# 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
It should be noted that the value of \(F_{\text{spac}}=2Q+1\) leads to the highest number of interference-free samples of the coupling matrix. If the above ICI-free condition is fulfilled, then, the received complex symbol at a given sub-carrier can be expressed as follows  
\[\underline{\chi}_{d}= \begin{cases}\underline{\textbf{B}}(d,k_{d})\alpha_{k_{d} }+\underline{W}_{d},&\text{if }\mathcal{I}_{d}\cap\mathcal{P}=\{k_{d}\}\\ \underline{W}_{d},&\text{if }\mathcal{I}_{d}\cap\mathcal{P}=\emptyset\end{cases}  \tag{15}\]  
In view of (15) it can be noticed that there can be sub-carrier locations, \(d\), where not pilot tone is received, only noise, when the intersection of the pilot sub-carrier locations, \(\mathcal{P}\), and \(\mathcal{I}_{d}\) is the empty set. In addition, it can be observed that if \(\mathcal{I}_{d}\cap\mathcal{P}\
eq\emptyset\), then there is a single sub-carrier location, \(k_{d}\), that contributes to the received signal. Therefore, it is possible to obtain interference-free samples of the coupling matrix with a LS criterion as:  
\[\underline{\textbf{B}}(d,k_{d})=\frac{\underline{\chi}_{d}}{\alpha_{k_{d}}}= \underline{\textbf{B}}(d,k_{d})+\frac{W_{d}}{\alpha_{k_{d}}},\text{with }k_{d}\in\mathcal{P}\cap\mathcal{I}_{d}\
eq\emptyset. \tag{16}\]  
**Remark 2**. : _Let us consider that pilot locations are defined according to \(\mathcal{P}\) and ICI-free conditions as per Remark 1 are fulfilled, i.e., \(F_{\text{spac}}>2Q\); then, from (16), \(|\mathcal{P}|\) columns of **B** are sampled, whose locations are also defined according to \(\mathcal{P}\). The overall number of samples of the coupling matrix, \(N_{s}\), can be expressed as_  
\[N_{s}=\sum_{p=0}^{\left\lfloor\frac{N_{d}-1-k_{p}}{\text{spac}}\right\rfloor} \left(\min\bigl{(}N_{A}-1,k_{p}+Q\bigr{)}-\max\bigl{(}0,k_{p}-Q\bigr{)}+1\right) \tag{17}\]  
Figure 5: Sketch of proof for Remark 1: minimum spacing between pilot tones to avoid Inter-Carrier Interference (ICI). _where \(k_{p}=k_{0}+p\cdot F_{\text{spac}}\)._  
Proof. : If ICI-free conditions are fulfilled, then, a received complex symbol on sub-carrier, \(d\), satisfying \(\mathcal{I}_{d}\cap\mathcal{P}\
eq\emptyset\), represents a weighted noised sample of a single element of the coupling matrix, \(\mathbf{B}\), as per remark 1 and (10). On the other hand, the received OFDM symbol in the frequency domain can be written as follows  
\[\underline{\mathbf{Y}}=\underline{\mathbf{B}}_{1}\cdot\underline{X}_{1}+ \underline{\mathbf{B}}_{2}\cdot\underline{X}_{2}+\cdots+\underline{\mathbf{B }}_{N_{A}-1}\cdot\underline{X}_{N_{A}-1}=\sum_{k=0}^{N_{A}-1}\underline{ \mathbf{B}}_{k}\cdot\underline{X}_{k} \tag{18}\]  
where \(\underline{\mathbf{B}}_{k}=[\underline{b}_{0,k},\underline{b}_{1,k},\cdots, \underline{b}_{N_{A}-1,k}]^{T}\) represents the \(k\)-th column of the coupling matrix \(\mathbf{B}\). Since the OFDM symbol in the frequency domain, \(\underline{\mathbf{Y}}\), is a pilot symbol, then \(\underline{X}_{k}=0,\forall k\
otin\mathcal{P}\), and thus  
\[\underline{\mathbf{Y}}=\sum_{p=0}^{\left\lfloor\frac{N_{A}-1-k_{0}}{F_{\text {spac}}}\right\rfloor}\underline{\mathbf{B}}_{k_{p}}\cdot\underline{X}_{k_{p}} \tag{19}\]  
where \(k_{p}=k_{0}+p\cdot F_{\text{spac}}\). As can be observed from (14), \(|\mathcal{P}|=\left\lfloor\frac{N_{A}-1-k_{0}}{F_{\text{spac}}}\right\rfloor+1\) columns of \(\underline{\mathbf{B}}\) are sampled. The number of samples per column is determined by the sub-carrier locations of pilot tones, \(k_{p}\), which falls within the inteval \([0,N_{A}-1]\) and by the band structure of the coupling matrix as per (7), which involves that non zero elements fulfill \(|d-k_{p}|\leq Q\).