# A Pilot-Based Integration Method of Ranging and LS Channel Estimation for OFDM Systems
## 3 Ranging
### Tracking of Ranging Pilot  
The acquisition accuracy is only within \(\pm 0.5\) code chips after the preliminary acquisition, and a more accurate code chip offset can be obtained by the tracking process.  
From the above analysis, it can be seen that each ranging code can be regarded as a single direct sequence spread spectrum signal without carrier interference. Therefore, the paper adopts an equal sampling tracking loop based on the direct spread baseband pseudocode DLL tracking loop. Compared with commonly used tracking loops, it eliminates the complex implementation of the loop numerically controlled oscillator (NCO) and the impact of the uncertainty of the NCO initial phase on the pseudocode tracking accuracy. The structure of the tracking loop is shown in Figure 3, which will track each ranging pilot in parallel. As shown in Figure 3, each loop has the same DLL structure, and \(\hat{p}_{i}(k)\) is the \(i\)th ranging pilot signal after stripping the sub carrier from the receiving end. In addition, \(i\) is the serial number of the ranging pilot, which indicates that there are \(i\) channels of pilot utilized for ranging. Moreover, \(I-D\) is the integral and accumulated result, \(E\) is the leading correlation result, \(P\) is the current correlation result, and \(L\) is the lagging correlation result. Furthermore, \(E\) and \(L\) utilize a pseudo-random code that exceeds or lags behind the current \(P\)-branch pseudocode by one sample point, respectively. The \(E\), \(P\), and \(L\) of the \(i\)th pilot can be expressed as  
\[\left\{\begin{array}{l}E_{i}=\sum\limits_{k=0}^{N}\hat{p}_{i}(k)\cdot PL_{i} (k+1)\\ P_{i}=\sum\limits_{k=0}^{N}\hat{p}_{i}(k)\cdot PL_{i}(k)\\ L_{i}=\sum\limits_{k=0}^{N}\hat{p}(k)_{i}\cdot PL_{i}(k-1)\end{array}\right. \tag{13}\]  
where \(PL_{i}(k)\) is the corresponding generated local pseudo-random code signal after acquisition and \(k\) represents the current phase.  
The phase detector of the loop adopts the normalized point product power phase detection algorithm, and the output error \(e(k)\) of the phase detector is  
\[e(k)=\frac{L-E}{P} \tag{14}\]  
which will control the local regenerative pseudo code generator to update a new set of pseudo code phase output \(PL_{i}(a)\), \(a=k+1,k,k-1\). The loop filter \(F(z)\) can make the local pseudocode phase shift smoothly. This paper adopts an ideal second-order loop, and the digital filter function is  
\[F(z)=C_{1}+\frac{C_{2}}{1-z^{-1}} \tag{15}\]  
where \(C_{1}\) and \(C_{2}\) are constant gains.  
In the simulation analysis of this paper, in order to simplify the complex implementation of a traditional NCO, we adopt a baseband pseudocode digital tracking loop. The phase shift direction of the pseudorandom code is controlled by the new NCO of the tracking loop, and its correlation spacing is the duration of two sampling points. Its specific principle is to set a suitable threshold \(\varepsilon\) to compare the output of \(F(z)\). When the output of  
Figure 3: Ranging pilot parallel tracking loop.  
\(F(z)\) is within the range of \((-\varepsilon,\varepsilon)\), the tracking is locked. Otherwise, when the output of \(F(z)\) is greater than \(\varepsilon\) or less than \(-\varepsilon\), the phase of the pseudocode updates the direction to the left or right. The tracking accuracy of this tracking method is related to the sampling frequency of the system. Moreover, the minimum code phase offset that the tracking loop can lock is the offset of the code slice corresponding to one sampling point. Since the receiver has acquired \(\pm 0.5\) chips before tracking, the tracking process will be finished after a limited number of shift correlations.