# A Survey of Candidate Waveforms for beyond 5G Systems
## 4 Performance Results and Discussion
### Computational Complexity  
The next figure of merit that is included in this paper is the computational complexity regarding the implementation of the different transceiver presented for every waveform. In order to perform a fair comparison the complexity is evaluated in terms of the number of complex multiplications for each MC scheme. The same signal conditions, regarding sub-carriers \(N\) and sub-symbols \(N_{s}\) employed in transmission are assumed in this analysis. Table 2 shows the computational complexity of the 5G and beyond candidate waveforms in terms of total number of complex multiplication employed in transmission and reception. It is clear that most of the complexity effort comes from the receiver operations. Figure 7: BER performance of 25% CP-OFDM and F-OFDM together with GFDM and TIBWB-OFDM with and without WTO considering the different proposed receivers when \(\beta=0.5\). The complexity of OFDM schemes comprises the IFFT and FFT complexity employed at the transmitter and receiver, respectively. For the F-OFDM waveform, the overall complexity includes the multiplication operations performed in OFDM added by complexity due to transmit and receive filters. It is assumed that the filtering operation employed in transmission is performed once for the CP samples [67]. The complexity analysis of OFDM transmitter is based on the low complexity transmitter implementation proposed in [42], where the first term originates from the \(N_{x}\)-sized FFTs of \(N\) sub-carriers, the second one arises from the filtering operation of \(N\) sub-carriers and the last one from the \(N_{s}N\)-sized IFFT that converts the signal back to time domain. The receiver complexity analysis is based on [54] and includes the multiplications presented in both the MF operation and the SIC algorithm. In [64] the TIBWB-OFDM with WTO receivers complexity was analysed and it was concluded that:  
* Receiver A includes only a direct path, where includes a \(N_{x}\)-sized FFT upon signal reception, a MMSE equalization algorithm and BWB-OFDM unformatting with WTO compensation. * Receivers B and D include the direct path but instead of MMSE equalization, the IB-DFE algorithm is employed. The feedback path wherein the BWB-OFDM block formatting is performed is also included. For \(L\) iterations, the number of multiplications will increase proportionally by \(L\) in the direct path and \(L{-}1\) in the feedback path. * Receiver C is similar to receiver A in the first iteration, performing all the operation in the direct path, while including the BWB-OFDM formatting operation in the feedback path. For \(L>1\), the direct path only includes the BWB unformatting with WTO compensation and the feedback path includes the BWB-OFDM formatting. The TIBWB-OFDM receivers A and B are identical to the ones explained for the TIBWB-OFDM with WTO with the only difference being the removal of the WTO compensation block. Therefore the multiplications employed in this block are not considered in these cases. Table 2 presents the overall transceiver complexity for the same number of carriers, sub-symbols and roll-off employed in previous subsections. Also in this case we considered that the filter length is \(I=1032\), the number of iterations is \(L_{GFDM}=1\) and \(N_{x}=(Ns+\beta)\) (we can discard the ZP here since it represents only a very small fraction of the overall signal). The number of multiplications for the TIBWB-OFDM standard case can be considered discarding the term \(2\beta N_{s}N\) and considering that \(N_{x}=N_{s}N(1+\beta)\). The complexity regarding the simpler GFDM receivers with the MF and ZF detectors can be obtained by discarding the term \(L_{GFDM}(2\log_{2}(N_{s})+1)\).