# A Survey of Candidate Waveforms for beyond 5G Systems
## 4 Performance Results and Discussion
### PSD and Spectral Efficiency  
In order to compare the waveforms regarding their spectral efficiency, we will evaluate the required bandwidth taking as reference the total transmission time of the overall symbol sequence, i.e., the useful bit-rate \(R_{b}\), assuming that all the sub-carriers are modulated by symbols from the same constellation. This means that the waveforms that transmit a signal with larger overhead require a larger bandwidth in order to achieve the same total transmission of the symbol sequence. Figure 5: PAPR’s CCDF of the OFDM, TIBWB-OFDM with and without WTO, GFDM and F-OFDM. For the TIBWB-OFDM with WTO waveform, the overhead includes the temporal extension of the sub-blocks and the ZP, but this is negligible when compared to the total signal length. This means that the total bandwidth required is proportional \(\frac{R_{b}}{2}(1+\frac{\beta N+N_{Zp}}{NN_{s}})\approx\frac{R_{b}}{2}\), where \(R_{s}\) denotes the symbol rate and relates to the bit-rate by \(R_{s}=\frac{R_{b}}{2\log_{2}(M)}\). On the other hand, the standard TIBWB-OFDM, although providing a higher degree of spectrum confinement, does not deal with the temporal extension of the sub-symbols and the overhead in this case is proportional to the window roll-off, \(\beta\), leading to an increase of the required bandwidth, \(\frac{R_{b}}{2\log_{2}(M)}\Big{(}1+\beta+\frac{N_{Zp}}{NN_{s}(1+\beta)}\approx \frac{R_{b}}{2\log_{2}(M)}(1+\beta)\). A similar analysis can be made to GFDM since the overall overhead includes a single CP. However, the CP is only added per GFDM block and then can be discarded. This way, the total required bandwidth is also proportional can also be approximated by the symbol rate i.e., \(\frac{R_{b}}{2\log_{2}(M)}\Big{(}1+\frac{N_{Zp}}{NN_{s}}\Big{)}\approx\frac{R _{b}}{2\log_{2}(M)}\). The CP-OFDM and F-OFDM cases the overhead will include the CP per symbol, which in these cases, represents a significant percentage of the overall signal, granting an increase of the required bandwidth proportional to the CP percentage, i.e., \(\frac{R_{b}}{2\log_{2}(M)}\Big{(}1+\frac{N_{CP}}{N}\Big{)}\). Table 1 presents a resume of the total bandwidth for arbitrary values of the CP and window roll-off, while in Figure 6 it is presented a comparison on the PSD normalized to the information bit rate \(R_{b}\), of CP-OFDM, F-OFDM, GFDM and TIBWB-OFDM with and without WTO, assuming a baseband QPSK-modulated transmitted symbol sequence. Once more, the strict window \(\beta=0.5\) is considered for both the TIBWB-OFDM and GFDM cases, and the CP employed in CP-OFDM and F-OFDM techniques is assumed to be 25% of the overall signal's length. From Figure 6, it is clear the considerable improve on spectral efficiency gains of the new proposed TIBWB-OFDM with WTO against all the other techniques. Although the F-OFDM waveform is the one that provides the lowest OOB radiation due to the employment of the filtering operation, it relies on a CP (just like OFDM) to deal with the channel's impairments and also requires a large ZP (in the order of the filter length), which limits its spectral efficiency. A similar OOB radiation to F-OFDM is obtained with the conventional TIBWB-OFDM, but also in this case the spectral efficiency is limited due to the temporal extension of the symbols due to the windowing operation. Regarding GFDM this presents a poor OOB rejection, that compares to CP-OFDM schemes [42]. The amount of spectrum saved for the TIBWB-OFDM with WTO (when compared to conventional TIBWB-OFDM) is proportional to the temporal growth of the OFDM-based blocks, i.e., to \((1+\beta)\). Also, the use of a ZP per group of OFDM-based symbols allows this technique to save spectrum when compared to CP-OFDM and F-OFDM schemes.