## 1 Introduction
Any errors accumulated in a given SIC process are repeated to the next iteration and amplified. Another notable weakness of the MGSTC is the wireless links between the antenna pairs in the descending order of the transmitter's antenna array (e.g., antennas 5 and 6 in our simulation) to all antenna arrays of the receiver. These paths suffer more attenuation regarding the SNR or Signal-Interference Ratio (SIR) and rely on diversity gain. Consequently, when facing interfering signals, deep-fading effects, or high levels of AWGN, an imbalance in the trade-off between higher diversity gain and a lower SNR occurs, leading to increased values of the BER. The second method we propose is divided into two sub-approaches. The first sub-approach uses the MGSTC-OSTBC at the transmitter and the AP-HSIC decoding algorithm at the receiver instead of the MGSTC decoding algorithm in the first method. In the second sub-approach, we use the MGSTC-OSTBC at the transmitter and the LM-HSIC at the receiver. This hybrid scheme provides immunity against various interference scenarios, as shown in the simulation results below and in [56]. Additionally, it achieves a more efficient and accurate process of parallel spatial decoding of the series of simultaneously transmitted symbol blocks compared to the first scheme we described. We separately analyze and simulate the two methods under challenging interference Rayleigh deep-fading scenarios in the same spatial domain. It is important to emphasize that in both methods, we simultaneously transmit all the transmission sub-blocks, \(S=\left[S_{c_{1}}^{T},S_{c_{2}}^{T},...,S_{c_{l}}^{T}\right]^{T}\), in parallel. However, in the first MGSTC method, the MGSTC decoding algorithm decodes the sub-blocks serially, while in the second method that we propose, the receiver, based on the AP-HSIC and on the LM-HSIC algorithms, decodes all the sub-blocks at the same time, based on parallel iterations of offsetting the noise/interfering part in the channel, as we will present later. We start the first and essential analysis with the Rayleigh deep-fading case. In that case, the communication standard model, \(Y=\sqrt{PHS}+Z\) changes (see [4]):  
\[Y=\sqrt{P}H_{TR}S+\sqrt{\frac{P}{SNR}}Z, \tag{1}\]  
where  
\[H_{TR}=R^{\frac{1}{2}}HT^{\frac{1}{2}} \tag{2}\]  
where, essentially, the Kronecker correlation structure of the MIMO-transmission-receiver matrix is \(H_{TR}\) with dimension \([N_{r}\times N_{l}]\)[4; 55]. The elements of \(H\) are the conventional MIMO channel matrix, where \(R\) and \(T\) are the receiving and transmitting Rayleigh deep-fading self-correlation matrices. Let  
\[ \begin{cases}\Delta T=I-T^{\frac{1}{2}}\\ \Delta R=I-R^{\frac{1}{2}}\end{cases}  \tag{3}\]  
where \(\Delta T\) and \(\Delta R\) are the changes in the Rayleigh deep-fading self-correlation matrices on the transmitter and receiver sides, respectively, concerning the matrix \(H\). Next, \(Y\) is the received signal matrix, with dimensions \([N_{r}\times k]\), where \(k\) is the number of sample symbols per frame and \(S\) is the matrix of OSTBC transmission modulation symbols of the desired transmission, with dimensions \([N_{l}\times k]\). Finally, the \(SNR\) is the signal-to-noise ratio, \(P\) is the total transmission power, and \(Z\) is the independent complex Gaussian random variable noise. We can rewrite (2) by placing (3) into (2) to obtain the following:  
\[ \begin{split} H_{TR}&=(I-\Delta R)H(I-\Delta T)\\ &=H-\Delta RH-H\Delta T+\Delta RH\Delta T\\ &=H+\Delta H,\end{split}  \tag{4}\]  
where  
\[\Delta H=-\Delta RH-H\Delta T+\Delta RH\Delta T. \tag{5}\]The new form of the MIMO model is, therefore,  
\[Y=\sqrt{P}(H+\Delta H)S+\sqrt{\frac{P}{SNR}}Z. \tag{6}\]  
In the following, we would write  
\[Y=(H+\Delta H)S+Z, \tag{7}\]  
where the normalizing factors \(\sqrt{P},\sqrt{\frac{P}{SNR}}\) were suppressed into the related variable for convenience.