# Review of Channel Estimation for Candidate Waveforms of Next Generation Networks
## 5 Simulations and Discussion of Results  
All simulations reported in this paper were implemented using MATLAB programming language. Simulations for the filter bank OFDM-OQAM were implemented assuming NPR and Non-PR based on the system model described in Figures 16 and 19. Firstly, we show the plot comparing the FIR of the prototype filter assuming NPR and Non-PR, as shown in Figure 21.  
Thus, the performances of the linear and adaptive estimators are analysed and compared with one another through Monte-Carlo simulations with 10,000 trials per SNR. Uncoded 16-QAM scheme is assumed for the OFDM-OQAM based transceiver with \(M=512\) sub-carriers under frequency selective Rayleigh fading channel with Doppler rates \(f_{d}T_{s}=0.001\) and \(f_{d}T_{s}=0.02\), corresponding to both slow and fast fading channel conditions, respectively. The speed of the moving mobile is selected as 360 km/h and 18 km/h, correlating with fast and slow channels, respectively. The overlapping factor is chosen as \(K=4\), while the initial value of the filter coefficient \(P[k]\) is selected to be \(\tilde{P}[1]=0.97195983\)[131]. Cyclic prefix overhead is not considered for the OFDM-OQAM system in contrast to the GFDM-OQAM system. In a similar procedure, using similar parameters as summarised in Table 6, the GFDM-OQAM is implemented with the addition of CP overhead of 25% of symbol duration for NPR filter bank. For the adaptive CE algorithms, the positive step size parameter is selected as \(\mu=0.35\) and a regularisation factor \(\delta=0.001\) is chosen to initialise the inverse autocorrelation matrix of Equation (42). Additionally, a forgetting factor \(\lambda=0.998\) is selected in the simulation as it is verified in [152] that this value yields a good results. The results are shown in terms of the NMSE at various levels of SNR. For the Non-PR simulations, the randomly introduced errors with values between \(-0.1\) and \(0.1\) are chosen to inject imperfection, as shown in Equation (23). Figure 22 demonstrates the curve at 5 dB SNR for the NMSE vs Non-PR error at different values of random errors. It can be seen from the figure that the higher the error value, the more the distortions and degradation in NMSE performance. Moreover, the RLS estimator offers the best performance in comparison with the other estimators, even for high error value of 1.5. Performance comparison in terms of the NMSE versus SNR assuming filter bank OFDM-OQAM under NPR and Non-PR for both slow and fast fading channel is shown in Figure 23a,b. Note that the ideal channel condition is used as a reference point to evaluate the performances of other CE algorithms. In Figure 23a, all CE schemes perform better assuming NPR filter design in comparison to Non-PR filter design. This is largely due to the random errors introduced in the Non-PR filter design. Furthermore, the superior performance of the adaptive schemes over the linear schemes is also demonstrated with RLS showing the best system performance. In Figure 23b, the performance of all the estimators plummets due to the increase in the fading rate (\(f_{d}T_{s}\)). In all practical scenarios, increase in the fading rate automatically reduces the performance of any estimator, as also shown in Figure 23b. This is because the channel fading distortion is more difficult to track in a fast fading channel. As such, there are irregularities in the channel estimate. Figure 21: Filter impulse responses. As mentioned above, we also present the NMSE versus SNR performance comparisons for GFDM-OQAM system assuming both NPR and Non-PR conditions. Figure 24a,b depicts the NMSE versus SNR performance for \(f_{d}T_{s}\) = 0.001 and \(f_{d}T_{s}\) = 0.02, respectively, assuming NPR condition. In the figures, the RLS CE offers the best performance in comparison with the other estimators. Besides, the adaptive estimators exhibit better performance as compared with the linear estimators. In particular, the performance of the linear estimators drops rapidly as the fading rate increases.