## 1 Introduction
Taking the expectation on both sides of (7), we obtain the following:  
\[E[Y]=E[H+\Delta H]E[S], \tag{8}\]  
since, by assumption, \(E[Z]=0\) and since \(S\) and \(H+\Delta H\) are assumed to be statistically independent. In the sequel, we will therefore write the following:  
\[Y=(H+\Delta H)S, \tag{9}\]  
for convenience. To demonstrate the weakness in the MGSTC serial decoding algorithm in the presence of the Rayleigh deep-fading phenomenon, the equation that presents the first iteration [7], where \(\widetilde{Y}_{c_{1}}\) is the received signal matrix for decoding the first symbol block \(S_{c_{1}}\), is as follows:  
\[\widetilde{Y}_{c_{1}} =\Theta_{c-c_{1}}Y=\Theta_{c-c_{1}}\Bigg{[}\sqrt{P}HS+\sqrt{P} \Delta HS+\sqrt{\frac{P}{SNR}}Z\Bigg{]} \tag{10}\] \[=\sqrt{P}\Theta_{c-c_{1}}H_{c_{1}}S_{c_{1}}+\sqrt{P}\Theta_{c-c_ {1}}\Delta HS+\sqrt{\frac{P}{SNR}}\widetilde{Z}_{c-c_{1}},\]  
where \(H=\left[ \begin{array}{cccc}H_{c_{1}}&H_{c_{2}}&\cdots&H_{c_{l}}\end{array} \right]\) and \(\Theta_{c-c_{1}}\) is the null-space matrix relative to the decoding process of \(S_{c_{1}}\), that is \(\Theta_{c-c_{1}}\left[ \begin{array}{cccc}H_{c_{2}}&\cdots&H_{c_{l}}\end{array}  \right]=0\), and \(\widetilde{Z}_{c-c_{1}}\) is the independent complex Gaussian random variable noise multiplied with the null-space matrix \(\Theta_{c-c_{1}}\). The next step of MGSTC is to assume that  
\[\widetilde{Y}_{c_{1}}=\sqrt{P}H_{c_{1}}^{(eff)}S_{c_{1}}+\sqrt{\frac{P}{SNR}} \widetilde{Z}_{c-c_{1}}\]  
where \(H_{c_{1}}^{(eff)}:=\Theta_{c-c_{1}}H_{c_{1}}\), and then use it in order to compute the Maximum Likelihood (ML) approximation \(\widetilde{S}_{c_{1}}\) for \(S_{c_{1}}\) as  
\[\widetilde{S}_{c_{1}}=\frac{1}{\sqrt{P}}E\bigg{[}\bigg{(}H_{c_{1}}^{(eff)} \bigg{)}^{+}\widetilde{Y}_{c_{1}}\bigg{]}.\]  
Obviously, the computation of \(\widetilde{S}_{c_{1}}\) from the very first step contains errors since it does not take into account the part \(\sqrt{P}\Theta_{c-c_{1}}\Delta HS\). Note that the probability that \(\Theta_{c-c_{1}}\Delta H=0\) is negligible. The interference part \(\sqrt{P}\Theta_{c-c_{1}}\Delta HS\) that is received as an additive impact to the MGSTC decoding algorithm produces the most destructive effects, as it destroys the orthogonality of \(\Theta_{c-c_{1}}\) relative to the other part of the MIMO channel-response matrix, a destruction that the MGSTC algorithm cannot deal with (see [11]). Moreover, the classical solution of increasing the transmission power \(P\)[3; 45] also increases the interference part and thus makes it a bad solution. In this paper, we present a new method called the AP-HSIC, the generalized AP-HSIC, to deal with problems associated with interference estimation and digital interference cancellation in the presence of Rayleigh deep-fading effects. Our algorithm utilizes self-computational feedback at the receiver to decode any value of \(M\) in an M-QAM under challenging conditions of Rayleigh deep-fading channels. Additionally, we provide BER analysis and results for 16-, 32-, and 64-QAM for better understanding. To calculate \(\Delta H\) and \(S\) from Equation (6), we can initially use the standard model to estimate an approximate value for \(S\) based on the known channel-response matrix \(H\). We then iterate Equation (6) to find the best approximations for \(\Delta H\) and \(S\), as described in Section 3. Below, we provide computational feedback detailing how to nullify \(\Delta H\) by computing \(\Delta H\) and utilizing it to accurately decode \(S\) and to update the new MIMO channel-response matrix to be \(H+\Delta H\) without having to update the transmitter.