## 4 Proposed Suppression System Model
### Adaptive IR NF
#### 4.2.1 Adaptation of Parameter \(k_{od}\)  
In this subsection, the derivation of the new adaptive algorithm for controlling the coefficient of the update filter is utilized using a second IIR NF direct-form structure. Let us assume that the CW interference with the AWGN is added to the QPSK signal as described in Equation (4). \(y(n)\) denotes the output of the \(H_{d1}(z)\), as described in Equation (5), which can be expressed in the TD in terms of the received signal. Thus, the output signal and gradient signal of the second IIR NF are defined as follows:  
\[y(n)=x(n)+k_{od}x(n-1)+x(n-2)-\beta k_{od}(n-1)-\beta^{2}y(n-2) \tag{6}\]  
The filter's cost function can be given as in [54]:  
\[J(k_{od})=E\Big{\{}|y(n)|^{2}\Big{\}} \tag{7}\]  
where \(E\{*\}\) is an expectation operator. Replacing the expectation value in Equation (7) with the ensemble-averaged value as shown in Equation (8), we get:  
\[\hat{J}(k_{od})=\frac{1}{L}\sum_{n=0}^{L-1}y^{2}(n) \tag{8}\]  
where \(L\) is the length of the data. The goal of the filter is to minimize the cost function by adjusting the notch parameter (\(k_{od}\)) using the gradient-based steepest descent method:  
\[k_{od}\;(n+1)=k_{od}\;(n)-\mu y(n)\big{(}g(J(k_{od}))\big{)} \tag{9}\]  
Figure 6: Second IIR NF realized in direct form.  
where \(g(J(k_{od}))\) is the gradient signal of a second ANF based on a direct-form structure for the adaptation of \(k_{od}\), and can be expressed as follows:  
\[g(J(k_{od}))=\frac{\partial y(n)}{\partial k_{od}(n)}=x(n-1)-\beta y(n-1) \tag{10}\]  
Substituting Equation (10) into Equation (9):  
\[k_{od}(n+1)=k_{od}(n)-\mu y(n)(x(n-1)-\beta y(n-1)) \tag{11}\]  
Equation (11) is the new adaptive algorithm \"LMS algorithm\" for the proposed ANF that controls the update filter's coefficient; \(\mu\) is the step size and the parameter that controls the convergence speed. For stability, \(k_{od}\in[-2,2]\).  
Since the normalized notch frequencies, \(f_{N}\) and \(k_{od}\), are related by \(=-2\cos(\omega_{o})\), where \(\omega_{o}=2\pi f_{N}\), the estimated frequency at sample \(n\) can be calculated as follows:  
\[\hat{f}_{N}(n)=\frac{1}{2\pi}\;Cos^{-1}(-0.5\;k_{od}(n)\;) \tag{12}\]  
The estimated \(JSR\) can be calculated by subtracting the filter's output \(y(n)\) from the received signal \(r(n)\), and is given as follows:  
\[JSR_{dB}=10 log_{10}\Big{\{}|r[n]-y[n]|^{2}\Big{\}} \tag{13}\]  
To mitigate the CW interfering signal, a direct-form adaptive IIR NF is used, whose transfer function is given as follows:  
\[H_{d2}(z)=\frac{N_{d2}(z)}{D_{d2}(Z)}=\frac{1+k_{od}(k_{1})Z^{-1}+k_{1}Z^{-2}} {1+k_{od}(\beta k_{1})Z^{-1}+\beta^{2}k_{1}Z^{-2}} \tag{14}\]  
where \(k_{1}\) is the notch depth, and is a function of the estimated \(JSR\); \(\beta\) is the radius of the pole that determines the width of the second IIR NF, as shown in Figures 7 and 8. As stated previously, many implementation schemes can obtain \(H_{d2}(z)\), but in this work, the transfer function \(H_{d2}(z)\) is implemented by cascading all-pole and all-zero direct IIR NF. Note that the setting \(k_{1}=1\) in Equation (14) results in \(H_{d2}(z)\equiv H_{d1}(z)\), as in Equation (5); hence, the same structure can be used to implement \(H_{d2}(z)\). By copying the \(k_{od}\) of \(H_{d1}(z)\), which is tuned to the IF by Equation (11), the notch of Equation (14) can be placed on the IF. As shown in Figure 8, the amount of reduction is a function of the depth of the notch. In the case of \(k_{1}=1\), \(H_{d2}(z)\) is the same as \(H_{d1}(z)\), which has zero on the unit circle and an infinite notch depth, removing all interference caused by excluding some useful signals. Therefore, in order to reduce signal distortion, the depth of the notch \(k_{1}\) should be adjusted with respect to the estimated JSR. To maximize the output SNR, the parameters JSR, \(k_{od}\), and \(k_{1}\) are required when designing \(H_{d2}(z)\).