## 2 Materials and Methods
### Multiuser MIMO-OFDM System
#### 2.1.1 System Model  
We assumed a frequency-selective Rayleigh fading multipath channel between the \(i_{TR}{}^{th}\) sender side and \(i_{RE}{}^{th}\) receiver side. The BS has \(N_{R}\) receiver antennae with \(N_{I}\) users. Every antenna has a single omni-directional antenna. The number of antennae \(L_{A}\) used is greater than the number of users \(I\). The signal along with the additive white Gaussian noise (AWGN) and MAI are considered as the inputs. Let us consider the stream of bits \(B^{i}\)\(\forall\)\(1\leq i\leq I\) for \(I\) users. The number of transmitting antennae for \(I\) users is denoted as \(Q_{x}\), where \(1\leq q_{x}\leq Q_{x}\). Therefore, the input stream can be represented as \(B_{q_{x}}\), where \(q_{x}\) is the transmitting antenna. First, the stream of bits can be encoded using the turbo encoder \(T\). After encoding, the bitstream can be represented by \(B_{q_{x}}^{T}\), which can be passed through a pseudo-random interleaver. This interleaver is usually based on the block interleaving scheme, which is used with turbo codes to scramble the input sequence. The output of the interleaver \(I\) can be represented as \(B_{q_{x}}^{I}\). The output stream from the interleaver is collected as blocks of log\({}_{2}\)M bits. After this, the blocks are modulated as M-ary quadrature amplitude modulation (M-QAM) symbols. The modulated output can be denoted as \(\widetilde{X}_{q_{x}}\). That signal can be converted into parallel bits using the serial-parallel (S/P) converter. We assumed that the OFDM system has \(N\) subcarriers. After this, the OFDM symbols \(X_{q_{x}}[n,k]\) are obtained after embedding the pilot symbols. The transmitted bit stream of data is comprised of user information and pilot symbols. The transmitted user data is represented as \(B_{q_{x}}(n)\), \(n\in\Omega\), \(\Omega=1,2,\cdots N\) and the pilot symbols are denoted as, \(B_{q_{x}}(n)\), where \(n\in\Psi_{x}\) and \(\Psi_{x}=k_{x,1},k_{x,2},\cdots k_{x,p}\). In this case, the subcarrier \(K\) will be in the range \(1\leq k\leq K\), \(P\) subcarriers are assigned for pilot symbols and \(n\) represents the OFDM symbol index. The receiver knows about the pilot symbols of the frequency domain (FD) and its channel allocation for the initial CE. After this, the FD modulated signal can be converted into the time domain (TD) modulated signal \(x_{q_{x}}[n,\ k]\) in the \(K\)-point IFFT. The resultant sequence of bits is transmitted to the receiver side through the MIMO channel after inserting the cyclic prefix (CP) of \(K_{cp}\) samples. The length of CP must be chosen as \(K_{cp}\geq L_{CIR}\), where \(L_{CIR}\) represents the length of \(CIR\). When transmitting the signal through the channel, the signal is contaminated with the AWGN and interference channel. The received signal can be represented as \(y_{q_{x}}\). The number of receiving antennae for the \(j\)-th output is represented as \(Q_{R}\), where \(1\leq q_{r}\leq Q_{R}\). The signal is received in serial form and the CPs are discarded from each OFDM symbol. Following this, it is fed into the \(K\)-point FFT for FD representation at the receiver side. \(Y^{j}[n,\ k]\) is the received signal for the \(k\)-th subcarrier in the \(q_{r}^{th}\) receiver. The antenna element of \(n\)-th OFDM symbol is given by:  
\[Y_{q_{r}}[n,\ k]=\sum_{i=1}^{I}X_{q_{x}}[n,k]\ H_{q_{r}}^{i}[n,\ k]+W_{q_{r}}[n, \ k], \tag{1}\]  
where during the \(s\)-th OFDM symbol period, \(w_{q}[n,k]\) is AWGN, while \(H_{q}^{i}[n,k]\) represents the \(K\)-sparse channel impulse vector and FD channel transfer function of \(i\)-th user and the \(q_{r}^{th}\) receiver antenna element, which holds the CIR coefficients.