# A Pilot-Based Integration Method of Ranging and LS Channel Estimation for OFDM Systems
## 2 Signal Model  
In this paper, we adopt the OFDM system with parallel data transmission. After convolution coding and QPSK modulation, serial baseband transmission data are converted into \(N\) parallel transmission data symbol streams, which are modulated by subcarriers and transmitted [35], where \(N\) is the number of subcarriers. The basic principle block diagram is shown in Figure 1.  
Each of the \(N\) symbols that undergo serial-parallel transformation will be modulated by a different subcarrier. Let \(X_{l}[k]\) denote the \(k\)th transmitted symbol on the \(l\)th subcarrier, where \(l=0,1,2,...,\infty\), \(k=0,1,2,...,N-1\). The transmission time of each symbol is \(T_{s}\). Then, due to the serial-parallel transformation, the transmission time of \(N\) symbols is extended to \(NT_{s}\), which is the duration of a single OFDM symbol \(T_{sym}\). The time domain continuous baseband signal can be expressed as  
\[x_{l}(t)=\sum_{l=0}^{\infty}\sum_{k=0}^{N-1}X_{l}[k]e^{l2\pi f_{k}^{\prime}(t- IT_{sym})} \tag{1}\]  
At moments \(t=IT_{sym}+nT_{s}\), sampling the continuous baseband OFDM signal in Equation (1) yields the corresponding discrete-time OFDM signal:  
\[x_{l}[n]=\sum_{k=0}^{N-1}X_{l}[k]e^{l2\pi kn/N},\ \ \ \ \ n=0,1,\cdots,N-1 \tag{2}\]  
where \(T_{s}=T_{sym}/N\), \(f_{k}=k/T_{sym}\). Equation (2) can be obtained by N-point IDFT of the QPSK modulated signal \(\{X_{l}[k]\}_{k=0}^{N-1}\), or it can be quickly implemented by IFFT [36]. Without considering the channel and noise effects, the received baseband time domain continuous signal is \(y_{l}(t)=x_{l}(t)\). Also the sampled at time \(t=IT_{sym}+nT_{s}\), the discrete time domain form of the received signal considering the \(l^{th}\) OFDM symbol can be expressed as  
\[Y_{l}[k]=\sum_{n=0}^{N-1}y_{l}[n]e^{-2\pi kn/N} \tag{3}\]  
which is the \(N\)-point DFT of \(\{y_{l}(n)\}_{n=0}^{N-1}\), i.e., the demodulation of the received signal can also be implemented by the FFT algorithm [36]. To reduce the influence of the channel on the output signal recovery, the OFDM system adopts the means of channel estimation and equalization. In the previous section, we ignored the effect of the channel on the phase, amplitude, and frequency of the signal. However, the effect of the channel and noise is always present in the actual application  
Figure 1: Principle block diagram of OFDM modulation and demodulation. environment. Under the influence of the channel, the received signal is demodulated, and the time domain received signal after discrete sampling is  
\[Y[k]=H[k]X[k]+Z[k] \tag{4}\]  
where \(X[k]\), \(Y[k]\), \(H[k]\), and \(Z[k]\) denote the transmit symbol, receive symbol, frequency response of the channel, and frequency domain noise on the \(k\)th subcarrier, respectively. Equation (4) shows that the OFDM system can be regarded as the product of input symbols and channel response in the frequency domain. In the absence of intercarrier interference, the receiver can recover the original signal and remove the channel interference as long as it knows the channel response of each subchannel. To estimate and compensate for the channel, this paper adopts the LS channel estimation method based on the insertion of a comb pilot in the frequency domain. Moreover, this design utilizes the ranging code instead of the original pilot for ranging to solve the problem of low spectrum utilization of the traditional channel estimation. As shown in Figure 2, the transmitter inserts the pilots in the frequency domain at certain intervals. Different pseudo-random codes are inserted into OFDM signals in the frequency domain as pilots in Figure 2. After receiving the OFDM signal through the channel, the receiver first converts the serial signal to the parallel. Next, it removes the Cp and performs frequency offset estimation and equalization. Finally, the signal is transformed by FFT, and the result is a series of parallel sub-signals. The receiver will take out the pilots from these parallel sub-signals, which are utilized for channel estimation.