## 3 Mathematical Models
### UFMC Based SLM System  
The SLM algorithm is one of the efficient frequency-domain approaches to reduce the PAPR in OFDM waveform. Basically, the algorithm applies a mechanism to make \(B\) copies of the M-QAM modulated data vector, multiply elementwise, phase rotation vectors [32]. Based on the colored blocks in Figure 4, after phase rotation vector implementation, the X in (6) takes the form  
\[X_{r}^{b}=p^{b}\times X_{r} \tag{16}\]  
where  
\[X_{r}^{b}=\left[\begin{array}{c}X_{r,0}^{b}\\ X_{r,1}^{b}\\ \vdots\\ X_{r,N-1}^{b}\end{array}\right],\;\;p^{b}=\left[\begin{array}{cccc}p_{0}^{b }&0&\cdots&0\\ 0&p_{1}^{b}&\cdots&0\\ \vdots&\vdots&\ddots&0\\ 0&0&\cdots&p_{N-1}^{b}\end{array}\right],\;\;X_{r}=\left[\begin{array}{c}X _{r,0}\\ X_{r,1}\\ \vdots\\ X_{r,N-1}\end{array}\right] \tag{17}\]  
and b stands for the phase rotation vector p index. Then, the UFMC-based SLM can be expressed as  
\[x^{b}=\sum\limits_{r=1}^{R}\mathrm{F}_{r}\cdot\mathrm{W}_{r}\left[p^{b}\times X _{r}\right] \tag{18}\]  
or  
\[x^{b}=\sum\limits_{r=1}^{R}\mathrm{F}_{r}\cdot\mathrm{W}_{r}\cdot X_{r}^{b} \tag{19}\]  
In other words, UFMC-based SLM now produces B candidates, the candidate who experiences lower PAPR will be adopted for transmission. The QAM-Mapper produces data then copies the data multiple times to phase-rotate it. In the above scenario, shown in Equation (17), the computational complexity can be calculated as that of the conventional CP-OFDM. For instance, the number of complex multiplication operations of the CP-OFDM matches that of the UFMC, as well as the number of complex addition operations [47]. Hence, the number of complex multiplication operations \(\mu\) can be presented as Equation (20)  
\[\mu=\frac{N}{2}\log_{2}{(N)} \tag{20}\]where \(N\) refers to the number of samples, then according to [47], the number of complex addition operations can be formed as in Equation (21)  
\[\alpha=N\log_{2}\left(N\right) \tag{21}\]  
After implementing the PAPR/CM reduction approach, which is represented by the SLM scheme, the number of complex multiplication operations will be increased according to the number of candidates, B,  
\[\mu_{SLM}=B\frac{N}{2}\log_{2}\left(N\right) \tag{22}\]  
The \(\alpha_{SLM}\), calculated by multiplying the number of complex addition operations (as shown in Equation (21)) by \(B\), in which the \(B\) refers to the total number of SLM candidates:  
\[\alpha_{SLM}=BN\log_{2}\left(N\right) \tag{23}\]  
On the other hand, the SLM scheme is the probabilistic method, hence the BER is not affected. Although the computational complexity has been increased, the achieved outcome is of great interest. Moreover, SLM needs to transmit some side information to the receiver, such that the data can be recovered. The amount of side information is a few bits, which can be calculated as Equation (24):  
\[\delta=log_{2}(B) \tag{24}\]  
In other words, decreasing the number of candidates decreases the reserved bits as side information and reduces the computational complexity at the same time. Hence, it is worth controlling the number of candidates in order to obtain improved results. Thus, there is a trade-off between the reduction performance and the computational complexity. These facts are validated by conducting the mathematical simulations, and their outcomes are presented in Section 3.