# 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
#### 4.1.1.2 Channel Description  
The frequency domain channel transfer function (FD-CTF) fading coefficients \(H[k]\) can be described by two properties: power spectral density (PSD) and the autocorrelation function (ACF). The PSD is expressed by [123]:  
\[s|f|=\left\{ \begin{array}{ll}\frac{1}{\pi f_{d}\sqrt{1-(f/f_{d})^{2}}},&|f| \leq f_{d}\\ 0,&\text{elsewhere}.\end{array} \right. \tag{9}\]  
If \(L_{ch}\) represents the length of the channel, then the corresponding normalised discrete-time ACF is expressed by Equation (10) under the assumption that the channel is repeated for a frame [123; 135]:  
\[a[k]=J_{0}\left(2\pi f_{d}T_{s}\left(\frac{M}{L_{ch}}\right)|k|\right),\ \ \ \ \ k=0,1,2,\ldots,(L_{ch}-1). \tag{10}\]  
Figure 17: Plot of filter impulse response for different overlapping factor \(K\), \(M\) = 32. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Overlapping Factor (\(K\))** & \multicolumn{3}{c}{**4**} & \multicolumn{3}{c}{**3**} & \multicolumn{3}{c}{**2**} & \multicolumn{1}{c}{**1**} \\ \hline \multirow{2}{*}{Filter impulse (\(\bar{P}[c]\))} & \(\bar{P}[0]\) & \(\bar{P}[1]\) & \(\bar{P}[2]\) & \(\bar{P}[3]\) & \(\bar{P}[0]\) & \(\bar{P}[1]\) & \(\bar{P}[2]\) & \(\bar{P}[0]\) & \(\bar{P}[1]\) & \(\bar{P}[0]\) \\ \cline{2-11}  & 1 & 0.9720 & \(\frac{\sqrt{2}}{2}\) & 0.2351 & 1 & 0.9114 & 0.4114 & 1 & \(\frac{\sqrt{2}}{2}\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Different values of overlapping factors. Equation (9) is the u-shaped band limited Jakes spectrum, where \(f_{d}\) is the maximum Doppler frequency of the moving channel given as \(f_{d}\)= \(\frac{\gamma}{\lambda}\). The velocity of the moving mobile is denoted by \(v\), and \(\lambda\) is the wavelength of the carrier wave. A plot of the u-shaped Jakes power spectrum for \(f_{d}\) = \(10 Hz\) and \(f_{d}\) = \(200 Hz\) is given in Figure 18. In Equation (10), \(J_{0}(\cdot)\) is the zero-order Bessel function of the first kind, \(T_{s}\) is the symbol period. Note that \(f_{d}T_{s}\) is normally referred to as the normalised Doppler frequency (Doppler rate). The correlation matrix \((\mathbf{R}_{HH}[k])\) is given in its expanded form in Equation (11) as a function of ACF as [135; 136]:  
\[\mathbf{R}_{HH}[k]= \begin{bmatrix}a[0]&a[1]&\ldots&a[(L_{ch}-1)]\\ a[1]&a[0]&\ldots&a[(L_{ch}-2)]\\ \vdots&\vdots&\ddots&\vdots\\ a[(L_{ch}-1)]&a[(L_{ch}-2)]&\ldots&a[0]\end{bmatrix} . \tag{11}\]  
The FD-CTF fading channel coefficients \(H[k]\) can be generated via an \(L_{ch}\)th-order autoregressive (AR) process represented as \(AR(L_{ch})\). It can thus be expressed as [135; 136; 123]:  
\[H[k]=\sum_{i=1}^{L_{ch}}z[i]H[k-i]+\Omega[k],\ \ k=0,1,2,\ldots,(L_{ch}-1), \tag{12}\]  
where \(H[k]\) represents the channel fading coefficient over the \(k\)th sub-carrier. The notation \(z[i]\) is given by Equation (13):  
\[z[i]=-\varphi[i], \tag{13}\]  
and \(\varphi[i]\) is defined as:  
\[\varphi[i]= \begin{bmatrix}\varphi[1],\varphi[2],\ldots,\varphi[L_{ch}]\end{bmatrix}  ^{T}. \tag{14}\]  
In Equation (13), \(T\) is the transpose operator. Moreover, \(\Omega[k]\) in Equation (12) is a zero mean white Gaussian noise process with variance \(\sigma_{\Omega}^{2}\) given as [135; 136]:  
\[\sigma_{\Omega}^{2}=a[0]+\sum_{k=0}^{L_{ch}-1}\varphi[k]a[k], \tag{15}\]  
where \(\Omega[k]\) are the coefficients of the AR model parameters and can be obtained by computing the Yule-Walker (YW) equation defined by Equations (16) and (17). \[\varphi[k]=-R_{HH}^{-1}[k]b[k], \tag{16}\]  
Figure 18: U-shaped Jakes power spectrum. Based on the channel expression in Equation (12), the channel coefficients are realised to be equal for a given frame length \(F_{L}=\frac{M}{L_{\text{ch}}}\). For instance, the fading coefficients at \(H[0]\) is given as:  
\[H[0]=z[1]H[-1]+\Omega[0], \tag{18}\]  
where \(H[-1]\) in Equation (18) is initialised through random generation. The channel coefficients across all frequencies at a given time for \(F_{L}=\frac{M}{L_{\text{ch}}}\) are chosen to be the same for all \(M\) sub-carriers.