## 3 The Generalized AP-HSIC Algorithm for QAM Modulations
This means that the algorithm will work best when the sent information is distributed between the radii of the QAM modulation points so that the radius of the sent signal \(\frac{\|S_{*}\|_{F}}{\sqrt{nk}}\) is close to the expected radius \(r_{av}=\mathbb{E}\left[\frac{\|S\|_{F}}{\sqrt{nk}}\right]\), and the farther the radius is from the expected radius, the more errors the result will contain. Note, however, that \(S_{*}=s_{0}\mathbb{I}_{n\times k}\) and other \"corner\" points of \(QAM^{n\times k}\) are very rare, if not impossible, in real-world applications, and can be avoided by protocols [7]. ```
0: An algorithm for computing Moore-Penrose Pseudo-inverses, and an algorithm for computing \(P_{QAM}\). 0:\(H,Y,\epsilon>0,r_{av}=\mathbb{E}\left[\frac{\|S\|_{F}}{\sqrt{nk}}\right],n,k\). 0:\(\Delta H\) and \(S\) such that \(Y=(H+\Delta H)S\), where \(S\in QAM^{n\times k}\). 1.:\(\Delta H_{0}\gets 0\)
2.:\(S_{0}\gets H^{+}Y\)
3.:\(S_{0}\leftarrow\frac{S_{0}}{\|S_{0}\|_{F}}\cdot r_{av}\sqrt{nk}\)
4.:\(\Delta H_{1}\gets YS_{0}^{+}-H\)
5.:\(S_{1}\leftarrow(H+\Delta H_{1})^{+}Y\)
6.:\(S_{1}\leftarrow\frac{S_{1}}{\|S_{1}\|_{F}}\cdot r_{av}\sqrt{nk}\)
7.:\(t\gets 1\)
8:while\((\left\|\Delta H_{t}-\Delta H_{t-1}\right\|_{F}>\epsilon)\vee(\left\|S_{t}-S_{ t-1}\right\|_{F}>\epsilon)\)do
9.:\(t\gets t+1\)
10.:\(\Delta H_{t}\gets YS_{t-1}^{+}-H\)
11.:\(S_{t}\leftarrow(H+\Delta H_{t})^{+}Y\)
12.:\(S_{t}\leftarrow\frac{S_{t}}{\|S_{t}\|_{F}}\cdot r_{av}\sqrt{nk}\)
13.:endwhile
14.:\(\Delta H\leftarrow\Delta H_{t}\)
15.:\(S\gets P_{QAM}(S_{t})\)
16.:return\(t,\Delta H,S\)
```

**Algorithm 4** LM-HSIC: Receiver Self-Feedback Algorithm For QAM

## 5 Simulations and Numerical Results

In order to demonstrate the effectiveness of the proposed AP-HSIC method and assess the performance of LM-HSIC, we conducted simulations of a communication environment using

\[=\left\|Y\big{(}I-S^{+}S\big{)}\right\|_{F}^{2}+\left\|\big{(}YS^ {+}-(H+\Delta H)\big{)}S\right\|_{F}^{2},\]0 LM-HSIC algorithm flowchart. both the MATLAB and SIMULINK platforms. These simulations were based on three different methods described in Sections 2-4 with the same conditions and challenging scenarios. We compared the performance of these simulations using graphical and numerical aspects. Each simulation involves multiple stages that simulate various disturbance scenarios. We used the two architectures from the previous article [56] (with an extension to 16-, 32-, and 64-QAM modulation) and created a new architecture for LM-HSIC, as described below. To enhance the architectures' efficiency, the standard MGSTC transmitter encoder for all configurations integrates three OSTBC components. Each element is equipped with two transmission antennas, resulting in a total of six transmission antennas (\(N_{t}=6\)). On the receiving side, the signal matrix \(Y\) is captured through six receiving antennas (\(N_{r}=6\)). Furthermore, alongside the estimated MIMO channel matrix, the received signal is directed through the MGSTC decoding algorithm in the first architecture, the AP-HSIC algorithm in the second architecture, and the LM-HSIC algorithm in the third architecture. In the first architecture, the decoding process at the receiver is based on three Maximal-Ratio Combiners (MRC) components. We developed the environment of Rayleigh deep-fading using lotteries of the low values of the Rayleigh envelope (below 0.3), and we characterized the PDF of the SNR-Rayleigh distribution as a function of the SNR [58; 59]:

\[=\left\|Y\big{(}I-S^{+}S\big{)}\right\|_{F}^{2}+\left\|\big{(}YS^ {+}-(H+\Delta H)\big{)}S\right\|_{F}^{2},\]1

In addition, we ran the simulations based on the average SNR (\(\overline{SNR}\)). In the first simulation, we used an MGSTC transmitter comprising an information generator block, symbol modulation block, and OSTBC-encoder blocks. The MIMO channel is quasi-static flat Rayleigh deep-fading and can change the AWGN intensity parameter. The receiver blocks consist of an MGSTC decoding algorithm and MRC blocks.