# ROAD Statistics-Based Noise Detection for DME Mitigation in LDACS
## 4 Nonlinear Estimators
### OBE Enhanced Pulse Peak Processors  
Bayesian estimators are useful in any Gaussian source affected by any Gaussian-mixture noise [28]. The time domain OFDM signal \(x\) can be approximated by Gaussian pdf, \(f_{X}(x)=G\big{(}x;\sigma_{x}^{2}\big{)}=\frac{x^{2}/2\sigma_{x}^{2}}{\sqrt{2 }}\). The complex valued received signal \(r_{t}[n]\) at the receiver side has real and imaginary parts \(r_{t,R}[n]\) and \(r_{t,t}[n]\), respectively. Consider that \(r\) represents distinctly either the real or the imaginary part of \(r_{t}[n]\). When the received signal of interest is modeled or approximated as a Gaussian pdf or K-component pdf, the minimum mean square error Bayesian estimators can be effectively utilized along with the knowledge of the signal \(x_{t}\). By exploiting the statistical dependency between \(X\) and \(i\), it is possible to write \(f_{(r|X)}(y)=f_{i}(r-x)\) and \(G(r;\sigma_{rk}^{2})=G(r;\sigma_{x}^{2})*G(r;\sigma_{k}^{2})\). Here, \(*\) stands for convolution operation. Thus, the received noise power \(\sigma_{rk}^{2}\) is the sum of the signal power \(\sigma_{x}^{2}\) and \(k\)th Gaussian component noise power \(\sigma_{k}^{2}\)  
\[\sigma_{rk}^{2}=\sigma_{r}^{2}+\sigma_{k}^{2} \tag{21}\]  
The OBE enhanced pulse peak processors perform Bayesian estimation only when the received signal is identified as DME affected signal. The pulse peak processors included in this section are OBE enhanced pulse peak attenuator and OBE enhanced pulse peak limiters (Type I and Type II). The device attenuates the nonlinear input \(r\) when the amplitude of the received signal is above \(\alpha_{p}\). The mathematical expression of OBE enhanced pulse peak attenuator is as in (22). \[\hat{x}_{|kOPA|}(r)= \begin{cases}|r|&\text{if }|r|\leq\alpha_{p}\\ \beta_{o}(r)\cdot r&\text{otherwise.}\end{cases}  \tag{22}\]  
where  
\[\beta_{o}(r)\cdot r=\frac{\Sigma_{k=0}^{K-1}\rho_{k}p_{k}G(r;\sigma_{rk}^{2}) }{\Sigma_{k=0}^{K-1}p_{k}G(r;\sigma_{rk}^{2})} \tag{23}\]  
As the device is a pulse peak attenuator, the scaling factor \(\mu\) is \(\beta_{o}(r)\). It is possible to use this OBE enhanced PPA for processing complex valued signal as well. The mathematical statement for this operation is as given in (24). \[\hat{x}_{|kOPA|}(r)= \begin{cases}|r|&\text{if }|r|e^{jarg(r)}\leq\alpha_{p}\\ \beta_{o}(r)\cdot|r|\cdot e^{jarg(r)}&\text{otherwise.}\end{cases}  \tag{24}\]  
OBE enhanced PPL is an altered or upgraded form of OBE enhanced PPL, where it reduces the amplitude of the received signal unceasingly until the amplitude of the DME affected subcarrier reaches the threshold value. As the repeated attenuation is performed only for the subcarriers which exceed the threshold value, it will not disturb the subcarriers which are not affected with DME interference. The OBE enhanced PPL process the input signal \(r\) and deliver output \(x_{KOLP}(r)\) as stated in (25). \[\hat{x}_{|kOPL|}(r)= \begin{cases}|r|&\text{if }|r|\leq\alpha_{p}\\ \beta_{mod}(r)\cdot|r|&\text{otherwise.}\end{cases}  \tag{25}\]  
where  
\[\beta_{mod}(r)=\frac{\alpha_{p}}{|r|} \tag{26}\]  
Here, value of \(\beta_{mod}(r)\) is the modified scaling factor \(\mu\). The modification can be performed in two ways so that \(\beta_{mod}(r)\). \(|r|\) becomes equal to \(\alpha_{p}\). In one method, \(P\) multiples of \(\beta_{O}(r)\) is considered as \(\beta_{mod}(r)\). In this situation, the maximum value of \(P\) for limiting the output (\(P_{max}\)) can be expressed as in (27) [30]. \[P_{max}=\frac{\alpha_{p}}{\beta_{mod}(r)|r|_{max}} \tag{27}\]In the second method, the value of the noise power component \(\sigma_{k}^{2}\) is boosted \(R\) times so that the output of the nonlinear device is limited to the threshold value \(\alpha_{p}\). In this case, the modified scaling factor \(\beta_{mod}(r)\) can be expressed as in (28) [30].