## 4 Computer Simulations  
In this section, we provide computer simulations to verify the efficacy of the proposed CFO estimate technique. Consider an OFDM system with 256 subcarriers (\(N=256\)) that employs QPSK modulation and a CP of length 64 (\(N_{g}=N/4=64\)) over a multipath fading channel. For the simplicity of implementation, here, the 64 pilot symbols (\(K=64\)) are equally interlaced with an interval of four-subcarrier spacing in the span of frequency. The characteristic of the background channel is described as the standard wireless exponential decay model given in [21]. Suppose that the channel impulse response is composed of complex Gaussian samples with unit power, and the \(l\)-th delay path has an average power given by:  
\[\sigma_{l}^{2}=(1-e^{-1/\tau})e^{-l/\tau} \tag{42}\]  
where \(\tau\) is the root-mean square (RMS) of the delay spread. Assume that the channel order \(L=N_{g}=64\) is sufficiently large such that the impulse response tail can be neglected. The signal-to-noise ratio (SNR) is defined as \(SNR=-10\log_{10}\sigma_{N}^{2}\), where \(\sigma_{N}^{2}\) is the average power of the noise. In the following simulations, the CFO search granularity used in Equations (9) and (16) is set to be \(2^{-7}\) of the complete search interval. For the sake of comparison, the results of [PERSON]'s method [19] are also included while considering that the estimated channel order \(\hat{L}\) is selected out of the set \(\{2,16,62\}\). Without loss of generality, the parameter settings, including SNR = 30 dB, \(\tau=4\), \(\epsilon_{d}=0.3\) and \(J=5\), are used throughout this section. The performance of the proposed CFO estimation is evaluated by the mean square error (MSE), which is expressed by:  
\[\text{MSE}=\frac{1}{N_{s}}\sum_{i=1}^{N_{s}}\left|\hat{\epsilon}_{d}(i)- \epsilon_{d}\right|^{2} \tag{43}\]  
where \(\epsilon_{d}\) is the actual CFO, \(\hat{\epsilon}_{d}(i)\) is the estimated CFO obtained through the \(i\)-th trial and \(N_{s}\) is the overall number of Monte Carlo trials. To investigate the MSE performance of the proposed iterative CFO estimation, in Figure 1, we demonstrate the convergence characteristic of the proposed iterative CFO estimation while assuming \(\epsilon_{d}=0.3\). Although in Section 3, the iterative estimator is shown to cost much less complexity than the search-based scheme, the resultant expense is the reduction of the MSE convergence rate. When compared with the search-based scheme, the iterative method requires two more iterations to achieve the MSE convergence for all considered SNR values. It is noteworthy that at high SNR, say 40 dB, the approximation error in Equation (30) cannot be neglected in relation to the noise effect. This is why the performance of the iterative scheme is somewhat inferior to the search-based estimation in the convergence performance. Figure 2 shows the MSE _versus_ the iteration number for several specific CFO values. The figure indicates that the MSE using the iteration technique converges and approaches that for the search-based approach after three iterations when \(\epsilon_{d}=0.4\). The larger the actual CFO is, the more iterations for achieving an estimate close to that of the proposed search-based scheme there will be. In Figure 3, we evaluate the MSE while considering certain RMSs of the channel delay spread. For \(J>1\), it can be seen that the proposed iterative estimator provides distinguished robustness against the variation of the RMS of the delay spread and achieves a performance close to that of the search-based scheme. In addition, the MSE increases as the RMS becomes largerand larger, because a long delay spread will bring about a significant variation in the channel response between subcarriers. Figure 1: Convergence behavior of the proposed iterative CFO estimation for different SNRs. \(N\) = 256, \(K\) = 64, \(\epsilon_{d}\) = 0.3 and \(\tau\) = 4. Figure 2: Convergence behavior of the proposed iterative CFO estimation for different CFOs. \(N\) = 256, \(K\) = 64, SNR = 30 dB and \(\tau\) = 4.