## 6 Simulation Results and Discussions
### AWGN Channel Simulation Results  
The outcomes of traditional OFDM, GFDM, WHT-GFM [18] and the proposed WHT-LWT-GFDM methods are compared on the AWGN channel. The influences of various wavelet families of the varied wavelet families, for example bior1.3, bior2.6, coif1, Haar, db7, sym3, lazy and sym4, are examined on the proposed technique. The performance of the proposed waveform with various mother wavelets is shown in Figure 5.  
Figure 5 shows performance comparison whose conclusion is that the best mother wavelets in case of AWGN channel is sym4 and the worst mother wavelets in this case is the coif1 mother wavelet in this case. The performance of the proposed waveform in the  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Methods** & **Multiplication** \\ \hline OFDM Transmitter & \((K\log_{2}K-3K+4)\) \\ \hline OFDM Receiver & \((K\log_{2}K-3K+4)\) \\ \hline GFDM Transmitter [8] & \((KM)^{2}\) \\ \hline Matched Filter GFDM Receiver [8] & \(KM(\log_{2}KM+\log_{2}M+L)\) \\ \hline WHT-GFDM Transmitter & \((KM)^{2}+KM\log_{2}KM\) \\ \hline Matched Filter WHT-GFDM Receiver & \(KM(\log_{2}KM+\log_{2}M+L)+KM\log_{2}KM+KM\) \\ \hline WHT-LWT-GFDM Transmitter & \((KM)^{2}+KM\log_{2}KM+KM(P+U)/2\) \\ \hline Matched Filter WHT-LWT-GFDM Receiver & \(KM(\log_{2}KM+\log_{2}M+L)+KM\log_{2}KM+KM+KM(P+U)/2\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Computational complexities of the waveform for AWGN channel.  
case of the other mother wavelets is between the performance of the proposed waveform obtained using sym4 and coif1 mother wavelets.  
In Figure 6, the performance of the conventional OFDM, GFDM, WHT-GFDM [18] and the proposed WHT-LWT-GFDM using sym4 mother wavelets are given in the AWGN channel as BER-SNR (Signal to Noise Radio) curve comparisons for 1000 Monte Carlo simulations.  
Figure 6 illustrates the BER-SNR curves of traditional OFDM, GFDM, WHT-GFDM [18] and the proposed method in the AWGN channel. Analyzing the figure, it can be observed that the GFDM waveform converges to nearly 1 \(\times\) 10\({}^{-3}\) error floor. Although the WHT-GFDM method achieves the same performance as the GFDM up to roughly 6 dB of SNR value, it converges to the base of the 1 \(\times\) 10\({}^{-2}\) error floor after 6 dB of SNR value. The classical OFDM system overcomes the GFDM system and eliminates the error floor. On the  
Figure 5: The comparison of performance of the proposed WHT-LWT-GFDM obtained using different mother wavelets on the AWGN channel.  
Figure 6: Comparison of traditional OFDM, GFDM, WHT-LWT-GFDM, and proposed WHT-LWT-GFDM waveforms performance on AWGN channel.  
other hand, the proposed waveform eliminates the error floor by outperforming both the classical OFDM, WHT-GFDM [18] and GFDM waveform, and provides an approximately 5 dB of SNR gain versus the traditional OFDM for the \(1\times 10^{-3}\) BER worth. This enhancement is obtained as a consequence of the application of both Walsh-Hadamard transform and lifting wavelet transformation. Additionally, this improvement can be explained by using twice as much data in the proposed system with zero padding components compared to the OFDM system. This also reduces the effect of errors caused by noise and disturbing effects of the channel by spreading the sent data. It can also be explained by the fact that this improvement meets the excellent reconstruction feature since the wavelets have orthonormal structure.