# A Low-Complexity Pilot-Based Frequency-Domain Channel Estimation for ICI Mitigation in OFDM Systems
## 5 Numerical Results  
The performance of the proposed method is assessed in terms of BER for Quadrature Phase Shift Keying (QPSK) modulation, using the ITU-Vehicular A channel [32]. The carrier frequency is 2 GHz and the mobile speed is 540 km/h. We is consider an OFDM system with sub-carrier spacing of 15 kHz and a sampling rate of 1.92 MHz. Unit transmit power is considered for all the sub-carriers, i.e., \(\mathbb{E}[|X_{k}^{(m)}|^{2}]=1\), \(k\in[0,N_{A}-1]\). The rest of parameters are given in Table 1. The CP length at a sampling rate of 1.92 MHz yields 4.6875 \(\mu\)s of guard time, which is longer than the maximum delay spread of the channel, and thus, there is no ISI. With these parameters, the efficiency of the proposed pilot pattern as given from (9) is 2/3. It can be shown that the efficiency of comb-type pattern is, \(\eta_{\text{comb}}=\frac{1-F_{\text{space}}}{F_{\text{space}}}\), and thus, this pattern has also an efficiency of 2/3 with the above simulation parameters. Finally, the LTE CRS pattern has a high efficiency, since only four sub-carriers are reserved for pilot tones within a PRB and slot [28], leading to \(\eta_{\text{comb}}\approx 0.952\). To assess the performance of the proposed pilot pattern and channel estimation for ICI mitigation, its performance is compared with other techniques of small complexity. Considered pilot patterns are: (1) comb-type [33], with a frequency spacing between pilots of \(F_{\text{space}}=3\); (2) the LTE pilot pattern [32]; and (3) our proposed pilot pattern. With the comb and LTE patterns, it is only possible to estimate the main diagonal of the coupling matrix because of the ICI, and thus, a classical 1-tap ZF equalization is considered [7]. Finally, we also analyze the case in which the main diagonal of the coupling matrix is estimated ideally (i.e., without ICI nor any other estimation error), but the transmitted data is obtained through 1-tap ZF equalization. This case is labeled as \"\(H_{n^{\prime}k}\) ideal 1-tap ZF\" to clarify the equalization does not mitigate the ICI in this case, but the channel response is obtained without ICI. The frequency domain channel interpolation in all the cases is spline [33]. However, for the time domain interpolation, the following methods are used: (1) zero-order hold (ZOH); (2) linear interpolation; and (3) spline interpolation. In case of comb-type pattern, since every OFDM symbols contain pilots, time interpolation is not performed. Figure 7 illustrates the performance obtained with the methods mentioned above. The legend in the figure specifies the time-interpolation method and pilot-pattern, which is defined in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(N\) & FFT size & 128 \\ \hline \(N_{A}\) & Number of active sub-carriers & 72 \\ \hline \(G\) & CP length & 9 \\ \hline \(\Delta f\) & Sub-carrier spacing & 15 kHz \\ \hline \(f_{\mathrm{s}}\) & Sampling rate & 1.92 MHz \\ \hline \(\alpha_{k}\), \(k\in\mathcal{P}\) & Pilot tone amplitude & 1 \\ \hline \(\mathbb{E}[|X_{k}^{(m)}|^{2}]\), \(m\
otin\mathcal{M}\) & Power of data symbols & 1 \\ \hline - & Modulation order & QPSK \\ \hline \(k_{0}\) & Pilot sub-carrier offset (prop. pattern) & 0 \\ \hline \(Q\) & ICI spreading factor & 1 \\ \hline \(T_{\mathrm{spac}}\) & Symbol pilot spacing (prop. pattern) & 3 \\ \hline \(F_{\mathrm{spac}}\) & Freq. pilot spacing (prop. and comb patt.) & 3 \\ \hline \(f_{c}\) & Carrier frequency & 2 GHz \\ \hline \(v\) & Velocity & 540 km/h \\ \hline \(f_{d}\) & Normalized Doppler frequency & 0.067 \\ \hline - & Doppler Spectrum & Jakes \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 7: BER versus Signal-to-Noise Ratio (SNR) under ITU-Vehicular A channel model for different channel estimation and equalization techniques. It can be observed that the proposed method with spline interpolation in the time domain achieves the best performance in the high SNR regime (prop-SPL).