## 6 BER Analysis of OFDM Systems with Residual ICI
The Equation (41) becomes:  
\[\xi_{k} \approx\frac{\frac{E_{s}}{\sigma_{v}^{2}}(\sum_{l=0}^{N_{h}-1} \alpha_{k}^{2}(l))^{2}}{\frac{1}{2Q+1}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)+ \beta_{l}\frac{E_{s}}{\sigma_{v}^{2}}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)} \tag{43}\] \[=\frac{\frac{E_{s}}{\sigma_{v}^{2}}(\sum_{l=0}^{N_{h}-1}\alpha_{k }^{2}(l))^{2}}{\frac{1}{2Q+1}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)+\beta_{l} \frac{E_{s}}{\sigma_{v}^{2}}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)}\]  
According to [30], the PDF of \(X_{k}=\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)\) can be expressed as:  
\[f_{X}(x)=\frac{\zeta x^{L-1}e^{-\zeta x/\epsilon}}{(L-1)!\epsilon^{L}}, \tag{44}\]  
where \(\epsilon=2\sigma_{\alpha}^{2}b\), and the quantity \(\sigma_{\alpha}^{2}\) denotes the variance of \(\alpha_{k}(l)\). We define \(a=\frac{1}{2Q+1}\), \(b=\frac{E_{s}}{\sigma_{v}^{2}}\) and \(\zeta=\beta_{l}\frac{E_{s}}{\sigma_{v}^{2}}\). Additionally, Equation (43) is rewritten as:  
\[\xi=\frac{bx}{a+\zeta x}, \tag{45}\]  
\[x=\frac{\xi a}{b-\xi\zeta}, \tag{46}\]  
\[\frac{d\xi}{dx}=\frac{ab}{(a+\zeta x)^{2}}=\frac{(\xi\zeta-b)^{2}}{ab} \tag{47}\]  
Hence, the PDF of the SINR \(\xi\) can be expressed as [30]:  
\[f_{\xi}(y) =\frac{1}{|\frac{d\xi}{dx}|}f_{X}\left(\frac{ya}{b-y\zeta}\right)= \frac{ab}{(b-y\zeta)^{2}}f_{X}\left(\frac{ya}{b-y\zeta}\right) \tag{48}\] \[=\frac{ab(ya)^{L-1}}{(b-y\zeta)^{L+1}}\frac{e^{\frac{ya}{2(b-y \zeta)\sigma_{\alpha}^{2}b}}}{(L-1)!(2\sigma_{\alpha}^{2}b)^{L}}\]For the BER analysis of binary phase shift keying (BPSK) modulation, the conditional BER given the SINR \(\xi\) is given by [31]:  
\[P_{e}(y)=\mathcal{Q}(\sqrt{2y}), \tag{49}\]  
where the notation \(\mathcal{Q}()\) denotes the Q-function for the error probability calculation. The average BER can be given as:  
\[\bar{P}_{e}=\int_{0}^{+\infty}\mathcal{Q}(\sqrt{2y})f_{\xi}(y)dy, \tag{50}\]  
For the numerical evaluation of Equation (50), the average BER needs to be decomposed as follows:  
\[=\frac{\frac{E_{s}}{\sigma_{v}^{2}}(\sum_{l=0}^{N_{h}-1}\alpha_{k }^{2}(l))^{2}}{\frac{1}{2Q+1}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)+\beta_{l} \frac{E_{s}}{\sigma_{v}^{2}}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)}\]0 \[=\frac{\frac{E_{s}}{\sigma_{v}^{2}}(\sum_{l=0}^{N_{h}-1}\alpha_{k }^{2}(l))^{2}}{\frac{1}{2Q+1}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)+\beta_{l} \frac{E_{s}}{\sigma_{v}^{2}}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)}\]1  
where \(\phi(x)=\frac{2}{\sqrt{\pi}}\sum_{i=1}^{+\infty}\frac{(-1)^{i+1}}{(2i-1)(i-1)! }x^{2i-1}\) as in [32]. According to [29], we can obtain the following equation:  
\[=\frac{\frac{E_{s}}{\sigma_{v}^{2}}(\sum_{l=0}^{N_{h}-1}\alpha_{k }^{2}(l))^{2}}{\frac{1}{2Q+1}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)+\beta_{l} \frac{E_{s}}{\sigma_{v}^{2}}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)}\]2  
where:  
\[=\frac{\frac{E_{s}}{\sigma_{v}^{2}}(\sum_{l=0}^{N_{h}-1}\alpha_{k }^{2}(l))^{2}}{\frac{1}{2Q+1}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)+\beta_{l} \frac{E_{s}}{\sigma_{v}^{2}}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)}\]3  
and \(\Gamma(t)=\int_{0}^{\infty}x^{t-1}e^{-x}dx\) denotes the Gamma function. Hence, the average BER can be obtained by Equation (50) with Equations (52) and (53). The analytical result of the uncoded BER with different band widths \(Q2,4,6\) is plotted in Figure 4, with \(f_{d}T_{\text{OFDM}}=0.65\) the uniform power delay profile \(L=8\) (\(\beta_{l}\) is a constant for any \(l\)). It can be observed that the analytical curves are approximately in agreement with the simulated ones, but the curves with \(Q>6\) are not plotted due to the significant mismatch to the simulated ones. This is because more interference is removed from the desired signals that cause the inaccuracy of the autocorrelation function of the ICI approximation in Equation (42), the PDF of which does not follow the similar distribution as described in Equation (44). Although the mismatch exists in the analysis, the analysis will be more reliable in the low Doppler frequency scenarios with small band sizes \(Q\).