# Review of Channel Estimation for Candidate Waveforms of Next Generation Networks
## 4 System Model Description of FBMC OFDM-OQAM and GFDM-OQAM
### Gfdm-OQAM Transceiver
#### 4.2.1.1 Transmitter Description  
Figure 20 depicts a typical GFDM-OQAM system. In GFDM-OQAM based transceivers, the real valued \(N\) data symbols generated at the transmitter are decomposed into \(\tilde{M}\) sub-symbols, where the sub-symbols are being carried by a sub-carrier [149]. If \(\tilde{K}\) represents the total number of sub-carriers and \(\tilde{M}\) is the total number of sub-symbols per-sub-carrier, then the total number of data symbols to be transmitted is given as:  
\[N=\tilde{K}\tilde{M}. \tag{45}\]  
To implement OQAM for GFDM, let \(\vec{\tilde{d}}_{k,m}\) represent the real data obtained from the QAM symbols, where \(m\) is the sub-symbol index at the \(\hat{k}\)th sub-carrier of the GFDM-OQAM system, the entire symbols to be transmitted can be collected into a vector \(\vec{\tilde{\mathbf{d}}}\) given as [37]:  
\[\vec{\tilde{\mathbf{d}}}=[\vec{\tilde{d}}_{0,0},\vec{\tilde{d}}_{1,0},\ldots, \vec{\tilde{d}}_{\tilde{K}-1,0},\vec{\tilde{d}}_{0,1},\vec{\tilde{d}}_{1,0}, \ldots,\vec{\tilde{d}}_{\tilde{K}-1,1},\ldots,\vec{\tilde{d}}_{0,\tilde{M}-1}, \vec{\tilde{d}}_{\tilde{K}-1,\tilde{M}-1}]^{T}, \tag{46}\]  
for \(\hat{k}=0,1,\ldots,(\hat{K}-1)\) and \(\hat{m}=0,1,\ldots,\tilde{M}-1\). Each \(\vec{\tilde{d}}_{\hat{k},m}\) symbol of the GFDM system is transmitted with a corresponding pulse shape given as [150]:  
\[g_{\hat{k},m}[n]=p\Bigg{[}n-\frac{n\hat{K}}{2}mod\hat{K}\tilde{M}\Bigg{]}\exp \Bigg{(}j\frac{2\pi k}{\hat{K}}\bigg{(}n-\frac{(L_{p}-1)}{2}\bigg{)}\Bigg{)}, \tag{47}\]  
where \(n=0,1,\ldots,N-1\) and \(g_{\hat{k},m}[n]\) is the shifted version of the prototype filter \(p\) in the time and frequency domains as defined in Equation (7) and \(L_{p}\) is the prototype filter length given by Equation (5). Thus, the time domain transmit signal just before cyclic prefix insertion is given as:  
\[x[n]=\sum_{\hat{k}=0}^{\hat{K}-1}\sum_{m=0}^{\tilde{M}-1}\vec{\tilde{d}}_{ \hat{k},m}g_{\hat{k},m}[n]e^{i\phi_{\hat{k},m}}, \tag{48}\]  
where \(e^{i\phi_{\hat{k},m}}\) is used to ensure a \(\frac{\pi}{2}\) difference in phase between the real valued data \(\vec{\tilde{d}}_{\hat{k},m}\) in time and frequency domains, and \(\phi_{\hat{k},m}\) is given as in Equation (49).  
\[\phi_{\hat{k},m}=\frac{(\hat{k}+\hat{m})\pi}{2}. \tag{49}\]  
The operations described in this Section 4.2.1.1, can be written in a matrix form \((\vec{\tilde{x}})\) given as \(\vec{\tilde{x}}=\mathbf{A}\vec{\tilde{\mathbf{d}}}\)[151], where \(\vec{\tilde{\mathbf{d}}}\) is given in Equation (40), and \(\mathbf{A}\) is an \(N\times N\) transmitter modulation matrix containing elements of \(g_{\hat{k},m}[n]\) expressed as:  
\[\mathbf{A}=[\mathbf{g}_{0,0},\ldots,\mathbf{g}_{\hat{K}-1,0},\mathbf{g}_{0,1},\mathbf{g}_{1,1},\ldots,\mathbf{g}_{\hat{K}-1,1},\ldots,\mathbf{g}_{0,\tilde {M}-1},\mathbf{g}_{1,\tilde{M}-1},\cdots\mathbf{g}_{\hat{K}-1,\tilde{M}-1}]. \tag{50}\]  
where \(\mathbf{g}_{\hat{k},m}\) are vectors given as \(\mathbf{g}_{\hat{k},m}=\left[g_{\hat{k},m[n]}e^{i\phi_{\hat{k},m}}\right]^{T}\). Cyclic prefix of length \(N_{\mathcal{C}P}\) is then appended to \(\vec{\tilde{x}}\) so as to obtain the final transmit signal \(\vec{\tilde{x}}_{T}\) whose length is \((N+N_{\mathcal{C}P})\).