# A Multi-Carrier Waveform Design for 5G and beyond Communication Systems
## 4 Numerical Results  
This section presents the study of the proposed waveform based on the analysis using computer simulations in MATLAB(r). To investigate the performance analysis, random data are generated and then modulated by using various modulation schemes such as 4-QAM, 16-QAM, 64-QAM and 256-QAM, respectively. Table 1 presents different parameters used for simulations and analysis in detail. According to the Table 1, the FFT size is set to be 512 where the number of sub-bands are equal to 10 and sub-carriers are equal to 20. Cubic-metric (CM) or PAPR analysis is one of the most important parameters to measure system performance. We selected PAPR analysis instead of CM because it is widely adopted in the literature to analyze the multicarrier waveform. SLM-based GCL precoding is employed for phase-rotation to lower the PAPR, and the Dolph-Chebyshev filter is used with the length of the filter given as \(L=73\) with stop-band attenuation 40. The minimum PAPR of the selected time domain signal given in Equation (4) can be expressed as:  
\[PAPR=\frac{\textit{peak power of }x_{i}}{\textit{average power of }x_{i}} \tag{6}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Specification** \\ \hline FFT Size & 512 \\ Sub-Band Size & 20 \\ No. of Sub-Bands & 10 \\ Sub-Band Offset & 156 \\ Filter & Dolph Chebyshev \\ Filter Length & 73 \\ Stop-Band Attenuation & 40 \\ Modulation & 4-QAM, 16-QAM, 64-QAM and 256-QAM \\ Subcarrier Mapping & Localized \\ Bits per Sub-Carriers & 2, 4, 6 \\ Sampling Frequency & 15.36 MHz \\ PAPR Reduction Scheme & SLM, GCL Precoding \\ Channel Bandwidth & 20 MHz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. The complimentary cumulative distribution function (CCDF) is commonly used to evaluate the PAPR performance in the literature. It is a measure of the probability of the signal's instantaneous power being greater than a specified level over its average power. The CCDF of the signal can be mathematically expressed as:  
\[P(PAPR>PAPR_{0}=1-(1-e^{-PAPR_{0}})^{N} \tag{7}\]  
where \(PAPR_{0}\) denotes the level of clipping. Figures 3-6 present a CCDF comparison of the PAPR for NOMA-UFMC waveform, GCL precoding-based UFMC waveform, SLM-based GCL precoded power domain UFMC waveform using \(V\) = 2, SLM-based GCL precoded power domain UFMC waveform using \(V\) = 4, and SLM-based GCL precoded power domain UFMC waveform using \(V\) = 16 for 4-QAM, 16-QAM, 64-QAM and 256-QAM, respectively. Figure 4: PAPR analysis using 16-QAM. Figure 5: PAPR analysis using 64-QAM. Table 2 presents a summary of the CCDF comparison of the PAPR analysis at clip rate of \(10^{-3}\) in Figures 3-6 by employing 4-QAM, 16-QAM 64-QAM and 256-QAM, respectively. It can be concluded from Table 2 that if we increase the size of \(V\) in SLM, which is a phase rotation factor, the PAPR of the UFMC signal will decrease, associated with the cost of increasing the system's complexity. Furthermore, the modulation size also influences the PAPR reduction. The impact of using different QAMs is more obvious on the GCL precoded UFMC waveform and SLM-based GCL precoded power domain UFMC waveform using \(V\) = 2, 4, 16, respectively, as compared to the NOMA-UFMC waveform. PAPR performance is degraded with the higher order modulations. Hence, modulation size should be selected carefully. Furthermore, the massive multiple input multiple output (MIMO) can be combined with the proposed waveform to enhance the future requirements of 5G and beyond communication systems significantly, at the cost of complexity. The complexity is directly proportional to the number of sub-bands and their sizes, digital modulation schemes, and number of transmit and receive antennas, and their configurations. The complexity can also be reduced by employing smaller size FFTs and a reduced number of sub-bands [25]. Recently, a hardware solution has been proposed in [26] to reduce the complexity of UFMC based waveforms. The complexity can be reduced up to 4.5 times by using poly-phase implementations.