## 4 Proposed Suppression System Model
### Adaptive IR NF
#### 4.2.2 Optimal Notch Depth, Maximizing the Output SNR  
In this subsection, the output SNR of the second IIR NF is expressed as a function of its parameters. Moreover, we describe how to obtain the optimal notch depth value (\(k_{1}\)) that maximizes the \(SNR_{o}\)\(dB\), as follows:  
\[SNR_{o}dB=10\log_{10}(\frac{E\big{[}S^{2}(n)\big{]}}{E\big{[}(y(n)-S(n))^{2} \big{]}}) \tag{15}\]  
where \(y(n)\) is the output of the second IIR NF \(H_{d2}(z)\), which is expressed as follows:  
\[y(n)=H_{d2}(z)r(n)\triangleq S_{o}(n)+w_{o}(n)+J_{o}(n) \tag{16}\]where \(S_{o}(n)\), \(w_{o}(n)\), and \(J_{o}(n)\) are the output components of the desired QPSK-modulated signal, the AWGN, and the CWI, respectively. \(S_{o}(n)\) is a distorted version of the information signal \(S(n)\) caused by information removal at the notch frequency of \(H_{d2}(z)\). \[E\Big{[}\big{(}y(n)-S(n)\big{)}^{2}\Big{]}=E\Big{[}y^{2}(n)\Big{]}-2E[y(n)S(n) ]+E\Big{[}S^{2}(n)\Big{]} \tag{17}\]  
Since \(E\big{[}S^{2}(n)\big{]}=1\) is obtained after normalization, Equation (17) becomes:  
\[E\Big{[}\big{(}y(n)-S(n)\big{)}^{2}\Big{]}=E\Big{[}y^{2}(n)\Big{]}-2E[y(n)S(n) ]+1 \tag{18}\]  
Figure 8: Characteristics of the notch related to filter parameters: notch depth vs. \(k_{1}\). Figure 7: Characteristics of the notch related to filter parameters: notch width vs. \(\beta\). To express \(SNR_{o}\ dB\) as a function of the parameters of the second IIR NF, \(E\big{[}y^{2}(n)\big{]}\) and \(E\big{[}y(n)S(n)\big{]}\) can be expressed with respect to filter parameters by assuming that \(S_{o}(n),w_{o}(n),\ and\ J_{o}(n)\) are independent and expressing them as a function of the parameters of the second IIR NF:  
\[E\Big{[}y^{2}(n)\Big{]}=E\big{[}(S_{o}(n)+w_{o}(n)+J_{o}(n))^{2}\big{]} \tag{19}\]  
\[E\Big{[}y^{2}(n)\Big{]}=E\big{[}{S_{o}}^{2}(n)\Big{]}+E\Big{[}{w_{o}}^{2}(n) \Big{]}+E\Big{[}{J_{o}}^{2}(n)\Big{]} \tag{20}\]  
The components \((S_{o}(n),w_{o}(n),\ and\ J_{o}(n))\) are independent of one another; thus, other terms will be equal to zero. Equation (20) can therefore be expressed as follows:  
\[E\Big{[}y^{2}(n)\Big{]}=\sum_{k=0}^{\infty}{h_{k}}^{2}+{\sigma_{w_{o}}}^{2}+{ \sigma_{l_{o}}}^{2} \tag{21}\]  
where \(h_{k}\) is the impulse response of the second IIR NF, while \({\sigma_{w_{o}}}^{2}\) and \({\sigma_{l_{o}}}^{2}\) are variances of \(w_{o}(n)\ and\ J_{o}(n)\), respectively. \(E\big{[}y(n)S(n)\big{]}\) can be expressed as follows:  
\[E\big{[}y(n)S(n)\big{]}=E\big{[}(S_{o}(n)+w_{o}(n)+J_{o}(n))S(n)\big{]}=E[(S_{ o}(n)\ S(n)]=h_{o} \tag{22}\]  
Substituting Equations (21) and (22) into Equation (18):  
\[y(n)=H_{d2}(z)r(n)\triangleq S_{o}(n)+w_{o}(n)+J_{o}(n) \tag{16}\]0  
Then, substituting Equation (23) into Equation (15), note that \(h_{o}=1\)  
\[y(n)=H_{d2}(z)r(n)\triangleq S_{o}(n)+w_{o}(n)+J_{o}(n) \tag{16}\]1  
As stated previously, in order to express \(SNR_{o}\) as a function of the parameters of the second IIR NF in Equation (24), we first define \(u(n)\) as the output of \(\frac{1}{D_{d2}(Z)}\) with input signal \(w(n)\) :  
\[y(n)=H_{d2}(z)r(n)\triangleq S_{o}(n)+w_{o}(n)+J_{o}(n) \tag{16}\]2  
where \(D_{d2}(Z)\) is the all-pole of the second IIR NF, as in Equation (14). Let \(R_{uu}(m)\) be the autocorrelation of \(u(n)\), i.e., \(E[u(n)u(n-m)]\); \(R_{uu}(0)\), \(R_{uu}(1)\), and \(R_{uu}(2)\) can then be expressed as follows [54]:  
\[y(n)=H_{d2}(z)r(n)\triangleq S_{o}(n)+w_{o}(n)+J_{o}(n) \tag{16}\]3 \[y(n)=H_{d2}(z)r(n)\triangleq S_{o}(n)+w_{o}(n)+J_{o}(n) \tag{16}\]4 (27) \[y(n)=H_{d2}(z)r(n)\triangleq S_{o}(n)+w_{o}(n)+J_{o}(n) \tag{16}\]5  
where \(\sigma_{w_{o}}^{2}=E\big{[}w_{o}^{2}(n)\big{]}\). Since \(w_{o}(n)=N_{d2}(z)u(n)\), \(w_{o}(n)\) can then be expressed as follows:  
\[y(n)=H_{d2}(z)r(n)\triangleq S_{o}(n)+w_{o}(n)+J_{o}(n) \tag{16}\]6  
\[y(n)=H_{d2}(z)r(n)\triangleq S_{o}(n)+w_{o}(n)+J_{o}(n) \tag{16}\]7The variance of \(w_{o}(n)\) is given by \(\sigma_{w_{o}}^{2}=E\left[w_{o}^{2}(n)\right]\); thus:  
\[