# Maximal Ratio Combining Detection in OFDM Systems with Virtual Carriers Over V2V Channels
## 2 System Model  
The OFDM system employs a total number of \(N=N_{d}+N_{g}+N_{p}+1\) subcarriers, comprising \(N_{d}\) data subcarriers, \(N_{g}\) guard subcarriers, \(N_{p}\) pilot subcarriers, and an additional subcarrier for the DC component. The \(k\)th OFDM symbol transmitted in the time domain (TD) \(x^{k}[n]\), not including the cyclic prefix (CP), is defined by  
\[x^{k}[n]=\frac{1}{\sqrt{N}}\sum_{m=0}^{N-1}s^{k}[m]e^{j2\pi\frac{\pi m}{N}}, \quad n=\{0,1,\ldots,N-1\}, \tag{1}\]  
where \(N\) denotes the length of the OFDM symbol and \(s^{k}[m]\) is the \(m\)-th data symbol belonging to an \(M\)-order-quadrature amplitude modulation (M-QAM) constellation. Once the CP exclusion is performed, the received signal for the \(k\)-th OFDM symbol in its complex baseband representation can be described by the discrete circular convolution defined as:  
\[y^{k}[n]=\sum_{l=0}^{L-1}h^{k}[n,l]x^{k}[\langle n-l\rangle_{N}]+w^{k}[n], \tag{2}\]  
where \(n=\{0,1,...,N-1\}\), \(l=\{0,1,...,L-1\}\), \(h^{k}[n,l]\) is the CIR of the \(k\)-th symbol at time \(n\) for an impulse input at the previous \(l\) samples, and \(w^{k}[n]\) is the complex additive white Gaussian noise (AWGN) with zero mean and variance \(\sigma_{w}^{2}=N_{0}/2\). The circular convolution between the channel impulse response (CIR) and \(x^{k}[n]\) can be rewritten in its matrix form as:  
\[\mathbf{y}^{k}=\mathbf{H}^{k}\mathbf{x}^{k}+\mathbf{w}^{k}, \tag{3}\]  
where:  
\[\mathbf{y}^{k}=\big{[}\begin{array}{cccc}y^{k}[0],&y^{k}[1],& \cdots,&y^{k}[N-1]\end{array}\big{]}^{T},\] \[\mathbf{x}^{k}=\big{[}\begin{array}{cccc}x^{k}[0],&x^{k}[1],& \cdots,&x^{k}[N-1]\end{array}\big{]}^{T},\] \[\mathbf{w}^{k}=\big{[}\begin{array}{cccc}w^{k}[0],&w^{k}[1],& \cdots,&w^{k}[N-1]\end{array}\big{]}^{T},\]  
additionally, \(\mathbf{H}^{k}\) is the channel matrix of dimension \(N\times N\) whose elements are formed by the coefficients of the CIR using  
\[\Big{[}\mathbf{H}^{k}\Big{]}_{n,n^{\prime}}=h^{k}\big{[}(n,\langle n-n^{\prime }\rangle)_{N}\big{]}. \tag{4}\]  
The received OFDM symbol in the frequency domain (FD) is obtained by multiplying both sides of Equation (3) by the normalized discrete Fourier transform (DFT) matrix defined as:  
\[[\mathbf{F}]_{n,n^{\prime}}=\frac{1}{\sqrt{N}}e^{(-j2\pi nn^{\prime}/N)}, \tag{5}\]  
obtaining the following product equation:  
\[\mathbf{u}^{k}=\mathbf{F}\mathbf{H}^{k}\mathbf{x}^{k}+\mathbf{z}^{k}, \tag{6}\]  
where \(\mathbf{u}^{k}\) is the OFDM symbol in the FD and \(\mathbf{z}^{k}\) is the DFT of the noise vector. Since the matrix \(\mathbf{F}\) is unitary, Equation (6) can be rewritten as\[\mathbf{u}^{k} = \mathbf{FH}^{k}\mathbf{F}^{H}\mathbf{F}\mathbf{x}^{k}+\mathbf{z}^{k}, \tag{7}\] \[= \mathbf{FH}^{k}\mathbf{F}^{H}\mathbf{g}^{k}+\mathbf{z}^{k},\] \[= \mathbf{G}^{k}\mathbf{g}^{k}+\mathbf{z}^{k}, \tag{8}\]  
where \(\mathbf{g}^{k}\) is the DFT of the data vector and \(\mathbf{G}^{k}=\mathbf{FH}^{k}\mathbf{F}^{H}\) is the channel matrix in the frequency domain (CMF). Suppose the CIR is time-varying. This time-varying causes the matrix \(\mathbf{G}^{k}\) to be non-diagonal, translated, and interpreted as a system with ICI.