# Performance Evaluation of GFDM Channel Estimation Using DFT for Tactile Internet Application
## 2 GFDM Systems and Channel Model
### LS and DFT-Based Channel Estimation  
The channel coefficient at pilot symbol can be estimated as follow:  
\[\hat{\mathbf{H}}_{\text{LS}}[k,m]=\ \frac{Z[k,m]}{\widetilde{d}\left[k,m\right]} \,\ 0\leq k\leq K-1\ and\ 0\leq m\leq M-1 \tag{9}\]  
in which \(\mathbf{Z}\) with size \(K\times M\) is matrix representation of \(\overrightarrow{z}\) with \(KM\times 1\) dimension and \(\widetilde{d}\left[k,m\right]\) is pilot symbol position. Note that \(\hat{\mathbf{H}}_{\text{LS}}[k,m]\) is an element matrix of \(\hat{\mathbf{H}}_{\text{LS}}\) which consist of \(K\) row and \(M\) column where the channel information located only at pilot position. In an attempt to obtain channel response for all \(KM\) data, interpolation is applied in time-frequency grid to obtain the estimated channel frequency response. In order to measure the accuracy of this estimation technique, the mean-square error (MSE) of the LS channel shall be estimated. MSE can be written as  
\[\text{MSE}_{\text{LS}}=\text{E}\Big{\{}(\mathbf{H}-\ \hat{\mathbf{H}}_{\text{LS}}) ^{H}(\mathbf{H}-\ \hat{\mathbf{H}}_{\text{LS}})\Big{\}}. \tag{10}\]  
The performance of LS channel estimation can be enhanced by using channel estimation using DFT technique. This can be obtained by eliminating the effect of interference and noise outside the maximum channel delay since channel information only concentrated in the first \(L\) samples. In this case, it is assumed that a priori knowledge of CIR length \(L\) is known. The process flow of proposed method is shown in Figure 3.  
The estimated sample of CIR is written as  
\[\begin{array}{rcl}\hat{\mathbf{h}}_{\text{LS}}\left[n\right]&=&\text{IDFT}_ {N}\big{\{}\hat{\mathbf{H}}_{\text{M}op}[k]\big{\}}\,\ 0\leq n\leq N-1\\ &=&h[n]+i[n]+\widetilde{w}[n]\end{array} \tag{11}\]  
Figure 3: DFT-based channel estimation block diagram.  
where IDFT\({}_{\lambda}\) shows N-point IDFT. Note that an interference \(i[n]\) and noise \(\widetilde{w}[n]\) terms emerge in channel frequency response. It causes error floor in the estimated CIR. If the length of \(L\) is known, the Equation (11) can be rewritten as  
\[\hat{h}_{\text{LS}}\left[n\right]=\ \left\{\begin{array}{c}h[n]+i[n]+\ \widetilde{w}[n],\ 0\leq n\leq L-1\\ i[n]+\widetilde{w}[n],\ \text{otherwise}.\end{array}\right. \tag{12}\]  
The channel impulse response is contained in the first \(L\) samples. Removing samples outside the \(L\) samples can reject the interference and noise. Expressing this process, we obtain  
\[\hat{h}_{\text{IDFT}}\left[n\right]=\ \left\{\begin{array}{c}h[n]+i[n]+b[n]+ \ \widetilde{w}[n],\ 0\leq n\leq L-1\\ 0,\ \text{otherwise}.\end{array}\right. \tag{13}\]  
From Equation (13), the channel estimation is expressed as  
\[\hat{\mathbf{H}}_{\text{DFT}}[k]=\ \text{DFT}_{\text{N}}\Big{\{}\hat{h}_{ \text{DFT}}\left[n\right]\Big{\}},\ 0\leq k\leq N-1 \tag{14}\]  
The accuracy of estimated channel can also be expressed in MSE performance. After obtaining the estimated channel, equalization can be performed. The equalization process is expressed as  
\[\begin{array}{rcl}\overset{\rightarrow}{d}&=&\frac{\mathbf{Z}}{\mathbf{H} }\\ &=&\frac{\left(\overset{\rightarrow}{d}+\overset{\rightarrow}{d}\right) \mathbf{H}+\overset{\rightarrow}{w}}{\mathbf{H}}\\ &=&\left(\overset{\rightarrow}{\overset{\rightarrow}{d}+\overset{ \rightarrow}{d}}\right)+\overset{\rightarrow}{w}\end{array} \tag{15}\]  
After equalization process, the recovered data then demultiplexed to separate the data from pilot symbol. The received data \(\overset{\rightarrow}{d}\) then is processed to demapper to convert it into bits.