# A Survey of Candidate Waveforms for beyond 5G Systems
## 3 Candidate Waveforms
### Generalized Frequency Division Multiplexing
#### 3.3.2 GFDM Receiver  
If we assume that the CP inserted in the GFDM multi-symbol is larger than the channel's delay spread, the GFDM received symbol at frequency domain can be expressed as  
\[Y_{k}=S_{k}H_{k}+W_{k}, \tag{9}\]  
where \(H_{k}\) denotes the channel frequency response at kth sub-carrier and \(W_{k}\) represents the complex additive white Gaussian noise (AWGN) sample with variance \(E[|W_{k}|^{2}]\), while the total length of \(x_{n}\) is \(N_{x}=KN_{s}\). In order to compensate the influence of the channel on the received signal, a FDE method can be applied to the received signal [43]. Assuming perfect channel estimation and synchronization of the received signal, the minimum mean square error (MMSE) algorithm can be employed as follows [43]  
\[\tilde{S}_{k}=Y_{k}F_{k}\quad\text{with}\quad F_{k}=H_{k}^{*}\Big{/}\Big{(} \gamma^{-1}+|H_{k}|^{2}\Big{)}, \tag{10}\]  
where \(\gamma\) denotes the signal-to-noise (SNR) ratio.  
Afterwards, a \(N_{x}\)-sized inverse DFT (IDFT) is applied to the estimated signal (included in the FDE in Figure 2(b)), resulting in \(\tilde{s}_{n}\). Thereafter, the time domain signal is fed into a detector as follows:  
\[\mathbf{\tilde{d}}=\mathbf{B}\mathbf{s_{n}} \tag{11}\]  
with \(\mathbf{B}\) being the detection matrix and \(\mathbf{\tilde{d}}\) is the resulting vector which contains the estimated data symbols.  
Furthermore, as a consequence of self-created interference between adjacent sub-carries and time-slots, worse BER performances are observed [54]. Thus, it is necessary to cancel this interference to achieve an acceptable performance [57]. Recently, it has been proposed many equalization and self-cancellation techniques that aim to tackle this multi-user interference in several realistic scenarios [58; 59; 60].  
In [61], a double sided successive interference cancellation (DSSIC) method was proposed and applied on the detected symbols, subtracting the interference to adjacent sub-carriers in an iterative fashion.  
Four set of different detectors have been proposed, materialized in a different \(\mathbf{B}\) matrix. The ones mentioned in this paper follow the ones presented in [43; 53] and more details can be found in these references. They are divided as:* Match Filtering (MF), where the same filter included in the transmitter is now applied to each received block, i.e., \(\mathbf{B_{MF}}=\mathbf{A^{H}}\), where H is the Hermitian operator (conjugate and transpose). This maximizes the SNR ratio per sub-carrier, but introduces self-interference when a non-orthogonal transmit pulse is employed [43].
* Zero Forcing (ZF), where the inverse of matrix \(\mathbf{A}\), presented in (7), is applied to recover the data symbols, i.e., \(\mathbf{B_{ZF}}=\mathbf{A^{-1}}\). This approach completely removes any ICI at the cost of enhancing the influence of the noise in the detected symbols.
* DSSIC, which although being based on the MF detector, tries to minimize the ICI between neighboring sub-carriers. The basic idea is to subtract the ICI presented in the received signal at \(k\)th sub-carrier and caused by \((k+1)\)th and \((k-1)\)th sub-carriers. More details can be found in [43; 53; 61].
* SIC, which is similar to DSSIC but only the interference from the \((k-1)\)th sub-carrier is compensated.  
The detection of the data symbols ends with a downsampling of the signal presented in (11) by \(K\) (included in block \(\mathbf{B}\) in Figure 3b), followed by a demapper and decoder complementary to those used in the transmission process, in order to obtain an estimate of the original binary sequence, \(\bar{b}\).