## 3 System Description  
Figure 1 shows the block diagram of the proposed adaptive modulation with STBC for a \(2\times 2\) MIMO-OFDM system that invokes the CAZAC-TD and CAZAC-FD SNR estimators. The input data are initially modulated using the \(M\)-PSK scheme. The modulated signal is then converted to the time domain via the Inverse-Fast Fourier Transformation (IFFT). Then, a cyclic prefix \(C_{p}\) is appended to each OFDM symbol, resulting in the OFDM signal vector, \(S=(S_{1},S_{2},S_{3},\ldots,S_{i})\), where \(i\) represents the indexes of each OFDM symbol. A block of two scalar OFDM-modulated symbols \(S_{1}\) and \(S_{2}\) is STBC-encoded in two consecutive time slots according to the following code matrix:  
\[\mathbf{G}\ =\ \begin{bmatrix}S_{1}&S_{2}\\ -{S_{2}}^{*}&{S_{1}}^{*}\end{bmatrix}  \tag{1}\]  
This matrix \(\mathbf{G}\) consists of two rows that represent two time slots and two columns that represent two antennas. In the first time slot, the first transmit antenna \(T_{x1}\) transmits symbol \(S_{1}\) while symbol \(S_{2}\) is transmitted from the second transmit antenna, \(T_{x2}\). In the second time slot, symbol \(-{S_{2}}^{*}\) is transmitted from the first transmit antenna \(T_{x1}\), and the second transmit antenna \(T_{x2}\) transmits symbol \({S_{1}}^{*}\). Here, symbols \({S_{1}}^{*}\) and \({S_{2}}^{*}\) denote the complex conjugates of \(S_{1}\) and \(S_{2}\), respectively. In STBC, the complex conjugates of symbols are utilized to form the \(\mathbf{G}\) matrix, also known as the encoding matrix, which ensures orthogonality between the signals transmitted from different antennas. This orthogonality is essential for reducing interference between antennas and enhancing the receiver's ability to recover the transmitted signals. As a result, the structure of the matrix enables accurate symbol decoding and improves the resistance of the system to fading and interference. For transmission over a Rayleigh fading channel, the received signals are impacted by the channel matrix \(\mathbf{H}\), and the received signal \(\mathbf{Y}\) matrix at each receiver antenna can be represented as:  
\[\mathbf{Y}=\mathbf{H}\mathbf{G}+\mathbf{N}\mathbf{c} \tag{2}\]  
where \(\mathbf{H}= \begin{bmatrix}H_{11}&H_{12}\\ H_{21}&H_{22}\end{bmatrix} \) is the channel matrix that models the fading effects of the channel and \(\mathbf{N}\mathbf{c}= \begin{bmatrix}N_{11}&N_{12}\\ N_{21}&N_{22}\end{bmatrix} \) is the noise matrix, which represents the noise introduced at each receiver antenna. The received signal matrix \(\mathbf{Y}\), with scalar elements \(Y_{11}\), \(Y_{21}\), \(Y_{12}\), and \(Y_{22}\), represents the received signals at the respective antennas. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Year** & **Estimation** & **Algorithm and Adaptive** & **Contribution** & **Challenges** \\
[Ref.] & **Domain** & **Criteria** &  
More specifically, at the receiver, during the first time slot, signal \(Y_{11}\) is received at the first receiver antenna, \(R_{x1}\), and \(Y_{21}\) is received at the second receiver antenna, \(R_{x2}\), as described below:  
\[Y_{11}=H_{11}S_{1}+H_{12}S_{2}+N_{11} \tag{3}\]  
\[Y_{21}=H_{21}S_{1}+H_{22}S_{2}+N_{21} \tag{4}\]  
where \(H_{11}\) and \(H_{21}\) are the coefficients of channel matrix \(\mathbf{H}\) for transmission from \(T_{x1}\) to \(R_{x1}\) and \(R_{x2}\), respectively, while \(H_{12}\) and \(H_{22}\) are the coefficients of channel matrix \(\mathbf{H}\) corresponding to the transmissions from \(T_{x2}\) to \(R_{x1}\) and \(R_{x2}\), respectively. The noise signals at \(R_{x1}\) and \(R_{x2}\) during the first time slot are denoted as \(N_{11}\) and \(N_{21}\), respectively.