# ROAD Statistics-Based Noise Detection for DME Mitigation in LDACS
## 4 Nonlinear Estimators
### Pulse Peak Processors
\begin{cases}|r|&\text{if }|r|\leq\alpha_{p}\\ \rho_{k}\cdot|r|&\text{otherwise.}\end{cases}  \tag{12}\]  
where \(\rho_{k}=\frac{\sigma_{x}^{2}}{\sigma_{x}^{2}+\sigma_{k}^{2}}\)  
\[And,\sigma_{k}^{2}=(1+\frac{k}{A\Gamma})\sigma_{0}^{2}=\frac{k/A+\Gamma}{1+ \Gamma}\sigma_{i}^{2}=\frac{k}{A\Gamma}\sigma_{I}^{2}+\sigma_{0}^{2} \tag{13}\]  
As the device is a pulse peak attenuator, the scaling factor \(\mu\) is \(\rho\). The scaling factor changes at each instant \(t\) as it is a function of \(r_{t}\) and \(\pi_{t}\). GAE enhanced PPA has better performance than the pulse blanking method as it attenuates the DME affected signal rather than losing the data by blanking. The device is well suited to process complex data signals as in LDACS with the modified equation as follows:  
\[\hat{x}_{|kPA|}^{*}(r)= \begin{cases}|r|e^{\text{i}arg(y)}&\text{if }|r|\leq \alpha_{p}\\ \rho_{k}\cdot|r|e^{\text{i}arg(r)}&\text{otherwise.}\end{cases}  \tag{14}\]  
Figure 5: General block diagram for pulse peak attenuator. Figure 6: General block diagram for pulse peak limiter. GAE enhanced PPL is a modified or improved form of GAE enhanced PPA. It employs a decision device to update the scaling factor to attenuate the received signal continuously until the amplitude of the received signal reaches the threshold value (\(\alpha_{p}\)). The repeated attenuation will not affect other subcarriers which are not affected with DME noise. This nonlinear device has better performance at high SNR values compared to the GAE enhanced PPA. The GAE enhanced PPL processes the input signal \(r\) and delivers output \(\hat{x}_{kPL}(r)\) as stated in (15). \[\hat{x}_{|kPL|}(r)= \begin{cases}r&\text{if }|r|\leq\alpha_{p}\\ \rho_{mod}\cdot r&\text{otherwise.}\end{cases}  \tag{15}\]  
where \(\rho_{mod}=\frac{\sigma_{x}^{2}}{\sigma_{x}^{2}+N\cdot\sigma_{k}^{2}}\). Here, the value of \(N\) varies directly with the difference in power of received signal and threshold peak detection value at each instant. The maximum value of \(N\) occurs when the resulting signal holds no values greater than \(\alpha_{p}\). With this knowledge, the value of \(N_{max}\) is as in (16). The algorithm to support this derivation is from [30]  
\[N_{max}=\frac{\sigma_{x}^{2}(|r|_{max}-\alpha_{p})}{\alpha_{p}\sigma_{k}^{2}} \tag{16}\]  
Similar to GAE enhanced PPA, GAE enhanced pulse peak limiters also reduce the drawback of the pulse blanking method with less complexity. The same operation can be performed in another way as in (17)  
\[\hat{x}_{|kPLs|}(r)= \begin{cases}r&\text{if }|r|\leq\alpha_{p}\\ M\cdot\rho_{k}\cdot r&\text{otherwise.}\end{cases}  \tag{17}\]  
where the maximum value of value of \(M\) is derived as in (18) [30]  
\[M_{max}=\frac{\alpha_{p}}{\rho_{k}\cdot|r|_{max}} \tag{18}\]  
From Equations (15) and (17), the updated scaling factors (\(\mu\)) for GAEPPL Type I and Type II are identified as \(\rho_{mod}\) and \(M\cdot\rho_{k}\). Both the methods are applicable to perform scaling of complex valued data with a slight change in Equations (15) and (17) resulting in (19) and (20), respectively. \[\hat{x}_{|kPL|}^{*}(r)= \begin{cases}|r|e^{jarg(r)}&\text{if }|r|\leq\alpha_{p}\\ \rho_{mod}\cdot|r|e^{jarg(r)}&\text{otherwise.}\end{cases}  \tag{19}\]  
where \(\rho_{mod}=\frac{\sigma_{x}^{2}}{\sigma_{x}^{2}+N\cdot\sigma_{k}^{2}}\). \[\hat{x}_{|kPLs|}^{*}(r)= \begin{cases}|r|e^{jarg(r)}&\text{if }|r|\leq\alpha_{p}\\ M\cdot\rho_{k}\cdot|r|e^{jarg(r)}&\text{otherwise.}\end{cases}  \tag{20}\]  
In both cases, the definition for \(\rho_{mod}\) and \(M\) remains the same as that used in Equations (15) and (17).