal, in-band pre-existing strong NBIs, MAI and additive white Gaussian noises, and \(h_{1}(n)\) is the adaptive complex-coefficient of the basic one order adaptive notch filter unit. Due to the above-mentioned constrained relationship, the coefficient of notch filter can be reexpressed as:  
\[h_{1}(n)=\mid h_{1}(n)\mid\exp\{j\arg\{h_{1}(n)\}\}=\exp\{j\chi_{1}(n)\}, \tag{28}\]  
where \(\chi_{1}(n)\) denotes the instantaneous phase relating to angular frequency for the \(n\)-th moment. Figure 1: Normalized frequency response of one order complex-coefficient adaptive notch filter unit. The detailed implementation of the one order complex-coefficient adaptive notch filter unit with a single constrained zero-pole pair is depicted in Figure 2. Once the frequency angle \(\chi_{1}(n)\) has converged to the objective central frequency of NBI, the adaptive notch filter unit will cut off the corresponding in-band strong NBI component. In this place, we define the error function \(e_{1}(n)\) as:  
\[e_{1}(n)=x_{1}(n)-h_{1}(n)x_{1}(n-1)+\rho h_{1}(n)y_{1}(n-1). \tag{29}\]  
According to adaptive filter theory, the objection is to minimize the mean square error (MSE) of \(e_{1}(n)\); hence, the cost function becomes:  
\[\mathcal{C}_{1}(n)=E\{|\ e_{1}(n)\ |^{2}\}\approx\frac{1}{N}\sum_{n=1}^{N}\{e_{1 }(n)e_{1}^{*}(n)\}, \tag{30}\]  
where * denotes the complex conjugate. The adaptive weight coefficient of the one-order adaptive notch filter with a single zero-pole pair is determined by minimizing the cost function \(\mathcal{C}_{1}(n)\).