## 1 Introduction  
The next-generation wireless communication system's key features include spectral efficiency, high data rate and power consumption with the simple transceiver structure [1, 2]. The systems can support non-synchronized users at numbers about 100 times greater than allowed by recent 4G wireless communication networks that have different demands [3, 4]. Real-time communication systems, such as tactile Internet, suffer from latency for short transmission time, synchronization for different demands of users and computational complexity issues for power consumption [5, 6]. The waveform used in current communication systems is not able to meet the new demands of the 5G and beyond the communication system. The orthogonal frequency division multiplexing (OFDM) system can prevent interference between symbols and carriers using a cyclic prefix and guard band, respectively. However, any time and frequency variations on the users' transmission channel and synchronization problems cause the interference between subcarriers and symbols. The new waveforms such as filter bank-based multicarrier (FBMC), generalized frequency division multiplexing (GFDM), universal filtered multicarrier (UFMC) are proposed to solve the drawbacks of the OFDM. Accordingly, FBMC is the waveform that utilizes the low-pass filters, which are shifted in the frequency domain [7, 8]. Unlike FBMC, more than one subcarriers are filtered instead of filtering every subcarrier. Moreover, UFMC is another waveform with a lower out-of-band (OOB) emission level similar to GFDM [9; 10; 11]. GFDM outperforms OFDM in terms of inter-user interference between asynchronous users with frequency and time misalignments [12]. Up to now, the related waveforms are mainly based on the rectangular lattice structure with adopting a pulse shaping. It is known that the pulse shaping methods suffer from filter tailing on time and frequency domain [13]. Therefore, two main solutions are: (i) using a pulse with a good time-frequency localization; and (ii) leaving space between the symbols in the time-frequency lattice points. These solutions can be adopted on a proposed waveform that benefits from the hexagonal lattice points and Gaussian pulses [14]. Accordingly, hexagonal multicarrier transmission outperforms OFDM with respect to robustness against the effects of the channel dispersion. Thus, the hexagonal lattice structures have better performance than conventional OFDM against ISI/ICI effects because of lower energy perturbation [15; 16]. Therefore, waveforms containing hexagonal lattice structures have also been proposed for next-generation communication systems. An optimum transceiver structure of FBMC waveform with hexagonal time-frequency allocations is proposed in [17]. Accordingly, hexagonal FBMC is more robust to frequency synchronization errors and offers fewer OOB emissions concerning rectangular FBMC. Moreover, hexagonal Hermite waveforms have many potential benefits for the unsynchronized user at multi-user communications under doubly dispersive channel conditions [18]. On the other hand, fractional Fourier transform, which has equivalent computational complexity with Fourier transform (FT), is used to design a hexagonal frequency lattice structure to provide robustness on noise [19]. Hexagonal nonorthogonal multicarrier modulation is robust when using the Gaussian pulse which is the best time-frequency localized pulse shape [14]. Besides the pulse shaping, time and frequency allocation is an important issue concerning the robustness on double dispersive channels. The Euclidean distance between the transmitted symbols on the time-frequency lattice is crucial for the robustness of the channel effects [20]. Thus, a transmission system is more robust when transmitted symbols are more separated. The time-frequency allocations of the GFDM allow for decreasing out-of-band (OOB) emission and peak average power ratio (PAPR) that are the main shortcomings of OFDM. One limitation of the GFDM is the computational complexity caused by filtering operation.