## 2 System Model
### Linear Interpolation for CPR
This noise is assumed to follow a normal distribution. The smoothed phase value at \(n\), denoted by \(\hat{q}[n]\), is obtained after applying the smoothing process. The weight function \(W[n]\) is adjusted based on the non-negative index \(n\), increasing or decreasing accordingly. The weights \(w_{\ell}[n]\) decrease as the distance between \(n_{\ell}\) and \(n\) increases. The weights \(w_{\ell}[n]\) are assigned to each burst, considering the neighboring symbols \(-n_{\ell}\leq\ell\leq n_{\ell}\) and \(\ell\
eq 0\). This process centers the weight matrix \(W\) at \(n\) within the burst and scales it so that \(W\) becomes zero at the \(r\)-th nearest neighbor of \(n\). The fitted value \(\hat{q}[n]\) for each \(n\) is obtained by fitting a polynomial of degree \(d\) to the data using weighted least squares, with weights \(w_{\ell}[n]\). This yields the initial set of fitted values. The residuals \(\delta[n]\) are then computed as

\[\delta[n]=q[n]-\hat{q}[n]. \tag{14}\]

In this framework, larger residuals result in smaller weights, while smaller residuals result in larger weights. This weighting process can be adjusted by manually assigning values at the pilot positions, where the information is theoretically more reliable. Consequently, the weights at the first and last positions of the burst, corresponding to known reference points, can be assigned to higher values. The methodology above is reapplied iteratively to update the fitted values using recalculated weights based on \(\delta[n]\), continuously refining both the weights and the fitted values to achieve an improved fit. The smoothing process \(g\) is designed to generate a robust approximation of \(\hat{q}[n]\) by using neighboring points around \(n\). The _span_, which defines the percentage of neighboring points that contribute to \(\hat{q}[n]\), typically ranges from 10% to 80%. In the absence of specific guidance for selecting the _span_, a default value of 50% is often used. In Algorithm 2, the _span_ is governed by the parameter \(\lambda\), restricted within the range \([0.65,0.95]\). The procedure for computing the robust weighting is as follows: Let \(h[n]\) denote the distance from \(n\) to its \(r\)-th nearest neighbor. Specifically, \(h[n]\) is the \(r\)-th smallest distance among \(|n-\ell|\) and \(\ell\
eq n\). Then,

\[w_{\ell}[n]=W\Big{(}h[n]^{-1}|n-\ell|\Big{)}. \tag{15}\]

Locally weighted regression (LWR) and robust locally weighted regression (RLWR) are characterized by the following sequence of operations:

1. For each \(n\), compute the estimates \(\hat{\beta}_{j}[n]\) for \(j=0,\ldots,d\), representing the parameters of a polynomial regression of degree \(d\) fitted to the data \((\ell,q[n+\ell])\). This regression is performed using weighted least squares, with weights \(w_{\ell}[n]\) for each data pair \((\ell,q[n+\ell])\). The estimates \(\hat{\beta}_{j}[n]\) are the values that minimize the following objective function: \[\sum_{\ell}w_{\ell}[n]\Big{(}q[n+\ell]-\beta_{0}-\beta_{1}\ell-\ldots-\beta_{ d}\ell^{d}\Big{)}^{2}.\] (16)

The smoothed point at \(n\) using a locally weighted polynomial regression of degree \(d\) is \((n,\hat{q}[n])\), where \(\hat{q}[n]\) represents the fitted value of the regression at \(n\). Thus,

\[\hat{q}[n]=\sum_{j=0}^{d}\hat{\beta}_{j}n^{j+1}. \tag{17}\]
2. Define the bisquare weight function \(B\) as follows: \[B[n]= \begin{cases}(1-n^{2}),&\text{for }|n|<1,\\ 0,&\text{for }|n|\geq 1.\end{cases} \] (18) Define the residuals based on the current fitted values as follows:\[e[n]=q[n]-\hat{q}[n]. \tag{19}\]

Let \(s\) represent the median of the absolute residuals, \(|e[n]|\). The robustness weights are then defined as follows:

\[\delta[n+\ell]=B\bigg{(}\frac{e[n+\ell]}{6s}\bigg{)}. \tag{20}\]
3. For each \(n\), compute the new \(q[n]\) by fitting a polynomial of degree \(d\) using weighted least squares, where the weights for each pair \((\ell,q[\ell])\) are given by \(\delta[\ell]w_{\ell}[n]\). 4.