## 3 The Generalized AP-HSIC Algorithm for QAM Modulations
\[=\left\|Y\big{(}I-S^{+}S\big{)}\right\|_{F}^{2}+\left\|\big{(}YS^ {+}-(H+\Delta H)\big{)}S\right\|_{F}^{2},\]5 Average number of iterations for each average SNR value of AP-HSIC under 16-QAM modulation in the presence of Rayleigh deep-fading and high-level AWGN. performance of the LM-HSIC in 16-, 32-, and 64-QAM constellation order, respectively. It can be seen from Figure 7 that both the AP-HSIC and LM-HSIC attain comparable low BER values for each modulation. This means that both the AP-HSIC algorithm and the LM-HSIC algorithm converge in all three scenarios to a BER performance level that enables proper and accurate decoding. However, it is important to note that the AP-HSIC algorithm outperforms the LM-HSIC algorithm in terms of BER performance and achieves higher diversity gains across all three constellation orders. Furthermore, the graph of LM-HSIC shifts to the right relative to the graph of AP-HSIC, and this shift diminishes as the SNR increases. That is, both the AP-HSIC algorithm and the LM-HSIC algorithm converge in all three constellations to a BER performance level that enables a proper and accurate decoding process. However, it can be noticed that the BER performance of the AP-HSIC algorithm is better than the BER performance of LM-HSIC, and it achieves a higher diversity gain in all three constellation orders. Figure 8 illustrates the number of iterations of both algorithms for each SNR value under 16-QAM modulation. We observe that the number of iterations of LM-HSIC remains constant throughout the graph, even when the SNR value is low. In contrast, AP-HSIC's number of iterations increases as the SNR decreases. Hence, we conclude that the LM-HSIC algorithm runs faster than the AP-HSIC algorithm. However, this phenomenon is related to the LM-HSIC algorithm's local convergence against the AP-HSIC algorithm's global convergence. In summary, in simulations with each of the modulations, the LM-HSIC algorithm has a better diversity gain than AP-HSIC, but the AP-HSIC's performance is better. In addition, the LM-HSIC algorithm runs faster than AP-HSIC. \[=\left\|Y\big{(}I-S^{+}S\big{)}\right\|_{F}^{2}+\left\|\big{(}YS^ {+}-(H+\Delta H)\big{)}S\right\|_{F}^{2},\]6 Comparisons of an average number of iterations for each average SNR value between LM-HSIC and AP-HSIC under 16-QAM modulation in the presence of Rayleigh deep-fading. \[=\left\|Y\big{(}I-S^{+}S\big{)}\right\|_{F}^{2}+\left\|\big{(}YS^ {+}-(H+\Delta H)\big{)}S\right\|_{F}^{2},\]7 Plot graphs of BER performance vs. the average SNR for three systematic iterations using the LM-HSIC algorithm and AP-HSIC algorithm. Each iteration is represented by 16-QAM, 32-QAM, and 64-QAM modulations, respectively. The performance was compared under the presence of Rayleigh deep-fading and high-level AWGN. In Figure 9, we present comparisons between the BER performances of AP-HSIC under the scenarios of Rayleigh fading in the Single-Input Single-Output (SISO) case with 16-, 32-, 64-QAM to the theoretical 16-, 32-, and 64-QAM SISO AWGN only. We also present comparisons between the Rayleigh fading SISO case scenarios with 16-, 32-, and 64-QAM and no error correction algorithm. It is clear from Figure 9 that the numerical result of the AP-HSIC algorithm is situated between the trade-off region of the theoretical graphs to the system with no error correction algorithm. These results illustrate the reliability of the AP-HSIC algorithm and the decoding improvement capabilities in the parallel decoding mode under the assumptions of Rayleigh fading, including AWGN. ## 6 Discussion, Conclusions, and Future Directions

Dealing with interference offset demands precise spatial selectivity and accurate channel estimation through channel sharing and receiver-transmitter feedback. An intelligent algorithm upgrade for the receiver is crucial for achieving high interference cancellation capabilities and enabling cognitive and collaborative radio communications.