## 5 The Proposed WHT-LWT GFDM Waveform  
The transmitter and the receiver block scheme of the wireless telecommunication scheme using the offered WHT-LWT-GFDM waveform is illustrated in Figure 3 [27]. In this study, the WHT-LWT-GFDM waveform is offered as alternative waveform to OFDM due to the conveniences of LWT and the benefits of WHT, for the 5G and beyond waveform research studies.  
In the transmitting region of the WHT-LWT-GFDM scheme depicted in Figure 3, after randomly produced serial input bits are matched via the I-Q mapping unit, they are spread in the WHT unit with the help of Equation (7). In the ILWT block, Inverse Lifting Wavelet Transform is applied to the spread data with the help of Equation (28).  
The Inverse Lifting Wavelet Transform of a sequence \(\{Y(z)\}\), can be given by Equation (28).  
\[y_{ILWT}[n]=\sum_{q=0}^{Q-1}\sum_{z=0}^{Z-1}Y(z)^{q}2^{\frac{q}{2}}\psi(2^{q}n -z) \tag{28}\]  
Here, \(n\) is the number of sub-carriers \((0\leq n\leq K\)-1), \(Y(z)\) demonstrates the data in the wavelet domain and \(\psi(.)\) denotes the mother wavelets with scaling factor \(q\) and shift \(z\) for each subcarrier.  
Inverted LWT transformed bits are applied in GFDM modulator unit to produce WHT-LWT-GFDM data streams. After the acquired WHT-LWT-GFDM data streams are transmitted through the ISI channel and corrupted with AWGN, they arrive to the receiver. Then the deformed data are reconstructed with the proper channel equalization schemes, the recovered data are applied to the GFDM De-Modulator unit. At the output of the GFDM-De-Modulator block, the obtained data are subjected to LWT operation with the assistance of Equation (29).  
The Lifting Wavelet Transform of a sequence of \(\{y[n]\}\), can be given by Equation (29).  
\[Y_{LWT}^{q}(z)=\sum_{n=0}^{K-1}y[n]2^{\frac{q}{2}}\psi[2^{z}n-q],\ z=0,1,2, \ldots,K-1,\ q=0,1,2,\ldots,K-1 \tag{29}\]  
Here, \(y[n]\) denotes the data in the time domain.  
After the data arrives at the output of the LWT unit, the data spread to sub-carriers is combined with the help of Equation (8) in the IWHT unit. After the acquired data at the output of the IWHT block are de-modulated in the reverse I-Q mapping unit, the desired benchmark metric, as for example the bit-error rate (BER) or the SER, are computed by trying to acquire the information sent by the transmitter.  
The choice of both \(K\) and \(M\) even is known to be problematic in GFDM, this choice along with a high roll-off is even worst [28; 29; 30]. When a new GFDM-based system is to be designed for the future, it is necessary to correct the negative effect of high roll-off values on GFDM to make accurate analyzes according to different roll-off values within the existing conditions. This is one of the aims of this study. Therefore, it is very important to increase the performance of the GFDM waveform at high roll-off values. The main contribution of this study is to solve this issue with the use of LWT. The effects of different roll-off values in the RC pulse shaping filter on the BER performance of the GFDM waveform are shown in Figure 4.  
When Figure 4 is examined, it is seen that the performance deteriorates as the roll-off value of the RC pulse shaping filter used in the GFDM waveform increases. Increasing the performance of GFDM and WHT-GFDM waveforms with the proposed method at high roll-off values is the biggest contribution of this study.