# A Partitioned IRS-Aided Transmit SM Scheme for Wireless Communication
## 4 Detection Complexity Analysis  
In this section, we compare the computational complexity of PIRS-TSSK, PIRS-TSSSK, and the proposed PIRS-TSM scheme during ML detection, as shown in Table 1. In addition, we make a detection complexity analysis of the PIRS-TSM scheme with different numbers of transmit antennas \(N_{t}\) and modulation orders \(M\) at the same SE. The computational complexity is investigated with the number of real multiplications (RMs) and real additions (RAs) in ML detection. First, for the PIRS-TSSK scheme, the computational complexity is derived from the expression of ML detection ([20], Equation (14)). There is one group of reflecting elements that completely cancel out the channel phases, resulting in \(\frac{N}{N_{t}}\) RM terms: \(\alpha_{t,i}\beta_{i}\). Meanwhile, the remaining \((N_{t}-1)\) groups of reflecting elements cannot completely cancel out the channel phases, resulting in \((N_{t}-1)\frac{N}{N_{t}}\) multiplications of three complex numbers: \(h_{t,i}e^{i\phi_{t,i}}S_{i,i}\), each of which requires eight RMs and four RAs. Similarly, the computational complexity of PIRS-TSSSK is analyzed from the ML detection expression ([22], Equation (3)). In SSSK here, we assume that \(C=2^{a}\) combinations are utilized, with defining \(a=\left\lfloor\log_{2}\binom{N_{t}}{N_{t}}\right\rfloor\), \(\mathrm{SE}=a\ \mathrm{bit}/\mathrm{s}/\mathrm{Hz}\).  
On the other hand, the computational complexity of the proposed PIRS-TSM scheme is derived from the ML detection expression Equation (3), which has an additional product term of complex number \(x\), resulting in an added complex multiplication for each reflecting element. Then, it can be obtained that each of the reflecting element that perfectly cancels out the channel phases requires three RMs, while each of the remaining reflecting elementsthat cannot completely cancel out the channel phases requires four complex multiplications, each of which requires twelve RMs and six RAs.  
Table 1 summarizes the RMx and RA formulas for the PIRS-TSSK, PIRS-TGSSK, and PIRS-TSM schemes, demonstrating that the computational complexity primarily depends on the values of \(N_{t}\) and \(M\) under the same number of reflecting elements \(N\). Moreover, a computational complexity comparison bar is provided in Figure 2 for an example of SE = 4 bit/s/Hz and \(N=128\). The PIRS-TSM scheme is configured as three scenarios: \(N_{t}=8\), \(M=2\); \(N_{t}=4\), \(M=4\); and \(N_{t}=2\), \(M=8\). The computational complexity of PIRS-TGSSK is lower than that of PIRS-TSSK under the same SE because PIRS-TGSSK requires fewer \(N_{t}\). Among the three configuration scenarios of PIRS-TSM (the three green bars), the configuration with \(N_{t}\) = 2 and \(M=8\) has the lowest computational complexity. It can be seen that under the same SE, when \(N_{t}\) is smaller and \(M\) is larger, the computational complexity of PIRS-TSM is lower. This is because fewer transmit antennas implies fewer partitions on the IRS, resulting in lower computational complexity. When compared with PIRS-TSSK, PIRS-TSM with the configuration of \(N_{t}=2\) and \(M=8\) has a lower computational complexity. However, the computational complexities of the configurations with \(N_{t}=4\) and \(M=4\), and \(N_{t}=8\) and \(M=2\), are both higher than that of PIRS-TSSK. It is worth mentioning that the BER performance of PIRS-TSM with \(N_{t}=2\) and \(M=8\) is also optimal, which will be illustrated in the next section.