# Cognitive Interference Cancellation with Digital Channelizer for Satellite Communication
## 5 Performance Analysis  
In this section, the performance of the proposed cognitive interference detection algorithm is analyzed. First, we derive the detection and false alarm probabilities of the simple method with 1024 subchannels. To this end, we obtain the distribution of the decision statistics for both hypotheses \(\mathcal{H}_{0}\) and \(\mathcal{H}_{1}\). The decision statistic in each subchannel is given by Equation (3) and its distribution for both hypotheses is expressed by  
\[T(n) \sim\left\{\begin{array}{ll}\mathcal{N}\left(\sigma_{\text{s}}^ {2}+\sigma_{\text{w}}^{2},\frac{2\sigma_{\text{s}}^{4}}{L}\right),&\text{ under }\mathcal{H}_{1},\\ \mathcal{N}\left(\sigma_{\text{w}}^{2},\frac{2\sigma_{\text{s}}^{4}}{L} \right),&\text{under }\mathcal{H}_{0},\end{array}\right. \tag{11}\] \[T(m) \sim\mathcal{N}\left(\sigma_{\text{w}}^{2},\frac{2\sigma_{\text{ w}}^{4}}{L}\right),\text{ for }m\
eq n, \tag{12}\]  
where \(\sigma_{\text{s}}^{2}\) denotes the CW tone interference power and \(\sigma_{\text{w}}^{2}\) is the variance of \(\bar{x}(\cdot,\cdot)+\bar{w}(\cdot,\cdot)\), i.e., the sum power of the desired signal and AWGN components in each subchannel. It is also assumed that the desired signal in the frequency domain, \(\bar{x}(\cdot,\cdot)\), follows Gaussian distribution by the central limit theorem because it is the weighted sum of \(N\) modulated symbols in the time domain [26; 27].  
Figure 6: Overall structure of the proposed interference detector where \(N=1024\).  
Figure 7: Detailed structure of the continuous wave (CW) tone interference detection block in Figure 6 where \(N=1024\).  
Based on the above distribution, we can derive the probability to detect the interference correctly with a threshold \(\lambda\) in each subchannel as follows:  
\[p_{d} =Pr\left\{T(\cdot)\geq\lambda|\mathcal{H}_{1}\right\}\] \[=Q\left(\frac{\lambda-(1+\gamma)\sigma_{w}^{2}}{\sigma_{w}^{2} \sqrt{2/L}}\right), \tag{13}\]  
where \(\gamma=\frac{\sigma_{w}^{2}}{\sigma_{w}^{2}}\) represents SNR with respect to the CW tone interference [19; 20]. Similarly, the false alarm probability is given by  
\[p_{fa} =Pr\left(T(\cdot)<\lambda|\mathcal{H}_{0}\right)\] \[=Q\left(\frac{\lambda-\sigma_{w}^{2}}{\sigma_{w}^{2}/\sqrt{2/L}} \right). \tag{14}\]  
With the detection and false alarm probabilities for a single subchannel, we can evaluate the overall performance of the interference detection considering all subchannels.  
\[P_{d} =Pr\left(T(n)\geq\lambda|\mathcal{H}_{1}\right)Pr\left(T(m)< \lambda|\mathcal{H}_{0},\text{ for all }m\
eq n\right)\] \[=p_{d}(1-p_{fa})^{1023} \tag{15}\] \[P_{fa} =Pr\left(T(m)>\lambda|\mathcal{H}_{0},\text{ for any }m=0,\cdots,1023\right)\] \[=1-Pr\left(T(m)<\lambda|\mathcal{H}_{0},\text{ for all }m=0,\cdots,1023\right)\] \[=1-(1-p_{fa})^{1024}. \tag{16}\]  
For the improved approach, the decision statistics are distributed as follows:  
\[T_{im}(n) \sim\left\{\begin{array}{ll}\mathcal{N}\left(4\sigma_{s}^{2}+2 \sigma_{w}^{2},\frac{8\sigma_{w}^{4}}{L}\right),&\text{ under }\mathcal{H}_{1},\\ \mathcal{N}\left(2\sigma_{w}^{2},\frac{8\sigma_{w}^{4}}{L}\right),&\text{ under }\mathcal{H}_{0},\end{array}\right. \tag{17}\] \[T_{im}(m) \sim\mathcal{N}\left(2\sigma_{w}^{2},\frac{8\sigma_{w}^{4}}{L} \right),\text{ for }m\
eq n, \tag{18}\]  
because two 512-point FFT outputs with even- and odd-indexed inputs are added before measuring the signal power. Therefore, the detection and false alarm probabilities for each subchannel are given by  
\[p_{d}^{\prime} =Q\left(\frac{\lambda^{\prime}-2(1+2\gamma)\sigma_{w}^{2})}{2 \sigma_{w}^{2}\sqrt{1/L}}\right), \tag{19}\] \[p_{fa} \tag{20}\]  
where \(\lambda^{\prime}\) is a predefined threshold in the improved method. By considering the lower or upper band, the overall detection and false alarm probabilities of interference detection are expressed by  
\[P_{d}^{\prime} =p_{d}^{\prime}(1-p_{fa}^{\prime})^{511}, \tag{21}\] \[P_{fa}^{\prime} =1-(1-p_{fa}^{\prime})^{512}. \tag{22}\]