# Performance Evaluation of GFDM Channel Estimation Using DFT for Tactile Internet Application
## 2 GFDM Systems and Channel Model  
The GFDM system is shown in Figure 1. The GFDM block has a length of \(N=MK\) in which \(M\) sub symbols are transmitted on \(K\) subcarriers. The entries vector \(\overset{\rightarrow}{d}\in\mathbb{C}^{\mathrm{Nx1}}\) consists of data symbol and pilot symbol which have been multiplexed before modulated by pulse shaping filter. Vector \(\overset{\rightarrow}{d}\) carries data samples and \(\overset{\rightarrow}{d}\) includes the pilot samples. Vector \(\overset{\rightarrow}{d}\) contains zeros at pilot positions, meanwhile vector \(\overset{\rightarrow}{d}\) has zeros at data positions. Vector \(\overset{\rightarrow}{d}\) is filtered through circulatory time and frequency shifted versions of a prototype filter \(g[n]\). The vector \(\overset{\rightarrow}{d}\) can also be represented as matrix \(\mathbf{D}\), which consists of the \(K\) row and \(M\) column. Filter impulse response is defined as  
\[g_{k,m}[n]=g\;[(n-mK)\;\mathrm{mod}\;N]\exp\left[j2\pi\frac{k}{K}N\right]. \tag{1}\]  
The superposition of pulse shaped data symbols provides a GFDM transmit sample:  
\[x[n]=\;\sum_{k=0}^{K-1}\sum_{m=0}^{M-1}g_{k,m}[n]d_{k,m},\;n=0,\;\ldots,\;N-1 \tag{2}\]  
where \(d_{k,m}\) is the symbol transmitted on subcarrier \(k\) and subsymbol \(m\). In Equation (2), circular filtering is achieved via the modulo operation. Equation (2) can also be represented as  
\[\overset{\rightarrow}{x}=\mathbf{A}\overset{\rightarrow}{d} \tag{3}\]  
Figure 1: GFDM channel estimation block diagram. \[\text{in which }\overset{\rightarrow}{d}=\ \left[\overset{\rightarrow}{d}_{0}^{T},\ \ldots,\overset{ \rightarrow}{d}_{K-1}^{T}\right]^{T}\ \in\ \mathbb{C}^{N}\text{ is vectorized version of }\mathbf{D}\text{ while for each subcarrier }\text{ }k\text{ its corresponding }\overset{\rightarrow}{d}_{k}\text{ vector is given by }\overset{\rightarrow}{d}_{k}=\ \left(d_{k}[m]\right)^{T}_{(m=0,\ldots,M-1)}\text{. The GFDM transmit matrix }\mathbf{A}\text{ follows:} \tag{4}\]  
with column vector \(\overset{\rightarrow}{g}_{k,m}=\ \left(g_{k,m}[n]\right)^{T}_{n=0,1,\ldots,N-1}\). The signal \(\overset{\rightarrow}{x}\) is then protected by a CP against multipath propagation. Figure 2 illustrates pilot structures in time-frequency grid. Pilot subcarrier spacing is denoted as \(\Delta k\) and the number of pilot symbol used in different time slots is donated as \(Mp\). As can be seen, the pilot insertion with small \(\Delta k\) reduces the effective data rate by the factor  
\[\eta=\frac{\text{no. of data samples}}{\text{no. of total samples}}=\ 1-\frac{Mp\left(\frac{K}{\Delta k}\right)}{MK}. \tag{5}\]  
The CP is removed at the receiver. Assuming the time and synchronization are perfect, the received signal \(\overset{\rightarrow}{y}\) becomes  
\[\overset{\rightarrow}{y}=\ \overset{\rightarrow}{x}\ \overset{ \rightarrow}{u}\ \overset{\rightarrow}{h}+\overset{\rightarrow}{w}. \tag{6}\]  
The received signal \(\overset{\rightarrow}{y}\) in time is the circular convolution of transmitted signal \(\overset{\rightarrow}{x}\) and CIR \(\overset{\rightarrow}{h}\) as well as AWGN process \(\overset{\rightarrow}{w}\). In (6), all the channels are assumed to have shorter length \(L\) compared to the CP length. The CIR \(\overset{\rightarrow}{h}\) is defined as  
\[\overset{\rightarrow}{h}\ \triangleq\ \sqrt{\mathbf{P}}\overset{ \rightarrow}{g} \tag{7}\]  
where \(\mathbf{P}\ \in\mathbb{R}^{NxN}\) is the diagonal matrix of normalized power delay profile and \(\overset{\rightarrow}{g}\ \in\mathbb{C}^{Nx1}\) is the vector of zero mean complex Gaussian processes with unit variance. The received signal then goes to GFDM demodulator. There are three demodulator which is provided: matched filter (MF), zero forcing (ZF) and minimum mean square error (MMSE) [20]. Under the assumption that ZF is employed, the output of GFDM demodulator is  
\[