## 5 HRIS-DMA Novel Algorithm
As in the conventional T/R RIS, the power term is divided into two parts, one for transmission and the other for reflection, and as stated in (1), this means \(P\) can be rewritten as performed in (12). To model the signal power in a hybrid manner, the location of the user in regard to the HRIS is addressed in the power calculation; thus, (12) is expressed as it is in (13), where the value of the factors \(D_{p(trans,ref)}\), depending on the user location derived from the HRIS, will make the other signal power equal to zero. Substituting these values in (12) and (13) and extracting the power signal on the opposite side to \(P_{opposite}\) equals \((SA_{e}A_{t})\) in the case of reflection and \((SA_{e}A_{r})\) in the case of transmission. As a result of remodelling the signal power using the suggested power allocation, the power for transmission and reflection will follow (14), in which the full signal power will be allocated for either the transmitted or reflected signal simultaneously, keeping in mind the fact that the received signals, as in (9) and (10), will direct the signal for the intended user depending on \(\theta_{t}\) and \(\theta_{r}\). \[P=P_{t}+P_{r} \tag{12}\]  
\[P=(S\times\mathrm{A_{e}}\times D_{ptrans}\times A_{t})+(S\times\mathrm{A_{e}} \times D_{pref}\times A_{r}) \tag{13}\]  
\[ \begin{array}{l}P_{t}=(S\times\mathrm{A_{e}}\times D_{ptrans}\times A_{t})+P _{opposite}\\ \text{ and }\\ P_{r}=(S\times\mathrm{A_{e}}\times D_{pref}\times A_{r})+P_{opposite},\end{array}  \tag{14}\]  
where \(S=\frac{\left|E_{t}\right|^{2}}{2\pi_{air}}\), and \(A_{e}=U_{AG}\frac{\lambda^{2}}{4\pi}\), in which the signal electric field \(E_{r}\) and the normalised radiation power \(NP\) are calculated in (15) and (16), respectively. \[E_{r}=NPR_{M}\sqrt{2Z_{air}P_{s}}U_{AG}^{H}A_{HRIS}G_{HRIS} \tag{15}\]\[NP=\frac{\sqrt{NP^{BS}NPHRIS,BSNPHRIS,UNPU}}{4\pi l_{HRIS}^{BS}l_{HRIS}^{UI}} \tag{16}\]  
The proposed HRIS-DMA scheme introduces several innovations that contribute to efficient performance gains; the integration of DMA within the HRIS allows for precise and real-time control over signal reflection, significantly enhancing signal strength and coverage. The novel iterative algorithm continuously updates the HRIS configuration based on real-time user location data, optimising the signal path and reducing power loss and latency. The ability to adapt to changing network conditions and user locations ensures consistent and reliable performance, even in dynamic environments. Additionally, the proposed scheme minimises power consumption by focusing signal energy precisely where it is needed, thereby improving overall energy efficiency. In order to determine the complexity equation for the HRIS-DMA method, we will analyse each individual component and then integrate them to create a comprehensive equation as follows:  
1. Initialization Phase: complexity \(O(1)\), this phase involves setting initial parameters, which is a constant time operation. 2. User Location Detection: complexity \(O(n)\), this phase involves updating the locations of \(n\) users in real-time. 3. Signal Propagation and Reflection: complexity \(O(M)\), this phase involves adjusting \(M\) elements in the HRIS to optimize signal reflection. 4. Real-Time Adjustment: complexity \(O(n)\times O(M)\), this phase involves updating the HRIS configuration based on the real-time location data of \(n\) users and \(M\) HRIS elements. Considering the most computationally intensive part of the algorithm (real-time adjustment), the overall complexity of the HRIS-DMA is dominated by the \(O(n)\times O(M)\) term. Therefore, the complexity equation can be summarized as  
\[O=O(1)+O(n)+O(M)+O(n)\times O(M) \tag{17}\]  
Since \(O(n)\times O(M)\) is the most significant term, the overall complexity simplifies to  
\[O=O(n)\times O(M) \tag{18}\]