## 2 System Model
### Linear Interpolation for CPR
A notable improvement in the algorithm's performance is achieved by replacing the previously discussed hard-decision filter with a soft-decision filter that uses a cosine function for outlier discrimination. This enhanced filter incorporates information from both pilot symbols and data symbols interspersed between pilots. The design of the soft-decision filter resulted from extensive experimental studies, which initially evaluated a Gaussian filter response. However, testing revealed that the cosine-based filter produced the most accurate results. This soft filter applies cosine-based corrections to severe outliers, ensuring smoother phase adjustments. Furthermore, the burst length is determined solely by the size of the pilot block, eliminating the need to rely on the entire signal or a large portion of it to correct the phase. Unlike the hard-decision filter, the standard deviation \(\sigma_{w}\) is directly derived from the phase noise and does not depend on the total length of the pilot sequence. Instead, it is calculated on the basis of the burst length, which corresponds to the interval between pilot samples. The parameter \(\varphi\) represents the coarse phase estimate. The equation for detecting and adjusting potential outliers at pilot positions, where \(n=k\) indicates the pilot indices, is given by:

\[w_{f}[n]= \begin{cases}\frac{1}{2}\left[1+\cos\left(\frac{\pi\sigma_{w}}{2(Lk+ k)}\right)\right]\varphi[Lk+k],&\text{if }n=Lk,\\ \frac{\beta}{2}\left[1+\cos\left(\frac{\pi\sigma_{w}}{2(n+k)}\right)\right] \varphi[n+k],&\text{if }n\
eq Lk,\end{cases}  \tag{11}\]

in which \(w_{f}[n]\) represents weights derived from a cosine function, with a default burst length of \(L\). This length can be adjusted on the basis of the proportion of pilot symbols or the percentage of resources allocated to phase recovery, providing flexibility in configuration. These weights are applied to update the local weights of the process during the a-posteriori step. In Equation (11), \(n\
eq Lk\) specifies the indices of symbols located between the pilots. The parameter \(\beta\) is a weighting factor chosen within the range \([0.65,0.95]\), experimentally determined as a function of the degradation of the phase noise \(\Delta\upsilon T_{S}\). Outliers, identified as values that exceed twice the standard deviation of phase noise (\(\varphi>2\sigma_{w}\)), are set to zero to minimize their influence on phase tracking. The pseudocode for the enhanced algorithm is presented in Algorithm 2, where outliers are identified and mitigated using the modified cosine weighting function and then applied to the smoothing function. After applying the soft-decision cosine filter for outlier discrimination, the robust locally weighted regression algorithm is applied. The received signal \(y[n]\) is processed to obtain the coarse phase at the pilot positions and between the pilots using the _burst division_. The coarse phase \(\varphi[n]\) is divided into _bursts_ of \(L\) samples each. The \(b\)-th burst, denoted as \(\varphi_{b}[n]\), can be expressed as:\[\varphi_{b}[n]=\varphi[n],\quad n\in\{kL,kL+1,\ldots,kL+L-1\}, \tag{12}\]

where \(k=0,1,2,\ldots,K\). The locally weighted regression algorithm is used to calculate the fitted phase values based on the input coarse phase within the _burst_ division \(\varphi_{b}[n]\). The signal \(q[n]=\varphi_{b}[n]\) represents the phase to be processed. Accordingly, the equation representing the input to the algorithm for the smoothing process is given by:

\[q[n]=g[n]+\epsilon[n],\ \forall\ kL\leq n<kL+L,\ 0\leq k<K. \tag{13}\]

in which \(g\) denotes a smooth function. Instead of processing the entire signal at once, the algorithm operates on smaller fragments of the burst segment. The range specified in Equation (12) defines the windowed segment within the burst length across the entire signal. The noise term \(\epsilon[n]\) added in Equation (13) represents the residual phase noise; as the coarse phase \(q[n]\) has not undergone any processing, it is the coarse phase received in _burst_ mode.