# 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  
In this section, the proposed pilot pattern is presented, which relies on the band structure of the coupling matrix to obtain interference free samples of it. To this end, our proposed pattern uses OFDM pilot symbols without data, thus allowing to perform a simple LS estimation, although at the expense of reducing the spectral efficiency. We assume that \(N_{A}\) sub-carriers are active, and thus, \(N_{v}=N-N_{A}\) sub-carriers are used as guard band. Hence, we can write \(\mathbf{X}=[\mathbf{0}_{1\times N_{f}/2},\mathbf{X}^{\mathsf{T}},\mathbf{0}_{1 \times N_{f}/2}]^{T}\), being \(\underline{\mathbf{X}}=\mathbf{X}(N_{v}/2:\mathbf{X}(N_{v}/2:\mathbf{X}(N_{v} /2:\mathbf{X}(N_{v}/2:\mathbf{X}(N_{v}/2:\mathbf{  
\(N_{\
u}/2+N_{A}-1\)) the transmitted vector in active sub-carriers. As can be observed, the sub-carrier indexes are shifted between the elements of these vectors, and hence, the \(k\)-th element of \(\underline{\mathbf{X}}=[\underline{X}_{0},...,\underline{X}_{N_{A}-1}]^{T}\) is expressed as \(\underline{X}_{k}=X_{k+N_{\
u}/2}\). From now on, the OFDM symbol index (\(m\)) is neglected for notation simplicity. Equivalently, the received vector in active sub-carriers can be written as \(\underline{\mathbf{Y}}=\mathbf{H}_{\text{df}}\underline{\mathbf{X}}+\underline {\mathbf{W}}\), where \(\mathbf{H}_{\text{df}}\) is the \(N_{A}\times N_{A}\) matrix obtained by selecting the central block of \(\mathbf{H}_{\text{df}}\), i.e., \(\underline{\mathbf{H}}_{\text{df}}=\mathbf{H}_{\text{df}}\left(\frac{N_{ \
u}}{2}:\frac{N_{\
u}}{2}+N_{A}-1,\frac{N_{\
u}}{2}:\frac{N_{\
u}}{2}+N_{A}- 1\right)\). \(\underline{\mathbf{Y}}\in\mathbb{C}^{N_{A}\times 1}\) and \(\underline{\mathbf{W}}\in\mathbb{C}^{N_{A}\times 1}\) are the central blocks of vectors \(\mathbf{Y}\in\mathbb{C}^{N\times 1}\) and \(\mathbf{W}\in\mathbb{C}^{N\times 1}\), respectively. It has been shown in [16] and [5] that the sub-carrier coupling matrix of the channel has a band structure, and the interference that undergoes the \(k\)-th sub-carrier mainly comes from the 2Q neighboring sub-carriers. This means that most of the energy of the coupling matrix relies in the \(2Q+1\) central diagonals; hence, it can be accurately approximated by a band matrix. As proven in [5], this banded structure is the result of the assumption that the CIR varies in a linear fashion within an OFDM symbol. It is shown in [5] that this linear approximation is valid for normalized Doppler frequencies, \(f_{d}=\frac{\mathrm{p}f_{c}}{cM}\) up to 0.1, being \(v\) the relative velocity in m/s, \(c\) the light speed, \(f_{c}\) the carrier frequency in Hz, and \(\Delta f\) the OFDM sub-carrier spacing in Hz. In order to illustrate the banded structure of \(\mathbf{H}_{\text{df}}(d,k)\), Figure 3 shows the coupling matrix assuming a receiver speed of 500 km/h at a carrier frequency of 2 GHz. It can be shown that most of the energy is concentrated in a few bands close to the main diagonal. Hence, the coupling matrix can be well approximated as its banded version, either if we include or exclude the guard bands, i.e., \(\mathbf{H}_{\text{df}}\approx\mathbf{B}\) and \(\underline{\mathbf{H}}_{\text{df}}\approx\underline{\mathbf{B}}\), where  
\[\underline{\mathbf{B}}(d,k)= \begin{cases}\underline{\mathbf{H}}_{\text{df}}( d,k),&\text{if }|d-k|\leq Q\\ 0,&\text{if }|d-k|>Q\end{cases}  \tag{11}\]  
Figure 3: Ideal coupling matrix for a ITU Vehicular A channel model with \(v=500\) km/h at a carrier frequency of 2 GHz using FFTs of 128 points and 15 kHz of sub-carrier spacing.