# PAPR Reduction in UFMC for 5G Cellular Systems
## 3 The Proposed Dcomposed SLM PAPR Reduction Technique  
Assume a typical UFMC system, as in Figure 2 with \(N_{\text{sc}}\) sub-carriers. The UFMC system accepts a number of data bits, which are mapped into the baseband symbols using \(M\)-QAM/\(M\)-PSK mapper. Then, these complex symbols are packed to \(B\) parallel \(N\)-point sequences, \(\mathbf{S}_{b}\). The \(\mathbf{S}_{b}\) sequences are Dcomposed into their real \(\Re\left\{\mathbf{S}_{b}(k)\right\}\) and imaginary \(\Im\left\{\mathbf{S}_{b}(k)\right\}\) components for all \(k=0,1,..,N-1\). Subsequently, each component is multiplied with the corresponding element of the random candidate phase \(\mathbf{P}^{u}(k)\)\(\forall\)\(u\)\(\in\left\{0,1,...,U-1\right\}\), where \(U\) is the overall generated phase blocks and \(\mathbf{P}^{u}(k)\in\left\{-1,1\right\}\). The element-wise multiplication operation can be formulated in matrix form as \(\mathbf{S}_{r}^{u}=\Re\{\mathbf{S}_{b}\}\odot\mathbf{P}^{u}\)and \(\mathbf{S}_{i}^{u}=\Im\{\mathbf{S}_{b}\}\odot\mathbf{P}^{u}\) for the real and imaginary parts, respectively. The resultant signals \(\mathbf{S}_{r}^{u}\) and \(\mathbf{S}_{i}^{u}\) are the rotated vectors. An UFMC system is applied for each modified components, \(\mathbf{S}_{r}^{u}\) and \(\mathbf{S}_{i}^{u}\) in addition to the the original symbols \(\mathbf{S}_{b}\), such as: \(\mathbf{x}_{r}^{u}=\text{UFMC}[\mathbf{S}_{r}^{u}]\) and \(\mathbf{x}_{i}^{u}=\text{UFMC}[\mathbf{S}_{i}^{u}]\), \(\mathbf{x}_{org}=\text{UFMC}[\mathbf{S}_{b}]\), respectively, where the UFMC\(\left\{\right\}\) means the operations applied, as given in Equation (4). Subsequently, the candidate signal \(\mathbf{x}^{\alpha}\) is transmitted, where \(v\) denotes the vector of minimum PAPR. The definition of the transmitted signal \(\mathbf{x}^{\alpha}\) PAPR can be expressed as:  
\[\text{PAPR}=10\log_{10}\frac{P_{peak}}{P_{avg}}=10\log_{10}\frac{\max\limits_{ n}\left[\left|x_{n}^{v}\right|^{2}\right]}{\mathbb{E}\left[\left|x_{n}^{v} \right|^{2}\right]}. \tag{7}\]  
Figure 2: The proposed block diagram of Dcomp-selected mapping (SLM) UFMC peak-to-average power ratio (PAPR) reduction scheme. where \(P_{peak}\) is the maximum instantaneous power and \(P_{avg}\) is the average power of \(\mathbf{x}^{v}\). The performance of the Dcomp-SLM reduction technique is analyzed by calculating the CCDF of the lowest PAPR obtained for an UFMC symbol of Equation (7). Subsequently, the candidate UFMC symbol with the minimum PAPR is considered for transmission, such as:  
\[\mathbf{x}^{u^{\prime}}=\mathbf{x}^{u}\left\{\underset{0\leq u\leq U-1}{\arg \min}\left[\mathrm{PAPR}_{org},\mathrm{PAPR}_{r}^{u},\mathrm{PAPR}_{i}^{u} \right]\right\}. \tag{8}\]  
Theoretically, the CCDF of \(\mathrm{PAPR}_{u^{\prime}}\) is defined as the probability that the minimum \(\mathrm{PAPR}_{u^{\prime}}\) exceeds a given threshold \(\mathrm{PAPR}_{0}\), when applying \(U\) trails for both real and imaginary parts separately. Therefore, the CCDF will be approximated to:  
\[\mathrm{Pr}(\mathrm{PAPR}_{u^{\prime}}>\mathrm{PAPR}_{0})\approx\left(1- \left(1-e^{-\mathrm{PAPR}_{0}}\right)^{\alpha N}\right)^{U^{2}}, \tag{9}\]  
where \(N\) is the number of sub-carriers per UFMC block, and \(\alpha\) is introduced, so that this approximation matches practical results [41]. It is worth noting that at the receiver side the original phases can be recovered by implementing any appropriate selective mapping technique for a phase vector recovery. For convenience, in our work, the methods of [25; 26] are used, which provide solutions for original data phase recovery without explicit side information Finally, the modified energy per transmitted symbol of our proposed approach can be expressed as: \(E_{s}^{\prime}=E_{s}[1+(C-1)/(U+1)]\), where \(E_{s}\) is the original energy per symbol and \(C\) is the extended factor. For all the probabilistic techniques (SLM and PTS), the main source of computational complexity is originally owing to IFFT operations.