# A Survey of Candidate Waveforms for beyond 5G Systems
## 3 Candidate Waveforms
### Filtered Orthogonal Frequency Division Multiplexing
#### 3.2.1 F-OFDM Transmitter and Filter Design  
The F-OFDM transmitter generates the sub-band OFDM based signals, by assigning \(M\) consecutive sub-carriers in an OFDM symbol. This number \(M\) can be different for different sub-bands. However, for the sake of clarity, in the following description of F-OFDM, all the OFDM sub-band symbols are considered to have the same number of carriers. This means that the transmitter performs \(P\) IFFTs of size \(M\), with \(P\) sub-band OFDM symbols being included within an OFDM symbol period, \(N\geq M\times P\). Afterwards, each one of these new \(P\) \"symbols\" are added with a different CP of length, \(N_{Cp_{n}}\)[44].  
Figure 2 shows the schematic of a F-OFDM transceiver. Each one of the OFDM sub-band based signals is given by  
\[s_{p}[n]=s_{p_{n}}=\sum_{k=m}^{m+M-1}S_{k}w[n]e^{\frac{2\pi\mathrm{k}n}{N}}, \quad n=0,\cdots,M-1,p=1\cdots,P \tag{2}\]  
Figure 2: F-OFDM transmitter (**a**) and receiver (**b**).  
where \(S_{k}\) denote the modulated symbol from an constellation carried by the \(k\)th sub-carrier, \(k=m,\cdots,m+M{-}1\) from the assigned sub-carrier range. Each \(\mathbf{s_{p}}\) is added a CP adding a total of \(M+N_{cp_{m}}\) samples, resulting in \(x_{p_{n}}\). The overall \(N\)-sized OFDM, \(\mathbf{x_{n}}\), signal is written in vector (3) and includes all the \(P\) sub-band signals, \(x_{p_{n}}\),  
\[\mathbf{x_{n}}=[\mathbf{x_{1}}\cdots\mathbf{x_{P}}] \tag{3}\]  
The F-OFDM transmitted signal is obtained by filtering the CP-OFDM sub-signals, \(x_{p_{n}}\), through an appropriately designed spectrum shaping filter, \(f_{m}[n]\), resulting in \(\bar{x}_{p_{n}}\). In other words, in each sub-band the rectangular window, \(w[n]\), is replaced by a filter \(f_{m}[n]\), as follows  
\[\bar{x}_{p}[n]=x_{p}[n]*f_{m}[n] \tag{4}\]  
These filters are centered around the assigned sub-carriers with a bandwidth able to cover all the sub-carriers \(M\) assigned for that sub-band, and its time duration is a fraction of an OFDM symbol duration [44]. The overall F-OFDM signal, \(x_{F_{n}}\) can, thus, be written as  
\[\mathbf{x_{F}}=[\bar{\mathbf{x}}_{1}\cdots\bar{\mathbf{x}}_{P}] \tag{5}\]  
Usually, these filter can be different for each sub-band, with their sizes being dependent on \(M\). The filter design is one the most critical aspects of this technique, involving a trade-off between the time and frequency domain characteristics and complexity [45]. Usually, the the filter length is also allowed to exceed the CP length to achieve a better balance between the frequency and time localization [44].  
The ideal prototype filter has a rectangular frequency response, i.e., a sinc time domain impulse response which causes no distortion in the pass-band and no OOB radiation [44]. For practical implementations, the sinc impulse response is soft-truncated with a window, such as the Hanning or root raised cosine (RRC) windows. In this paper, the Hanning window was employed in the filtering stage. More details on filter design can be found in [52]. By adopting this approach, the impulse response of the obtained filters will have smooth transitions, avoiding abrupt jumps at the edges, presenting only a limited time domain energy spread [9], which limits considerably the ISI introduced between consecutive OFDM symbols [45].