# Field Trials of SC-FDMA, FBMC and LP-FBMC in Indoor Sub-3.5 GHz Bands
## 2 Waveform Models for SC-FDMA, FBMC, and LP-FBMC
### Lp-Fbmc  
In [8], it was reported that simply applying DFT spreading to FBMC cannot achieve the PAPR reduction effect, and there exist a special condition of phase coefficients to maximize the single carrier effect of DFT spreading for PAPR reduction. This condition is called the identically time-shifted multicarrier (ITSM) condition. In addition, in order to further reduce the PAPR, LP-FBMC generates four ITSM-conditioned signal candidates for each subframe and concatenates the ones with the lowest PAPR to form a frame. Figure 3 shows the transmitter structure for the \(k\)-th subframe of LP-FBMC. The four versions of signal candidates are generated according to two switching control bits \(S_{1}\) and \(S_{2}\), as shown in Table II of [8].  
In Figure 3, a complex vector \(\mathbf{X_{m}^{(k)}}=[X_{0,m^{\prime}}^{k},X_{1,m^{\prime}}^{k},...\,,X_{N-1,m} ^{k}]^{T}\) denotes the DFT output vector of complex data symbols, and the \(n\)-th element of \(\mathbf{X_{m}^{(k)}}\) is given, as follows:  
\[X_{n,m}^{(k)}=A_{n,m}^{(k)}+jB_{n,m}^{(k)},0\leq n\leq N-1,\ 0\leq m\leq M-1, \tag{4}\]  
where \(A_{n,m}^{(k)}\) and \(B_{n,m}^{(k)}\) denote the real and imaginary parts of \(X_{n,m}^{(k)}\), respectively. The transmitter structures of LP-FBMC and FBMC are almost the same, except that the DFT is performed on the modulated data symbols, and two switches were added, as shown  
Figure 3: Structure of the the \(k\)-th user’s low peak-to-average power ratio filter bank multicarrier (LP-FBMC) transmitter.  
Figure 2: Structure of the \(k\)-th user’s filter bank multicarrier (FBMC) transmitter.  
in Figures 2 and 3. After generating four transmission signal candidates according to the two switches, the candidate signal with the lowest PAPR is selected for transmission. In the \(l\)-th data block of the \(k\)-th user, the \(v\)-th candidate of transmit signal \(s_{l}^{(k)(v)}(t)\), \(l=0,1,2,\ldots,M/W-1\), can be expressed, as follows [10]:  
\[s_{l}^{(k)(v)}(t)=\sum_{n=0}^{N-1}\sum_{m=lW}^{W(l+1)-1}\Big{\{}A_{n,m}^{(k)} p_{n,m}^{(v)}(t)+B_{n,m}^{(k)}q_{n,m}^{(v)}(t)\Big{\}}e^{j\frac{2\pi n(n)}{T}t},\;v=1,2,3,4, \tag{5}\]  
where \(W\) denotes the number of offset quadrature amplitude modulation (OQAM) symbols in the \(l\)-th data block, and \(p_{n,m}^{(v)}(t)\) and \(q_{n,m}^{(v)}(t)\) are expressed, as follows [10]:  
\[p_{n,m}^{(v)}(t) =\begin{cases}h(t-mT)e^{jn\left(\frac{\pi}{T}t+\frac{\pi}{2}\right) }&\text{if $v=1$}\\ h(t-mT)e^{jn\left(\frac{\pi}{T}t-\frac{\pi}{2}\right)}&\text{if $v=2$}\\ jh\Big{(}t-\frac{\pi}{2}-mT\Big{)}e^{jn\left(\frac{\pi}{T}t+\frac{\pi}{2}\right) }&\text{if $v=3$}\\ jh\Big{(}t-\frac{\pi}{2}-mT\Big{)}e^{jn\left(\frac{\pi}{T}t-\frac{\pi}{2}\right) }&\text{if $v=4$},\end{cases} \tag{6}\] \[q_{n,m}^{(v)}(t) =\begin{cases}h(t-\frac{\pi}{2}-mT)e^{jn\left(\frac{\pi}{T}t+ \frac{\pi}{2}\right)}&\text{if $v=1$}\\ jh\Big{(}t-\frac{\pi}{2}-mT\Big{)}e^{jn\left(\frac{\pi}{T}t-\frac{\pi}{2}\right) }&\text{if $v=2$}\\ -h(t-mT)e^{jn\left(\frac{\pi}{T}t+\frac{\pi}{2}\right)}&\text{if $v=3$}\\ -h(t-mT)e^{jn\left(\frac{\pi}{T}t-\frac{\pi}{2}\right)}&\text{if $v=4$}.\end{cases} \tag{7}\]  
At the final stage, we successively concatenate the selected candidates. The concatenated waveform up to the \(l\)-th block of the \(k\)-th user \(c_{l}^{(k)}(t)\) can be expressed, as follows [10]:  
\[c_{l}^{(k)}(t)=c_{l-1}(t)+s_{l}^{(k)(u_{l}^{k})}(t),\;\;\text{with $c_{-1}(t)=0$}, \tag{8}\]  
where \(u_{l}^{k}\) is a candidate index that is selected in the \(l\)-th data block of the \(k\)-th user.