## 3 Mathematical Models
### UFMC System Model
Then, when conducting the 2N-DFT after that point and while selecting every other subcarrier, which is down-sampling, UFMC  
Figure 4: Universal Filtered Multicarrier (UFMC) baseband transceiver. can achieve M-QAM de-mapping to collect the received data, as shown in Figure 4. At this stage, PAPR can be formulated as follows  
\[PAPR=10\log_{10}\left[\frac{\max\left|x(k)\right|^{2}}{E\left[\left|x(k)\right|^{ 2}\right]}\right] \tag{7}\]  
where \(\text{E}[|x(k)|^{2}]\) is the expectation operation, which gives the average power of the signal \(x(k)\). The HPA reaches the saturation region if high peaks are fed to it, thus it causes degradation in the system. Therefore, such high peaks have to be reduced. Recently, it is found that the PAPR metric alone is not enough to predict the HPA power de-rating, thus the CM is proposed as a sporting metric in the 3 GPP [41]. \[CM=\frac{20\log_{10}\left\{rms\left[x_{norm}^{3}(k)\right]\right\}-20\log_{10} \left\{rms\left[x_{ref}^{3}(k)\right]\right\}}{K}\;\;dB \tag{8}\]  
where \(20\log_{10}\left\{rms\left[x_{norm}^{3}(k)\right]\right\}\) represents the raw Cubic Metric (rCM) of the signal \(x(k)\) and \(20\log_{10}\left\{rms\left[x_{ref}^{3}(k)\right]\right\}\) is a constant reference that represents the rCM of Wide-Code Division Multiple Access (W-CDMA) voice signals [41]. The reference signal is found to be 1.25 dB [40,41]. The denominator stands for the empirical slope factor, which is determined empirically for a wide range of signals proposed for LTE applications to be 1.56. Furthermore, \(rms(x)\) can be calculated from. \[\mathrm{x}=\mathrm{W}\mathrm{\cdot}\mathrm{X} \tag{4}\]0  
and  
\[\mathrm{x}=\mathrm{W}\mathrm{\cdot}\mathrm{X} \tag{4}\]1  
where \(x\) denotes the vector form of the signal of [2,6]. That is, HPA power back-off affects the system performance, which is called power de-rating (the HPA operation is described in [40]). The HPA nonlinearity produces Adjacent Channel Leakage Ratio (ACLR), which is the ratio of mean power assigned to a certain channel with the mean power assigned to the adjacent channel. In the specifications of the LTE systems, it is found that the ACLR is 30 dB-sufficient for proper operation [46]. \[\mathrm{x}=\mathrm{W}\mathrm{\cdot}\mathrm{X} \tag{4}\]2  
where \(G_{1}\) and \(G_{2}\) are the linear and cubic gain design factors of the power amplifier. The power de-rating of a power amplifier is the amount of back-off power required to meet the ACLR specification of the transmitting system. Thus, the second term of the last expression shows the nonlinearity source in the power amplifier, which is the main source of the ACLR. Assuming the input of the HPA in the last expression has two consecutive frequencies, then  
\[\mathrm{x}=\mathrm{W}\mathrm{\cdot}\mathrm{X} \tag{4}\]3  
where \(\theta_{1}=2\pi k\frac{\theta}{N}\) and \(\theta_{2}=2\pi k\frac{\theta}{N}\), in which \(a\) & \(b\) are two adjacent frequency indices, and \(A_{1}\) & \(A_{2}\) are the two corresponding constellation points data. Then, the last expression of the two adjacent frequencies becomes  
\[\mathrm{x}=\mathrm{W}\mathrm{\cdot}\mathrm{X} \tag{4}\]4  
Expanding the cubic term yields  
\[\mathrm{x}=\mathrm{W}\mathrm{\cdot}\mathrm{X} \tag{4}\]5where the first term stands for the fundamental signal frequencies which is a linear region that can be inferred from the PAPR value and the second term involves the nonlinear frequencies. The terms of the last expression ignore other the effect of other terms (\(3A_{1}^{2}A_{2}e^{i(2\theta_{1}+\theta_{2})}+3A_{1}A_{2}^{2}e^{i(\theta_{1}+2 \theta_{2})}\), which make an essential motivation to introduce the ACLR [40; 41; 46]. Thus, the intermodulation is happened due to the cubic terms, which produce \(2\theta_{1}+\theta_{2}\) and \(\theta_{1}+2\theta_{2}\) frequencies. Hence  
\[