# A Low-Complexity Channel Estimation Based on a Least-Squares Algorithm in OFDM Systems
## 5 Simulation and Discussion
The channel was no longer approximately invariant within an OFDM symbol when the speed is very high. Therefore, the simulation results at high speed are worse than the simulation results at low speed when SNR is high. The BER curves of the LS methods [10,11] and the method proposed in this paper are still overlapped in Figures 3 and 4. Figure 2: Simulated diagram of SNR to error rate when the carrier frequency is 2 GHz, the sampling frequency is 10 MHz, and the speed is 20 km/h for typical urban time-varying channel model [8,10,11]. Figure 4: Simulated diagram of SNR to error rate when the carrier frequency is 2 GHz, the sampling frequency is 10 MHz, and the speed is 200 km/h for the typical urban time-varying channel model [8,10,11]. Figure 3: Simulated diagram of SNR to error rate when the carrier frequency is 2 GHz, the sampling frequency is 10 MHz, and the speed is 100 km/h for the typical urban time-varying channel model [8,10,11]. Consider the number of multiplications after CFRs at pilot subcarriers estimation until the CIR estimation. The number of complex multiplications of the LS method [11] is \(0.5N_{p}N_{p}^{out}\). There are 6848 complex multiplications after the relevant parameters are substituted in. Meanwhile, the number of complex multiplications of the method proposed in this paper is \(N_{p}^{out}N_{p}^{I}+(N_{p}/2)\text{log}_{2}(N_{p})\). There are 2695 complex multiplications after the relevant parameters are substituted in. Both the LS method [11] and the proposed method have some complex numbers that need to be stored in the memory. The numbers of complex numbers that need to be stored in the memory are \(0.5N_{p}N_{p}^{out}\) and \(N_{p}^{out}N_{p}^{I}\), respectively. In this example, \(0.5N_{p}N_{p}^{out}=6848\) and \(N_{p}^{out}N_{p}^{I}=2247\). Table 5 shows the comparisons of different methods on computational complexity and memory resource usage. It is obvious that the method proposed in this paper has lower computational complexity and stored complex numbers than the LS method [9,11], while both have the same BER from simulations. Therefore, the method proposed in this paper is better than the LS method [9,11]. Let us consider a bad urban time-varying channel model [27] as shown in Table 6. The number of maximum channel delay points is equal to 100 and greater than \(N_{p}/2\). Therefore, the dimension of the estimated CIR for both LS methods [10,11] is set to the maximum value of the dimension, which is \(N_{p}^{out}\). But in this case, the matrices in both LS methods [10,11] are getting closer and closer to singular. The regularization parameters of [10,11] are set to 0.00005 and 0.05, respectively. For fair comparison, the proposed method, the DFT method, and [8] also take the first \(N_{p}^{out}\) points of the estimated channel impulse response as the final estimated channel impulse response. Figure 5 shows the simulation results with speed equal to 20 km/h. The six curves in Figure 5 are all becoming worse than those in Figure 2. The numbers of maximum channel delay points in bad and typical urban channel models are 100 and 50, respectively. The larger the maximum channel delay, the worse the channel estimation efficiency of linear interpolation will be. Therefore, the linear interpolation curve in Figure 5 is obviously much worse than that in Figure 2. Since the matrices are getting closer and closer to singular in the two LS methods, although the regularization parameter is used, it still makes the two LS curves in Figure 5 worse than those in Figure 2. The estimated CFRs at the virtual pilot subcarriers with our proposed method will be inaccurate when the number of maximum channel delay points is larger than \(N_{p}/2\).