## 3 Mathematical Models  
### UFMC System Model  
In 5G, UFMC is a novel multicarrier modulation of QAM type that represents an alternative to the OFDM and FBMC waveforms. Unlike self-subcarrier modulation in FBMC, a group of subcarrier modulation is performed in UFMC. The subcarrier grouping reduces the length of the filter compared with FBMC and also reduces the performance time. Starting from the traditional OFDM system, UFMC can be considered as a generalized version of the OFDM. Let the OFDM be formulated based on Equation (3)  
\[x(k)=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}X(n)e^{j2\pi\frac{k}{N}} \tag{3}\]  
where \(x\) is the OFDM signals that represent an IDFT of size \(N\); \(k\) stands for the time index; \(n\) represents the frequency-domain index; X set is generated based on quadrature multi-level amplitude modulation (M-QAM) mapping. The X set contains Identically Independent Distributed (IID) random variables. The summation in Equation (1) is either constructive or destructive. Thus, the coherence summation of data will result in a large peak compared with the average, which is called the peak to average power ratio. The CP-OFDM (i.e., cyclic-prefix free) that is adopted in LTE can be formed by adding the filter to the whole band, as in Equation (1). Executing the filter in Equation (4) gives the CP-OFDM  
\[\mathrm{x}=\mathrm{W}\mathrm{\cdot}\mathrm{X} \tag{4}\]  
where W is the IDFT matrix of size \(N\times N\), where \(N\) is the number of subcarriers. Subsequently, we formed Equation (5)  
\[\mathrm{x}=\mathrm{F}\mathrm{\cdot}\mathrm{W}\mathrm{\cdot}\mathrm{X} \tag{5}\]  
Figure 3: Simulation design. where F stands for the Finite Impulse Response (FIR) filter toeplitz matrix, which achieves the convolution operation over the band of \(N\). If the whole band subdivided to \(R\) sub-bands, then, the UFMC system can be expressed as Equation (6)  
\[\mathrm{x}=\sum_{r=1}^{R}\mathrm{F}_{r}\cdot\mathrm{W}_{r}\cdot\mathrm{X}_{r} \tag{6}\]  
In Equation (6), the M-QAM symbols entries of \(\mathrm{X}_{r}\) are first converted to the time-domain according to the corresponding column of IDFT matrix \(\mathrm{W}_{r}\) which corresponds to the specified sub-band \(r\) within the frequency band. \(\mathrm{F}_{r}\) is the corresponding sub-band filter. Figure 4 depicts the complete UFMC system that is derived from Equation (6). In the receiver end, the process starts at the zero-padding to achieve the two times up-sampling. Then, conducting the 2N-DFT, the subcarriers (down-sampling) of a juncture are selected, until the M-QAM de-mapping is implemented to collect the received data, as shown in Figure 4. The next section illustrates the integration of the selected mapping method with the UFMC waveform. Based on Figure 4, the filtering operation has a block-wise form of Physical Resource Block (PRB) to facilitate the flexibility of the design. The filter length has an essential role in the design of the system in which the filter length, \(\mathrm{P}\gg 1\) and \(\mathrm{X}\) contain one M-QAM symbol. As long as the filter length in the frequency-domain is long, the time-domain tail is going to be shorter (i.e., in the order of cyclic prefix length of the traditional CP-OFDM), which enables the network to support Transmits Short Messages (TSM). Additionally, the elimination of the sidelobes is conducted at the ends of the sub-bands not between the subcarriers, as in the case of the FBMC signal structure. The ramping up/down in the time-domain gives more shields to prevent inter-symbol interference (ISI). Unlike the FBMC, the UFMC keeps the complex orthogonality, thus the Multiple-Input-Multiple-Output (MIMO) mechanism of multiple users is efficiently implemented in the UFMC. On the other hand, the UFMC is more efficient than CP-OFDM in terms of spectrum construct, due to the UFMC being CP-free. At the receiving end, the process starts at the zero paddings to achieve a two times up-sampling.