# A Low-Complexity Channel Estimation Based on a Least-Squares Algorithm in OFDM Systems
## 5 Simulation and Discussion  
First, the simulation parameters were set up in Table 3. The signal was QPSK modulation and the speed was 20 km/h. Table 4 shows the typical urban time-varying channel model [27]. The three power spectral density function models were Jakes, Gauss I, and Gauss II (see COST 207 specification [27]). Forty channels were generated randomly with the channel model for simulation with different SNRs. Fifty OFDM symbols were transferred in one simulation. In terms of regularization parameters, the LS method [11] selects 0.01, while the method proposed in this paper selects \(\alpha_{c}=\alpha_{o}=0.02\). The dimension of the estimated CIR for the LS method [10] is also set to \(N_{p}/2\). The problem of the matrix approaching singularity also exists in [10], so we use a regularization parameter that is set to 0.00001 to avoid this issue. The regularization parameter selection process refers to [28]. To ensure a fair comparison of the conventional DFT-based channel estimation with other methods, we only use first \(0.5N_{p}\) points of the CIR estimated by conventional DFT-based channel estimation. All simulations were performed using MATLAB software. Figure 2 is the simulated diagram of SNR to the bit error rate (BER). It is observed that the curves of the LS methods [10,11] and the method proposed in this paper are \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Path No. (I)** & **Propagation Delay** & **Path Power** & **Path Power (dB)** & **PSD** \\ \hline
0 & 0.0 \(\upmu\)s & 0.5 & \(-\)3 & [PERSON] \\ \hline
1 & 0.2 \(\upmu\)s & 1 & 0 & [PERSON] \\ \hline
2 & 0.5 \(\upmu\)s & 0.63 & \(-\)2 & [PERSON] \\ \hline
3 & 1.6 \(\upmu\)s & 0.25 & \(-\)6 & Gauss I \\ \hline
4 & 2.3 \(\upmu\)s & 0.16 & \(-\)8 & Gauss II \\ \hline
5 & 5.0 \(\upmu\)s & 0.1 & \(-\)10 & Gauss II \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The typical urban time-varying channel model. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Specification** \\ \hline \(N\) & 2048 points \\ \hline CP length & 128 points \\ \hline \(\Delta p\) & 16 subcarriers \\ \hline \(N_{p}\) & 128 subcarriers \\ \hline \(N_{p}^{out}\) & 108 subcarriers \\ \hline \(N_{p}^{L}\) & 54 subcarriers \\ \hline \(N_{p}^{R}\) & 54 subcarriers \\ \hline \(N_{p}^{l}\) & 20 subcarriers \\ \hline Sampling frequency fs & 10 MHz \\ \hline Carrier frequency f\({}_{\text{c}}\) & 2 GHz \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation parameters. completely overlapped under the appropriate regularization parameter, meaning the BERs are very close. The BER curves of conventional DFT-based channel estimation and [8] are very close. This shows that the leakage effect was not improved by the method proposed in [8]. The conventional DFT-based channel estimation and [8] have lower computational complexity than the proposed method in this paper. However, the BER simulation showed that the proposed method has better effectiveness than the conventional DFT-based channel estimation and [8]. At low SNR, the effect of noise is higher than the leakage for the conventional DFT-based channel estimation. Therefore, the three curves of the conventional DFT, LS, and proposed methods are very close at low SNR. At high SNR, the effect of leakage is higher than the noise for the conventional DFT-based channel estimation. Therefore, the BER curve of the conventional DFT-based method is worse than the BER curves of the LS methods and the proposed method at high SNR. Linear interpolation has the lowest computational complexity but the worst BER curve. Figures 3 and 4 show the simulation results with speeds equal to 100 km/h and 200 km/h, respectively. At low SNR, the six curves did not change from Figures 2-4 as the effect of noise was higher than speed. At high SNR, the six curves from Figures 2-4 all deteriorated with increasing speed as the effect of speed was higher than noise. Since the channel changes were extraordinarily slow, it can be assumed that the channel is approximately invariant within an OFDM symbol when speed is very low.