# A Pilot-Based Integration Method of Ranging and LS Channel Estimation for OFDM Systems
## 3 Ranging
### Coarse Acquisition of Ranging Pilot  
In ranging communications, when a single pseudorandom code is utilized for ranging, the period of the pseudorandom code needs to be large enough to achieve the measurement results without ambiguity. However, the acquisition time of traditional code acquisition methods is proportional to the period of the pseudorandom code. Therefore, it is difficult to achieve rapid acquisition of the pseudorandom code when ranging with a single ranging code. To solve this problem, different short-period ranging codes are utilized in this paper instead of the original pilot. This approach utilizes the parallel transmission feature of the OFDM system as a composite method to synthesize different ranging sub-codes into a composite ranging code. The ranging code composed of these sub-codes has greater range ambiguity. Moreover, since there is a smaller period of each sub-code than the traditional method, the receiver can achieve the rapid acquisition and tracking of these sub-codes. OFDM is a multi-carrier parallel transmission system, and according to (5), we can see that in the absence of inter-carrier interference, each method of parallel transmission can be regarded as a separate way signal. After inserting the pilot in the frequency domain, for the \(k\) th pilot, according to (1), we can observe that the signal of the \(k\)th pilot in the time domain is  
\[p_{l}(t)=\sum_{l=0}^{\infty}P_{l}[h]e^{j2\pi f_{h}(t-IT_{sym})},\ \ \ \ \ (h-1)T_{s}+IT_{sym}<t< htT_{s}+IT_{sym} \tag{7}\]  
where \(h\) (\(0<h<N-1\) ) is the subcarrier position where the pilot is located, and \(P_{l}[h]\) is the \(l\)th pilot symbol on the \(h\)th subcarrier, \(l=0,1,2,...,\infty\). From Formula (6), we can regard the \(h\)th pilot as an independent subsignal on the \(h\)th subcarrier. The data bit duration of each subsignal has been extended by \(N\) times compared with the original. Thus, the data symbol with the original duration of \(T_{s}\) has been extended to \(T_{sym}\). Since each subsignal is independent in the absence of inter-carrier interference, the time-domain continuous signal in the corresponding subchannel for the \(h\)th pilot can be equated as  
\[\hat{p}(t)=p_{h}(t)e^{j2\pi f_{h}t} \tag{8}\]  
where \(p_{h}(t)\) is the ranging subcode loaded on the \(h^{th}\) subcarrier. Under the influence of the channel, the receiver strips the subcarrier of the received ranging signal to obtain  
\[ \begin{split}\hat{p}_{h}(t)&=(p_{h}(t-\tau)e^{j2\pi( \hat{f}_{h}-\hat{f}_{d})(t-\tau)}+n_{h}(t))e^{-j2\pi(\hat{f}_{h}-\hat{f}_{d})(t- \tau)}\\ &=p_{h}(t-\tau)e^{j2\pi(\hat{f}_{d}-\hat{f}_{d})(t-\tau)}+n_{h}(t) e^{-j2\pi(\hat{f}_{h}-\hat{f}_{d})(t-\tau)}\end{split}  \tag{9}\]  
where \(\hat{f}_{d}\) is the Doppler shift generated by the signal during propagation, \(\hat{f}_{d}\) is the frequency offset estimated by the receiver, \(\tau\) is the time delay generated by the signal, and \(\hat{\tau}\) is the time delay compensated after synchronization at the receiver. With perfect frequency offset compensation as well as complete synchronization, i.e., when \(\hat{f}_{d}=\hat{f}_{d}\tau=\hat{\tau}\), Equation (8) can be expressed as  
\[\hat{p}_{h}(t)=p_{h}(t-\tau)+n_{h}(t)e^{-j2\pi(\hat{f}_{h}-\hat{f}_{d})(t-\tau)} \tag{10}\]  
Sampling (10) at the sampling frequency \(f_{s}\),\(f_{s}\geq 2/T_{sym}\), the discrete form of the ranging pilot signal can be obtained as  
\[\hat{p}_{h}(i)=p_{h}(i-i_{s})+N(i) \tag{11}\]  
where \(i_{s}\) is the phase sampling point of the ranging code corresponding to the time delay offset, and \(N(i)\) is the sampling signal corresponding to the noise. The correlation operation on (11) yields  
\[R=\frac{1}{N_{h}}\sum_{i=0}^{N_{h}}\hat{p}_{h}(i)p_{h}(i-i_{k}) \tag{12}\]  
where  
\[p_{h}(i-i_{k})\]  
is the locally generated \(h\)th ranging subcode, the phase of which is shifted by \(i_{k}\), and \(N_{h}\) is the period of the \(h\)th ranging subcode.