# Symbol-Level Selective Channel Estimation in Packet-Based OFDM Systems
## 4 Selective Channel Estimation Based on Correlation Coefficient  
In this section, we propose a criterion for selecting the better channel estimation scheme between the two channel estimation schemes on a symbol-by-symbol basis. TRFI and DD-TT show good performance at high and low SNRs, respectively. Therefore, if the better channel estimation scheme can be selected for each symbol rather than an entire packet, then the channel variations from the beginning to the end of the packet can be accurately tracked. Thus, the performance can be better than that of conventional techniques. Figure 2 shows a flowchart of the proposed SLS scheme. First, in Step 1, the initial channel estimate of the \(m\)-th OFDM symbol in a packet is obtained as \(\hat{H}(m,k)\) by computing Equations (2) and (3). Subsequently, in Steps 2 and 3, cubic spline interpolation (CSI) and TT are performed on initial channel estimate \(\hat{H}(m,k)\) to obtain \(H_{CSI}(m,k)\) and \(H_{TT}(m,k)\), respectively. \(H_{CSI}(m,k)\) obtained through the CSI on the initial channel estimates is used to select the TRFI. To obtain \(H_{CSI}(m,k)\), first, we define the uniformly spaced subcarriers indexed as \(\mathcal{K}_{S}\triangleq\{\pm 2,\ \pm 5,\ \pm 7,\ \pm 10,\ \pm 14,\ \pm 18,\ \pm 21,\ \pm 24\}\). \(\mathcal{K}_{S}\) is designated as the pilot subcarrierindexes and data subcarrier indexes spaced at regular intervals from the pilot subcarriers in consideration of correlation with the initial channel estimate. For \(\forall k\in\mathcal{K}_{S}\), the channel estimates of subcarrier \(k\) are obtained by CSI using \(\hat{H}(m,k)\), \(\forall k\in\mathcal{K}\setminus\mathcal{K}_{S}\). Meanwhile, \(H_{TT}(m,k)\) is obtained by performing Equations (6), (7), and (10) on initial channel estimate \(\hat{H}(m,k)\). In Step 4, we calculate the correlation coefficient between \(\hat{H}(m,k)\) and \(H_{j}(m,k)\) for indexes \(\mathcal{K}_{S}\), where \(j\) represents \(CSI\) or \(TT\). The correlation coefficient \(\rho_{j}(m)\) is defined as follows:  
\[\rho_{j}(m)=\frac{1}{R-1}\left|\sum_{k\in\mathcal{K}_{S}}\left(\frac{\overline {\hat{H}(m,k)-\mu_{\hat{H}}}}{\sigma_{\hat{H}}}\right)\left(\frac{H_{j}(m,k)- \mu_{H_{j}}}{\sigma_{H_{j}}}\right)\right|, \tag{13}\]  
where \(\rho_{j}(m)\) denotes the correlation coefficient between \(\hat{H}(m,k)\) and \(H_{j}(m,k)\) at the \(m\)-th OFDM symbol, and \(\overline{(\cdot)}\) represents a complex conjugate. \(\mu_{\hat{H}}\) and \(\sigma_{\hat{H}}\) are the mean and standard deviation of \(\hat{H}(m,k)\), respectively, and \(\mu_{H_{j}}\) and \(\sigma_{H_{j}}\) are the mean and standard deviation of \(H_{j}(m,k)\), respectively. \(R\) is the number of indexes in \(\mathcal{K}_{S}\). Finally, in Step 5, if the correlation coefficient of CSI, \(\rho_{CSI}\), is higher than that of TT, \(\rho_{TT}\), then the final channel estimate \(H(m,k)\) of the \(m\)-th OFDM symbol is determined as the channel estimate \(H_{TRFI}(m,k)\) obtained by performing the TRFI (Steps 6 and 7). Otherwise, the final channel estimate \(H(m,k)\) is determined as the channel estimate \(H_{TT}(m,k)\). This process is performed sequentially on \(M\) OFDM symbols constituting one packet. Figure 2: Flowchart of the proposed SLS scheme. The correlation coefficient can determine the degree of correlation between two variables. \(H_{CSI}(m,k)\) and \(H_{TT}(m,k)\) are based on the initial channel estimate; thus, by analyzing the correlation coefficients between the initial channel estimate \(\hat{H}(m,k)\) and two channel estimates, i.e., \(H_{CSI}(m,k)\) and \(H_{TT}(m,k)\), the better channel estimation technique can be selected.