# A Survey of Candidate Waveforms for beyond 5G Systems
## 3 Candidate Waveforms
### Generalized Frequency Division Multiplexing
#### 3.3.1. GFDM Transmitter  
In a GFDM transmitter, every stream is firstly upsampled by \(K\geq N\), filtered, and then shifted to its carrier frequency. In the filtering stage the sub-carriers are circularly convoluted with a prototype filter, \(g[n]\), such as the RRC [53]. A GFDM symbol is obtained through superposition of the filtered data symbols belonging to all sub-carriers and time slots and can be represented as follows [42]  
\[s[n]=s_{n}=\sum_{m=0}^{N_{s}-1}\sum_{k=0}^{N-1}S_{m,k}g_{m,k}[(n-mK)mod(N_{s}K)] e^{\frac{2\pi\mathrm{ln}}{K}},\quad n=0,\cdots KN_{s}-1 \tag{6}\]  
where \(S_{m,k}\) is the complex modulated symbol on the \(k\)th sub-carrier and \(m\)th sub-symbol, and \(g[n]\) denotes the prototype filter. The choice of \(g[n]=h_{RRC}[n]\) can lead to lower ISI but higher ICI, which in turn deteriorates with higher roll-off [53].  
Equation (6) can also be written as  
\[\mathbf{s_{n}}=\mathbf{A}\mathbf{S} \tag{7}\]  
where \(\mathbf{S}\) is a vector of \(NN_{s}\) data symbols, \(S_{m,k}\) and \(\mathbf{A}\) is a \(KN_{s}\times NN_{s}\) complex valued modulation matrix with elements based on the parameters \(N_{s}\), \(K\), \(N\) and \(g[n]\), whose elements can be represented as [54]  
\[\mathbf{A}[(k+1)+mN]=g_{m,k}[(n-mK)mod(N_{s}K)]e^{j\frac{2\pi\mathrm{ln}}{K}} \tag{8}\]  
for \(m=0,\cdots,N_{s}-1\) and \(k=0,\cdots,N-1\).  
Among the main characteristics of GFDM is characterized by the following:  
* The adjustable filters used for pulse shaping are circularly convoluted over a defined number of symbols. This can result in non-orthogonal sub-carriers and both ISI and ICI might arise [43; 4].  
Figure 3. GFDM transmitter (**a**) and receiver (**b**).  
Since GFDM is a MC technique it can exhibit high PAPR values, although with the inclusion of adjustable pulse shapes, along with the possibility of employing a filter-bank, equivalent to a DFT-OFDM signal, these values can be reduced [4; 55; 56].
* The possibility of adjusting the sub-carrier spacing allows a reduction on the OOB emissions.
* In the GFDM block construction, the overhead needed to avoid IBI is relatively small. Instead of adding a CP to every symbol like CP-OFDM schemes, the GFDM transmitter includes the addition of a single CP for an entire block that includes multiple sub-symbols [43], resulting in \(x_{n}\). Windowing techniques can also be employed in the GFDM multi-symbols in order to avoid discontinuities due to tail-biting.  
Therefore, GFDM can achieve higher spectral efficiency than OFDM systems, at the expanse of self-introduced interference between sub-symbols and higher transceiver complexity. Additionally, the need for long filter lengths (narrow filter bandwidth) requires higher block processing which can lead to higher latency [6]. Several transmitter and receiver architectures have been proposed to reduce the overall system complexity, in order to address the needs of future wireless communication networks, such as in [43] which proposes a low complexity transmitter implementation based on the IFFT/FFT approach, just like OFDM, or in [54] which presents a receiver architecture that presents a sparse representation of the pulse-shaping filter in frequency domain, which also simplifies any interference cancellation algorithm.