## 3 System Model
### Proposed Hexagonal GFDM Model  
If the transmission pattern of the time and frequency resources is rectangular, transmission pulses are generated by using the filter \(g(t)\). Accordingly, the transmitted signal of the traditional GFDM is given as:  
\[\mathbf{x}(n)=\sum_{m=0}^{M-1}\sum_{k=0}^{K-1}d^{(k,m)}\mathbf{g}(n-mK)e^{j2 \pi\frac{k}{k}n} \tag{2}\]  
where \(d^{(k,m)}\) is the signal on the \(k\)th subcarrier and the \(m\)th subsymbol. The indices \(n\) of the \(g(n)\) filter is \(n=1,2,...MK\). The basic form of the GFDM structure is a kind of transmission with rectangular lattice structure. The transmitted signal of hexagonal GFDM is given as:  
\[\mathbf{x}(n)=\sum_{m=0}^{M-1}\sum_{k=0}^{K-1}d^{(k,m)}\mathbf{g}(n-mK)e^{j2 \pi\left[k+\frac{1}{2}\text{mod}(m,2)\right]\frac{1}{K}n} \tag{3}\]  
where \(\text{mod}(m,2)\) is the remainder operator. The variable is change with \(n=lK+s\) for \(s=0,1,\ldots K-1\) and \(l=-\infty\ldots\infty\) for the parallel operation by utilizing polyphase decomposition of the filter \(\mathbf{g}(n)\) for using Fourier transformation as:  
\[\mathbf{x}(lK+s)=\sum_{m=0}^{M-1}\sum_{k=0}^{K-1}d^{(k,m)}e^{j2\pi\frac{k}{k} s}\mathbf{g}(s+(l-m)K)e^{j\pi\text{mod}(m,2)\frac{k}{K}}. \tag{4}\]  
Thus, the transmitted signal becomes  
\[\mathbf{x}(lK+s)=\sum_{m=0}^{M-1}\mathbf{D}^{(m)}\mathbf{g}(s+(l-m)K)e^{j\pi \text{mod}(m,2)\frac{k}{K}} \tag{5}\]  
where \(D^{(m)}\) is the \(K\)-point inverse DFT of \(m\)th subsymbol as \(\mathbf{D}^{(m)}=\sum_{k=0}^{K-1}d^{(k,m)}e^{(j2\pi\frac{k}{K}s)}\). We can rewrite the equation with polyphase components as:  
\[\mathbf{x}_{s}(l)=\sum_{m=0}^{M-1}\mathbf{D}^{(m)}\mathbf{g}_{s}(l-m)e^{j\pi \text{mod}(m,2)\frac{k}{K}} \tag{6}\]where the operations on \(\mathbf{x}_{s}(l)\) and \(\mathbf{g}_{s}(l-m)\) are the same as serial-to-parallel operation with \(K\) parallel streams. Finally, the transmitted signal with convolution operation is followed:  
\[\mathbf{x}_{s}(l)=\left[\mathbf{D}^{(m)}\times\mathbf{g}_{s}(l)\right]e^{j2\pi \left[\frac{1}{2}\mathrm{mod}(m,2)\right]\frac{1}{k}s}. \tag{7}\]  
The transmitter block diagram of the hexagonal GFDM that involves Fourier transforms and the polyphase filters is given in Figure 2. Accordingly, the convolution of the symbols on frequency resources is done with the polyphase components of the filter with the phase-shifting dependent on the time slot indices. Here, the shifted version of the transmitted symbol \(d_{N}\) is upsampled with \(M\). Thus, the symbols are shared among \(K\) subcarriers. In each subsymbol block, the IDFT transformations and filtering operations are done, respectively. In addition, the filtering operations have fewer multiplications because of using polyphase components. The received signal at the \(m\)th subsymbol is given as:  
\[\tilde{d}^{(k,m)}=\sum_{k=0}^{K-1}\sum_{l}\mathbf{x}(l)\mathbf{g}(mK-l)e^{-j2 \pi\left[k+\frac{1}{2}\mathrm{mod}(m,2)\right]\frac{1}{k}l}. \tag{8}\]  
Substitute the variable with \(l=aK+b\)  
\[\tilde{d}^{(k,m)}=\sum_{k=0}^{K-1}\left\{\sum_{l}\mathbf{x}(aK+b)\mathbf{g}( mK-aK-b)\right\}e^{-j2\pi\frac{1}{k}b}\times e^{-j\pi\mathrm{mod}(m,2)\frac{1}{k }b} \tag{9}\]  
and re-arrange with DFT transformation to obtain  
\[\tilde{d}^{(k,m)}=DFT\left\{\sum_{l}\mathbf{x}(aK+b)\mathbf{g}(mK-aK-b)\right\} e^{-j\pi\mathrm{mod}(m,2)\frac{1}{k}b}. \tag{10}\]  
To write the equation in a simply way with polyphase structure, change the variables again with \(\alpha=\frac{mK-b}{K}\) and \(\beta=(mK-b)\mathrm{mod}(K)\).