# A Survey of Candidate Waveforms for beyond 5G Systems
## 3 Candidate Waveforms
### Time Interleaved Block Windowed Burst Orthogonal Frequency Division Multiplexing  
#### 3.4.1 TIBWB-OFDM  
The TIBWB-OFDM was proposed in [46; 47] and is based on BWB-OFDM [62]. This new waveform allows a better spectral efficiency, by employing transmitted signals with a PSD as compact as F-OFDM schemes and a better power efficiency when compared to conventional CP-OFDM schemes [62]. The greater spectral efficiency is obtained by increasing the spectral confinement and lowering the OOB emissions of the signal transmitted through the use of windowing techniques. The increase in power efficiency is obtained by concatenating a number of OFDM symbols, to which a single prefix of zeros is added, thereby eliminating the need of a CP [47]. Furthermore, this modulation technique performs a time-interleave operation between the samples of the various OFDM symbols that make up the overall BWB-OFDM multi-symbol [62]. Figure 4a depicts the TIBWB-OFDM transmitter. The time interleave operation is performed after the cyclic extension and windowing operation and causes the compression and replication of the spectral data over the occupied bandwidth, creating a diversity effect in the frequency domain, thus increasing the robustness of the method against deep fading of the communication channel, by allowing partial data recover from eventual unaffected spectrum replicas [46; 47].  
The TIBWB-OFDM original packing consists into the juxtapotion of a set of \(N_{s}\)\(N\) sub-carrier windowed OFDM-based symbols, generated according to a \(N\)-sized IFFT operation \(\mathbf{s_{i}}=\mathbf{IFFT}\{\mathbf{S_{k,i}}\}=[\mathbf{s_{0,i}},\cdots, \mathbf{s_{N-1,i}}]\), where \(S_{k,i}\) denote the modulated symbol from an m-ary constellation carried by the \(k\)th sub-carrier, \(k=0,\cdots,N-1\), of the \(i\)th OFDM symbol, \(i=1,\cdots,N_{s}\).  
Afterwards, the cyclic extension and windowing operations is applied to each one of \(s_{i}\), to perform spectral shaping [46]. Windowing employs a square root raised cosine (SRRC) pulse shape where \(\beta\leq 1\) represents the window roll-off as in [46; 47].  
Consequently, the new \(N_{s}\) windowed symbol are expressed by  
\[\mathbf{s_{w,i}}=[\mathbf{s_{i}}|\mathbf{s_{i}}]\odot\mathbf{h_{SRRC}}, \tag{12}\]  
where the operator \(\odot\) represents a point-wise Hadamard multiplication. The tailing zeros from the windowing operation are then discarded, resulting in (12) being a vector with length \(N_{symb}=N(1+\beta)\). The simple time domain juxtaposition of the component OFDM symbols forms a BWB-OFDM multi-symbol [62]. As mentioned, the TIBWB-OFDM approach performs a time domain interleaving operation between the samples of the BWB-OFDM multi-symbol [46; 47], resulting in a set of \(N_{s}\) interleaved symbols, as follows  
\[\mathbf{s_{\pi}}=\mathbf{s_{w}}\Pi^{(N_{s})} \tag{13}\]making up the TIBWB-OFDM multi-symbol, where \(\mathbf{\Pi}^{(\mathbf{N_{s}})}\) is the time-interleaved matrix with period \(N_{s}\) of size \(N_{\text{symb}}\times N_{\text{symb}}\), where the \(c\)th column has a \"one\" at row \(\lfloor\frac{c}{N_{s}}\rfloor+(cN_{\text{symb}}\mod N_{\text{symb}}N_{s})\).  
The final step of Figure 4a includes the insertion of a single \(N_{zp}\)-sized zero-pad (ZP), acting as a guard interval in order to deal with channel's delay spread and avoid ISI. This results in the transmitted block, with a total length of \(N_{x}=N_{symb}N_{s}+N_{zp}\).  
\[\mathbf{x_{n}}=[Z\text{P}\ \mathbf{s}_{\pi}] \tag{14}\]