# A Low-Complexity Pilot-Based Frequency-Domain Channel Estimation for ICI Mitigation in OFDM Systems
## 3 Proposed Pilot Pattern and Frequency-Domain Coupling Matrix Estimation
Hence, the set of natural numbers that represent the sampled rows, \(d\), of the \(k_{p}\)-th column of \(\underline{\mathbf{B}}\) can be expressed as follows  
\[\mathcal{S}_{p} =\left\{d\in\mathbb{N}:|d-k_{p}|\leq Q,k\in[0,N_{A}-1]\right\}\] \[\overset{\text{(a)}}{=}\left\{d\in\mathbb{N}:k_{p}-Q\leq d\leq k_{ p}+Q,0\leq d\leq N_{A}-1\right\}\] \[\overset{\text{(b)}}{=}\left\{d\in\mathbb{N}:\max\bigl{(}0,k_{p}- Q\bigr{)}\leq d\leq\min\bigl{(}N_{A}-1,k_{p}+Q\bigr{)}\right\} \tag{20}\]  
where (b) comes from combining the two inequalities from (a). Finally, computing the number of elements of \(\mathcal{S}_{p}\) and summing for all the sampled columns completes the proof. Figure 6 illustrates an sketch of the number of elements sampled per column as determined with Remark 2. Three pilot tones are shown: one located at \(k_{0}=0\) in the frequency domain; a second one located at \(k_{p}\); and a third one located in the last sub-carrier, \(N_{A}-1\). It is observed that the limits on the sub-carrier indexes affect the first and last pilot tone, reducing the number of sampled elements in the first and last sampled columns of the coupling matrix. Hence, the sampled rows of the first column range from \(0\) up to \(Q\), whereas the last column has sampled rows ranging from \(N_{A}-1\) down to \(N_{A}-1-Q\). The second sampled column is different in the example of the figure. In this case, the sampled column is far from the sub-carrier edges, and the sampled rows range from \(k_{p}-Q\) up to \(k_{p}+Q\). Example: To illustrate the coupling matrix estimation, let us consider the case \(N_{A}=7\), \(Q=1\) and \(k_{0}=0\). Hence, the locations of the pilots in the frequency domain are expressed as \(\mathcal{P}=\{0,3,6\}\) and the received vector of useful sub-carriers related to an OFDM symbol with pilots can be expressed as  
\[ \begin{pmatrix}\underline{Y}_{0}\\ \underline{Y}_{1}\\ \underline{Y}_{2}\\ \underline{Y}_{3}\\ \underline{Y}_{4}\\ \underline{Y}_{5}\\ \underline{Y}_{6}\end{pmatrix} = \begin{pmatrix}\underline{b}_{0,0}&\underline {b}_{0,1}&0&0&0&0&0&0\\ \underline{b}_{1,0}&\underline{b}_{1,1}&\underline{b}_{1,2}&0&0&0&0&0\\ 0&\underline{b}_{2,1}&\underline{b}_{2,2}&\underline{b}_{2,3}&0&0&0\\ 0&0&\underline{b}_{3,2}&\underline{b}_{3,3}&\underline{b}_{3,4}&0&0\\ 0&0&0&\underline{b}_{4,3}&\underline{b}_{4,4}&\underline{b}_{4,5}&0\\ 0&0&0&0&\underline{b}_{5,5}&\underline{b}_{5,6}\\ 0&0&0&0&\underline{b}_{6,5}&\underline{b}_{6,6}\\ \underline{\chi}_{6}\end{pmatrix}\begin{pmatrix}\alpha_{0}\\ 0\\ 0\\ \underline{\chi}_{3}\\ \underline{\chi}_{4}\\ \underline{\chi}_{5}\\ \underline{\chi}_{6}\end{pmatrix} + \begin{pmatrix}\underline{W}_{0}\\ \underline{W}_{1}\\ \underline{W}_{2}\\ \underline{W}_{3}\\ \underline{W}_{4}\\ \underline{W}_{5}\\ \underline{\chi}_{6}\end{pmatrix}  \tag{21}\]  
According to Remark 2, we obtain \(N_{s}=7\) samples of the coupling matrix for the 0-th, 3-th and 6-th columns of \(\mathbf{B}\): \(\underline{\hat{b}}_{0,0}=\underline{\chi}_{0}/\alpha_{0}\), \(\underline{\hat{b}}_{1,0}=\underline{\chi}_{1}/\alpha_{0}\), \(\underline{\hat{b}}_{2,3}=\underline{\chi}_{2}/\alpha_{3}\), \(\underline{\hat{b}}_{3,3}=\underline{\chi}_{3}/\alpha_{3}\), \(\underline{\hat{b}}_{4,3}=\underline{\chi}_{4}/\alpha_{3}\), \(\underline{\hat{b}}_{5,6}=\underline{\chi}_{5}/\alpha_{6}\), \(\underline{\hat{b}}_{6,6}=\underline{\chi}_{6}/\alpha_{6}\). Where the \((d,k)\)-th element of \(\mathbf{\hat{B}}\) is represented as \(\underline{\hat{b}}_{d,k}\). It should be noticed that it is straightforward to obtain more samples of the coupling matrix if the received symbol power of guard band sub-carriers in the neighborhood of the 0-th and \(N_{A}\)-th sub-carriers are estimated. However, the performance improvement is negligible when \(N_{A}\) is high. Channel interpolation: once \(N_{s}\) samples of the coupling matrix are estimated, an interpolation of the samples is performed to estimate the coupling matrix.