## 6 BER Analysis of OFDM Systems with Residual ICI  
First, the system model in Equation (2) needs to be rewritten in the following form without the ICI presence for the \(l\)-th path of the time-varying channels:  
\[r_{k}(l)=\upalpha_{k}(l)e^{j\uptheta_{k}(l)}s_{k}+v_{k}(l), \tag{37}\]  
where \(h_{k}(l)=\sum_{n=0}^{N_{s}-1}h_{tl}(n,l)e^{-j2 mk/N_{s}}\), the random variable (r.v.) \(\upalpha_{k}(l)=|h(k,l)|\) is the random channel gain for the \(l\)-th path at the \(n\)-th time slot, which follows the Rayleigh fading with the zero mean value and the variance \(\upsigma_{\upalpha}^{2}\), and \(\uptheta_{k}(l)=\arctan\frac{\Im(h_{k}(l))}{\Re(h_{k}(l))}\in[-\uppi,\uppi]\). The AWGN \(v_{l}(n)\) denotes the same channel. Note that we assume that the interference inside the band is perfectly eliminated by the MBMF canceling strategy for simplicity. The received signal for the \(k\)-th subcarrier can be given as:  
\[r_{k}=\sum_{l=0}^{N_{h}-1}r_{k}(l)P_{k}(l)+\sum_{m=0,m\
eq k}^{N_{s}-1}\sum_{l =0}^{N_{h}-1}r_{k-m}(l)P_{k}(l)+\sum_{l=0}^{N_{h}-1}v_{k}(l)P_{k}(l), \tag{38}\]  
where \(P_{k}(l)=e^{-j2\pi(k/N_{s})l}\), and thus, the output signals of the match filter for the decision variable can be given by:  
\[\hat{s}_{k} =\underbrace{s_{k}\sum_{q=-Q}^{Q}\sum_{l=0}^{N_{h}-1}\upalpha_{k- q}^{2}(l)P_{k-q}^{2}(l)}_{\text{desired signal}}+\underbrace{\sum_{m=0,\atop m\
ot\in[-D+q,k+D+q]}^{N_{s}-1} \sum_{q=-Q}^{Q}\sum_{l=0}^{N_{h}-1}\upalpha_{k-q}(l)e^{j\uptheta_{k-q}(l)} \upalpha_{k-m}(l)e^{j\uptheta_{k-m}(l)}P_{k}(l)s_{m}}_{\text{ICI}} \tag{39}\] \[+\underbrace{\sum_{l=0}^{N_{h}-1}\upalpha_{k}(l)e^{j\uptheta_{k} (l)}v_{k}(l)P_{k}(l)}_{\text{noise}}\] \[=s_{k}\sum_{q=-Q}^{Q}\sum_{l=0}^{N_{h}-1}\upalpha_{k-q}^{2}(l)P_ {k-q}^{2}(l)+\sum_{l=0}^{N_{h}-1}\sum_{q=-Q}^{Q}\upalpha_{k-q}(l)e^{j\uptheta_ {k-q}(l)}c_{k-q}(l)+v_{k}.\]  
From the equation above, we can observe that the PDF of the SINR becomes very non-trivial due to the superposition of desired signals and ICI from multiple subcarriers. If the channels vary rapidly in the time domain and the number of subcarriers becomes very large, the power of the desired signal (\(q\
eq 0\)) from the adjacent subcarrier will become very close to that from the center subcarrier (\(q=0\)) [4]. Thus, it is possible to approximate Equation (39) in another form:  
\[\hat{s}_{k}\approx s_{k}(2Q+1)\sum_{l=0}^{N_{h}-1}\upalpha_{k}^{2}(l)P_{k}^{ 2}(l)+(2Q+1)\sum_{l=0}^{N_{h}-1}\upalpha_{k}(l)\sum_{q=-Q}^{Q}e^{j\uptheta_{ k-q}(l)}c_{k-q}(l)+v_{k}, \tag{40}\]To obtain the PDF of the SINR with the aid of Equations (32) and (33), the mathematical expression of SINR can be written as [29]:  
\[\xi_{k}=\frac{\frac{E_{s}}{\sigma_{v}^{2}}(\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l))^ {2}}{\frac{1}{2Q+1}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)+\frac{E_{s}}{\sigma_{v }^{2}}\sum_{l=0}^{N_{h}-1}\alpha_{k}^{2}(l)\sum_{q=-Q}^{Q}e^{j\theta_{k}(l)} \mathbb{E}(c_{k}(l)c_{k-q}^{*}(l))e^{-j\theta_{k-q}(l)}}, \tag{41}\]  
where:  
\[\mathbb{E}(c_{k}(l)c_{k-q}^{*}(l))=\sigma_{Dl}^{2}\sigma_{l}^{2}\sum_{ \begin{subarray}{c}m\
ot\in[-D+q,D+q]\\ \cup[-D-r+q,D-r+q]\end{subarray} }\frac{1}{(1-e^{-j2\pi(k+r-q)/N_{s}})(1-e^{j2 \pi(k+r-q)/N_{s}})} \tag{42}\]  
In order to further simplify the analysis, the normalized Doppler frequencies and the power are the same for any channel tap. The quantity \(\beta_{l}\) is a constant for any \(l\).