# 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
If the coupling matrix follows the band structure of (11), then, the received symbol in the frequency domain can be expressed as  
\[\underline{Y}_{d} =\sum_{k\in\mathbb{I}_{d}}\mathbf{B}(d,k)\underline{X}_{k}+ \underline{W}_{d} \tag{12}\] \[\mathcal{I}_{d} =[\max(0,d-Q),\min(N_{A}-1,d+Q)]\]  
The set \(\mathcal{I}_{d}\) can be obtained as the intersection of the set of sub-carrier locations with non-zero elements in the banded matrix, \(\{k\in\mathbb{N}:|d-k|\leq Q\}\), and the range of active sub-carriers, \([0,N_{A}-1]\). Our proposed pilot pattern considers that OFDM symbols with pilots are spaced in \(T_{\text{space}}\) symbols, i.e., pilot symbols are transmitted periodically with period \(T_{\text{space}}\). Since pilot symbols do not carry data, the efficiency of this scheme, \(\eta_{\text{prop}}\), computed as the ratio between sub-carriers used for data and useful carriers within a period, is given by:  
\[\eta_{\text{prop}}=(T_{\text{space}}-1)/T_{\text{space}} \tag{13}\]  
For those OFDM symbols with pilots, the spacing between consecutive pilots is \(F_{\text{space}}\) sub-carriers in the frequency domain and the rest of sub-carriers in such symbols are reserved. Hence, being \(\underline{\mathbf{X}}=[\underline{X}_{0},\cdots,\underline{X}_{N_{A}-1}]^{T}\) an OFDM symbol with pilots, \(\underline{X}_{k}=\alpha_{k}\) if \(k\in\mathcal{P}\) and \(\underline{X}_{k}=0\) if \(k\
otin\mathcal{P}\), where \(\alpha_{k}\in\mathbb{C}\) is the pilot value and \(\mathcal{P}\) represents the set of pilot locations, which can be written as  
\[\mathcal{P}=\bigg{\{}k\in\mathbb{N}:k=k_{0}+p\cdot F_{\text{space}},p\in \bigg{[}0,\bigg{\lfloor}\frac{N_{A}-1-k_{0}}{F_{\text{space}}}\bigg{\rfloor} \bigg{]},k_{0}\in[0,N_{A}-1]\bigg{\}}. \tag{14}\]  
The symbol \(k_{0}\) represents the smaller sub-carrier index with a pilot in the frequency domain, i.e., the first sub-carrier with pilots, whereas \(p\) stands for the pilot tone index, ranging from 0 up to \(\Big{\lfloor}\frac{N_{A}-1-k_{0}}{F_{\text{space}}}\Big{\rfloor}\), since the sub-carrier must fall within the interval \([0,N_{A}-1]\). It should be noticed that the set of sub-carriers with pilots can be also expressed in the equivalent form: \(\mathcal{P}=\{k\in\mathbb{N}:\langle k-k_{0}\rangle_{F_{\text{space}}}=0,k_{0 }\in[0,N_{A}-1]\}\), where \(\langle a\rangle_{b}\) stands for the modulo operation. The number of pilots per OFDM symbol is \(|\mathcal{P}|=\lfloor\frac{(N_{A}-1-k_{0})}{F_{\text{space}}}\rfloor+1\). The proposed pilot arrangement, and the other two considered pilot patterns (comb-type and LTE CRS) are illustrated in Figure 4.  
Figure 4: Pilot patters: (**a**) proposed pilot pattern; (**b**) comb-type; (**c**) Long Term Evolution Cell Specific Reference Signal (LTE CRS) pilot pattern. **Remark 1**. : _The condition to obtain ICI-free samples of the channel coupling matrix according to (11) is \(F_{\text{spac}}>2Q\), and hence, the minimum pilot spacing is \(F_{\text{spac}}=2Q+1\)._  
Proof. : Figure 5 illustrates an sketch to prove the above remark. It can be observed that, due to the banded structure of the coupling matrix, the received symbol, \(\underline{\chi}_{d^{\prime}}\), depends on the weighted sum of transmitted complex symbols located at sub-carriers, \(k\in\mathcal{I}_{d}\), which is represented as the multiplication and sum of yellow and green dots. If a given pilot location, \(k_{p}=k_{0}+p\cdot F_{\text{spac}}\), falls within the interval \(\mathcal{I}_{d}\), then to avoid ICI, the next pilot tone at \(k_{p}+F_{\text{spac}}\) must be located outside \(\mathcal{I}_{d}\). The case that require the greater \(F_{\text{spac}}\) is the one where a pilot tone is placed at \(k_{p}=d^{\prime}-Q\), which is the case shown in the figure. In this case to avoid ICI, the following inequality must be fulfilled: \(k_{p}+F_{\text{spac}}>d^{\prime}+Q\), which imposes that the next pilot tone is located outside \(\mathcal{I}_{d}\). Solving the above inequality leads to the ICI free condition, \(F_{\text{spac}}>2Q\).