# PAPR Reduction in UFMC for 5G Cellular Systems
## 2 The UFMC System Model
### UFMC Transmitter  
As can be seen in Figure 1, the UFMC transmitter works as follows. First, the information bits are mapped into \(M\)-ary quadrature amplitude modulation (M-QAM) symbols (e.g., \(M=16\)). Subsequently, \(N_{sc}\) symbols are grouped to form the entries of vector \(\mathbf{S}\); that is,  
Figure 1: The typical block diagram of the universal filtered multi-carrier (UFMC) transceiver.  
\(\mathbf{S}=[S(0),S(1),...,S(N_{sc}-1)]^{T}\). The symbols of \(\mathbf{S}\) are divided into \(B\) sub-sequences, each is of length \(N_{b}=N_{sc}/B\), to form \(B\) parallel new sub-sequences \(\mathbf{S}_{b}\) (\(b=1,2,...,B\)), defined as:  
\[\bar{S}_{b}(k)=\begin{cases}0&\text{if }0\leq k\leq N^{\prime}+(b-1)\frac{N_{sc}} {B}-1\\ S(k),&N^{\prime}+(b-1)\frac{N_{sc}}{B}\leq k\leq N^{\prime}+b\frac{N_{sc}}{B}-1 \\ 0,&N^{\prime}+b\frac{N_{sc}}{B}\leq k\leq N-1\end{cases} \tag{1}\]  
where \(N^{\prime}=(N-N_{sc})/2\). Note that the length of \(S_{b}\) is \(N(\leq N_{sc})\). The sub-sequence \(S_{b}\) undergoes \(N\)-point IFFT to obtain the discrete-time signal \(s_{b}(n)\); that is,  
\[s_{b}(n)=\frac{1}{N}\sum_{k=0}^{N-1}S_{b}(k)e^{\frac{2\pi nx}{N}},\hskip 28.452756 pt 0\leq n\leq N-1;1\leq b\leq B \tag{2}\]  
It is obvious from Equations (1) and (2) that each symbol is allocated to one sub-carrier, and that the \(N_{sc}\) symbols are divided equally and carried over \(B\) parallel sub-bands. In Equation (2), \(n\) and \(b\) correspond to the sub-carrier and sub-band indices, respectively.  
The \(N\)-point IFFT output of each sub-band is filtered using digital filter of length \(L_{f}\) to form a UFMC signal of length (\(N+L_{f}-1\)), which can be written as:  
\[x(k)=\sum_{b=1}^{B}\sum_{l=0}^{L_{f}-1}s_{b}(k-l)f_{b}(l),\hskip 28.452756 ptk=0,1,...,N+L_{f}-1 \tag{3}\]  
where \(f_{b}(l)\) are FIR filter coefficients, for \(l=0,1,...,L_{f}-1\). Because of the CP and convolution operations carried out in the time domain, the length of sub-band signals increases from \(N\) to \(N+L_{f}+L_{cp}-1\) where \(L_{cp}\) is the cyclic prefix length. Hence, the transmitted UFMC signal \(\mathbf{x}\in\mathbb{C}^{(N+L_{cp}+L_{f}-1)\times 1}\)can be expressed in matrix form as:  
\[\mathbf{x}=\sum_{b=1}^{B}\mathbf{F}_{b}\bar{\mathbf{D}}\mathbf{S}_{b} \tag{4}\]  
where \(\mathbf{F}_{b}\in\mathbb{C}^{(N+L_{cp}+L_{f}-1)\times(N+L_{cp})}\) is a Toeplitz matrix composed of the filter impulse response of the \(b^{th}\) sub-band, with the coefficients of each sub-band filer are chosen to be normalized, such that \(\sum_{l=0}^{L_{f}-1}|f_{b}(l)|^{2}=1\). \(\bar{\mathbf{D}}=[\mathbf{D}_{cp}:\mathbf{D}]^{H}\in\mathbb{C}^{(N+L_{cp}) \times N}\), where \(\mathbf{D}\in\mathbb{C}^{N\times N}\) denotes the FFT matrix whose element is given by \(\frac{1}{\sqrt{N}}exp^{-j\frac{2\pi nx}{N}}\) and \((k,n=0,...,N-1)\) and \(\mathbf{D}_{cp}\) consists of the last \(L_{cp}\) columns of \(\mathbf{D}\) (where \(L_{cp}\) is denoting the CP length). \(\mathbf{S}_{b}\in\mathbb{C}^{N\times 1}\) is the symbol vector transmitted on the \(b^{th}\) sub-band.  
Different kinds of digital FIR filters can be used. These include Hamming, Hanning, Chebyshev, and Blackman filters. Digital FIR filters are used to solve the OOB emission problem and improve the PSD efficiency. In this paper, the optimal Dolph Chebyshev filter was considered for sub-band filtering [18; 39]. It has special characteristics, as it contains equi-ripple side lobe level (SLL) with narrow main lobe width for given ripple ratio and filter length \(L_{f}\)[40]. The characteristic of Chebyshev window is expressed, as follows [40].  
\[w(l)=(-1)^{l}\frac{\cos\left[L_{f}\cos^{-1}\left[\beta\cos(\pi l/L_{f})\right] \right]}{\cosh\left[L_{f}\cosh^{-1}(\beta)\right]}\times R(l),\hskip 28.452756 pt0 \leq l\leq L_{f}-1 \tag{5}\]  
where \(L_{f}\) is the Chebyshev window length and \(\beta\) is the side lobe attenuation, and \(R(l)\) is the rectangular sequence. Hence, the Dolph-Chebyshev filter will be expressed as \(f(l)=f_{d}(l)w(l)\) where \(f_{d}(l)\) is the ideal linear phase filter.