## 2 System Model MIMO-OFDM-IM
\[\mathbf{x}_{t}=\mathbf{F}^{H}\mathbf{s}_{t}, \tag{3}\]  
where \(\mathbf{F}\) is the matrix of the normalized discrete Fourier transform (DFT):  
\[[\mathbf{F}]_{n,n^{\prime}}=\frac{1}{\sqrt{N}}e^{(-j\pi nn^{\prime}/N)}, \tag{4}\]  
The complex baseband model of the received signal at the \(r\)-th receiving antenna at time instant \(n\) can be expressed as follows:  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{\(b\) Input Bits \(\mathbf{a}=[a_{0}\quad a_{1}\quad a_{2}\quad a_{3}]\)}} & \multicolumn{4}{c}{\(\mathbf{X}_{C}=2,M=4\)} \\ \cline{2-6}  & & & **Carrier** & **Transmit** & **Sub-Blocks** \\ \cline{1-1}  & & & **Index** & **Symbol** & \\ \hline \([0\quad 0\quad 0\quad 0]\) & & \(\{0,0\}\) & \(+1+j\) & \([+1+j\quad 0]^{T}\) \\ \([0\quad 0\quad 0\quad 1\quad 1]\) & & \(\{0,0\}\) & \(-1+j\) & \([-1+j\quad 0]^{T}\) \\ \([0\quad 0\quad 1\quad 0\quad 1\quad 0]\) & & \(\{0,0\}\) &\[y_{r}[n]=\sum_{t=0}^{N_{t}-1}\sum_{l=0}^{L-1}h[r,t;n,l]x_{t}[\langle n-l\rangle_{N }]+w_{r}[n], \tag{5}\]  
where \(n=\{0,1,\ldots,N-1\}\), \(r=\{0,1,\ldots,N_{r}-1\}\), \(x_{t}[n]\) represents the signal obtained from the \(t\)-th transmitting antenna, \(L\) denotes the number of channel taps, and \(w_{r}[n]\) represents the circular complex delta correlation function of the additive white Gaussian noise on the \(r\)-th receiver antenna. We can express Equation (5) in a matrix-vector form as shown below:  
\[\mathbf{y}_{r}=\sum_{t=0}^{N_{t}-1}\mathbf{H}(t,r)\mathbf{x}_{t}+\mathbf{w}_{r} \tag{6}\]  
where  
\[\mathbf{y}_{r} =\big{[} \begin{array}{cccc}y_{r}[0]&y_{r}[1]&\cdots&y_{r}[N-1] \end{array} \big{]}^{T},\] \[\mathbf{x}_{t} =\big{[} \begin{array}{cccc}x_{t}[0]&x_{t}[1]&\cdots&x_{t}[N-1] \end{array} \big{]}^{T},\] \[\mathbf{w}_{r} =\big{[} \begin{array}{cccc}w_{r}[0]&w_{r}[1]&\cdots&w_{r}[N-1] \end{array} \big{]}^{T},\]  
Additionally, \(\mathbf{H}(t,r)\) is the MIMO channel matrix that contains the channel impulse response (CIR) from the \(t\)-th transmitting antenna to the \(r\)-th receiving antenna, the elements of which are formed by the coefficients of the CIR as follows:  
\[\big{[}\mathbf{H}(t,r)\big{]}_{n,n^{\prime}}=h\big{[}t,r;n,\langle n-n^{\prime }\rangle_{N}\big{]}, \tag{7}\]  
Assuming that \(n,n^{\prime}=\{0,1,\ldots,N-1\}\), and the channel impulse response (CIR) is assumed to be zero for \(\langle n-n^{\prime}\rangle_{N}>L-1\), the received OFDM symbol in the frequency domain (FD) can be obtained by multiplying both sides of Equation (6) by the matrix \(\mathbf{F}\). This operation yields the following result:  
\[\mathbf{u}_{r}=\mathbf{F}\sum_{t=0}^{N_{t}-1}\mathbf{H}(t,r)\mathbf{x}_{t}+ \mathbf{z}_{r}, \tag{8}\]  
where \(\mathbf{u}_{r}\) is the DFT of the \(\mathbf{y}_{r}\) and \(\mathbf{z}_{r}\) is the DFT of the noise sequence. Since matrix \(\mathbf{F}\) is unitary, (8) can be expressed as:  
\[\mathbf{u}_{r}=\sum_{t=0}^{N_{t}-1}\mathbf{G}(t,r)\mathbf{s}_{t}+\mathbf{z}_{ r}, \tag{9}\]  
In the described system, \(\mathbf{s}_{t}=\mathbf{F}\mathbf{x}_{t}\) represents the OFDM symbol transmitted from the \(t\)-th transmitter antenna in the frequency domain. The matrix \(\mathbf{G}=\mathbf{F}\mathbf{H}\mathbf{F}^{H}\) is the channel frequency matrix (CFM). When there is insignificant Doppler propagation, \(\mathbf{G}\) becomes a diagonal matrix, resulting in an inter-carrier interference (ICI)-free system. However, in V2V (vehicle-to-vehicle) environments, where both the transmitter and the receiver experience high mobility, the Doppler dispersion becomes significant. As a result, the matrix \(\mathbf{G}\) contains energy in components outside the diagonal, leading to the presence of ICI. Figure 1 illustrates the block diagram of the proposed MIMO-OFDM-IM (multiple-input-multiple-output orthogonal frequency division multiplexing with index modulation) transmitter. This design integrates index modulation (IM) into a conventional MIMO-OFDM system.