# Review of Channel Estimation for Candidate Waveforms of Next Generation Networks
## 4 System Model Description of FBMC OFDM-OQAM and GFDM-OQAM
### OFDM-OQAM Transceiver
#### 4.1.1.1 Transmitter Description  
Consider the OFDM-OQAM transmitter model diagram of Figure 16. The complex QAM symbols at the input \((c[l,k])\) have their real and imaginary parts interleaved by a time offset of \(\frac{T}{2}\), where \(l\) represents the OQAM preprocessing input/post processing output sample index and \(n\) is the OQAM preprocessing output/post processing input sample index while \(T\) is the signalling period. After the bits are mapped into symbols, OQAM pre-processing is performed on the QAM symbols where the complex input symbols are mapped into real data symbols \(d[n,k]\) at \(k\) sub-carrier index (representing the in-phase components of \(c[l,k]\)) and are transmitted at a rate \(\frac{2}{T}\), where \(T=\frac{1}{\Delta f}\) and \(\Delta f\) is the sub-carrier spacing. The quadrature components of the input symbols \(c[l,k]\) are represented as \(d[n+1,k]\) in an entire process termed complex to real conversion (C2R) [123]. Figure 16: Structure of a filter bank OFDM-OQAM transmitter. Figure 15: Structure of a filter bank OFDM-OQAM system. The conversion of the complex input symbols to real symbols is mapped mathematically as [123; 131]:  
\[d[n,k] =\left\{ \begin{array}{ll}R_{e}(c[l,k]);&\quad k\;\;\text{even},\\ I_{m}(c[l,k]);&\quad k\;\;\text{odd}.\end{array} \right. \tag{1a}\] \[d[n+1,k] =\left\{ \begin{array}{ll}I_{m}(c[l,k]);&\quad k\;\;\text{even}, \\ R_{e}(c[l,k]);&\quad k\;\;\text{odd}.\end{array} \right. \tag{1b}\]  
In Equation (1), the notation \(R_{e}\) and \(I_{m}\) signifies the real and imaginary parts of the complex input symbols, respectively. To ensure orthogonality of sub-carriers, \(\theta[n,k]\) is multiplied by \(d[n,k]\) resulting in [35; 132]:  
\[x[n,k]=d[c,k]\theta[n,k], \tag{2}\]  
where \(x[n,k]\) is the OQAM preprocessing output symbols and \(\theta[n,k]\) is defined as:  
\[\theta[n,k]=j^{(k+n)}. \tag{3}\]  
Before implementing IFFT, \(x[n,k]\) undergoes a transformation process where it is multiplied by the beta carrier (\(\beta\)) in order to provide enhanced robustness to the samples as they are been prepared to propagate through the wireless channel. Possible loss of sub-carrier orthogonality due to pulse shaping at SFB is also minimised from this transformation, since the cyclic prefix of the conventional OFDM transceivers are replaced (in FBMC OFDM-OQAM systems) with well-designed prototype filters that have good localisation properties [133], in north the frequency and time domains. The beta carrier (\(\beta\)) is mathematically given as:  
\[\beta[n,k]=(-1)^{kn}exp\Bigg{(}-j\frac{2\pi k}{M}\bigg{(}\frac{L_{p}-1}{2} \bigg{)}\Bigg{)}, \tag{4}\]  
where \(M\) is the IFFT size or the total number of sub-carriers, \(k\) is the sub-carrier index for \(k=0,1,2,\ldots,M-1\), and \(L_{p}\) is the length of the FBMC prototype filter \(p[n,k]\) given as:  
\[L_{p}=KM. \tag{5}\]  
In Equation (5), \(K\) is the overlapping factor that determines the number of symbols that superpose each other in time. The use of complex modulated filter banks is employed to ensure high spectral efficiency. The sub-channel synthesis filter bank impulse response \(g[\dot{n},k]\) is expressed as [122; 134; 132]:  
\[g[\dot{n},k]=p[\dot{n}]exp\Bigg{(}j\frac{2\pi k}{M}\bigg{(}\dot{n}-\frac{L_{p} -1}{2}\bigg{)}\Bigg{)}, \tag{6}\]  
where \(\dot{n}\) in Equation (6) is the symbol index at the SFB output and the AFB input for \(\dot{n}=0,1,2,\ldots,L_{p}-1\). To achieve NPR prototype filters, the FIR of the low-pass prototype filter \(p[\dot{n}]\) is defined as [123; 134; 131]:  
\[p[\dot{n}]=\bar{P}[0]+2\sum_{k=1}^{\frac{L_{p}-1}{2}}(-1)^{k}\bar{P}[k]cos \Bigg{(}\frac{2\pi k}{KM}\bigg{(}\dot{n}+\frac{KM-(L_{p}-1)}{2}\bigg{)}\Bigg{)}, \tag{7}\]  
where \(w[k]\)= \(\frac{2\pi k}{KM}\) is the uniformly spaced frequency points. From Equation (7), \(\dot{n}=0,1,2,\ldots,L_{p}-1\) and \(\bar{P}[0]=1\). Moreover, \(p[\dot{n}]\) is designed such that it meets the condition in Equation (8):\[\left\{