# ROAD Statistics-Based Noise Detection for DME Mitigation in LDACS
## 4 Nonlinear Estimators
### Pulse Peak Processors  
As discussed in the system model, data estimation needs one more vector \(\pi_{t}\) along with \(r_{t}\) in performing the nonlinear function \(f(\pi_{t},\,y_{t})\). The detailing of the nonlinear operations performed by pulse peak attenuators and limiters are depicted in block diagrams Figures 5 and 6, respectively. The received data (after CP removal) \(r\) are passed through a 2-GMM estimation block to extract the vector \(\pi_{t}\). The parameters contained in vector \(\pi_{t}\) are further used to compute scaling factor \(\mu_{t}\) for each time epoch. From Figure 5, it is clear that pulse peak attenuator uses this parameter \(\mu\) for processing the signal \(r_{t}\). From Figure 6, it is to be noted that pulse peak limiters have one extra block compared to pulse peak attenuators. It is denoted as a decision device that holds the algorithm to change or update the scaling factor. The decision device determines if the scaling factor is needed to modify or not. When the scaling is performed with a modified scaling factor, pulse peak attenuators become pulse peak limiters. In this paper, we have included two types of pulse peak attenuators and four types of pulse peak limiters to compare the performance of proposed ROAD statistics. It is inevitable to discuss K-GMM modeling to understand the vector \(\pi_{t}\) in more detail. This section outlines how GAE and OBE enhanced pulse peak processors exploit 2-GMM estimation (2-GMM) to scale the signals affected with DME. Any random variable can be expressed as the combination of K-number mutually exclusive Gaussian variables with K-GMM modeling [37; 38]. Hence, this model can be effectively applied to any ImpN distribution (Class A, S-\(\alpha\)-S noises, etc. ), either estimated [39; 40; 41] and approximated by a K-GMM [42] or modeled with the actual equation [43]. The K-GMM model is mathematically expressed with pdf,  
\[f_{W}(i)=\Sigma_{k=0}^{K-1}p_{k}\cdot G(i,\sigma_{k}^{2}) \tag{11}\]  
where \(\{P_{k}\}_{k=0,1,.....K-1}\) with \(\Sigma_{k=0}^{K-1}p_{k}=0\) are the probability of occurrence of each Gaussian component \(k\). For the value \(k=0\), the component \(i_{0}\approx G(i_{0},\sigma_{0}^{2})\) represents the thermal noise with variance \(\sigma_{0}^{2}\) and with the probability of occurrence \(p_{0}\). For values \(k=1\) to \(K-1\), the statistical combinations of components characterize the impulse noise with the probability \(p_{I}=1-p_{0}\) and noise power \(\sigma_{I}^{2}\). The ratio of thermal noise to impulse noise is expressed as \(\Gamma=\frac{\sigma_{0}^{2}}{\sigma_{I}^{2}}\). For the value \(K=2\), this model will reduce to 2-GMM with a thermal noise component \(i_{0}\approx G(i_{0},\sigma_{0}^{2})\) and an impulse noise component \(i_{1}\approx G(i_{1},\sigma_{1}^{2})\). The pulse peak processors included in this section are GAE enhanced PPA and GAE enhanced PPLs (Type I and Type II). These pulse peak processors have better performance than GAE by utilizing other side information, such as impulsive noise arrival time or relationship of the impulsive noise [30]. The received signal \(r\) can be expressed as the sum of transmitted signal \(X\) and noise \(i|_{k}\), where \(X\) and \(i|_{k}\) are two zero-mean independent Gaussian random variables with variances \(\sigma_{X}^{2}\) and \(\sigma_{i}^{2}\). With any impulse noise, when modeled as properly weighted mutually exclusive Gaussian events, the GAE claims to know which is the \(k\)th Gaussian component of the (pdf), which actually affects the transmitted signal at each time epoch. Once the signal affected with DME interference is identified, all three GAE enhanced pulse peak processors use the same knowledge of GAE for pulse peak processing. The amplitude of the received signal is used for the detection of DME as in pulse blanking. The GAE enhanced PPA attenuates the nonlinear input \(r\) when the amplitude of the received signal exceeds the threshold value. The operation of GAE enhanced PPA can be expressed as follows:  
\[\hat{x}_{|kPA|}(r)=