# Polynomial Cancellation Coded DFT-s-OFDM for Low-PAPR Uplink Signaling
## 2 System Model  
Figure 1 shows a block diagram of a discrete baseband PCC-DFT-s-OFDM transceiver. In this study, we focus on localized FDMA [14] owing to the LTE/NR compatibility. Let \(\{s_{r}\}_{r=0}^{P-1}\), \(P\triangleq M/2^{d}\) be the input PSK/QAM symbols, where \(M\) is the number of allocated subcarriers and \(d\in\{0,\ 1,\ 2,\ \ldots\}\) is the order of PCC. The PCC-mapped symbols can be expressed as:  
\[v_{m}=s_{m/2^{d}}(-1)^{m\bmod 2^{d}},\ 0\leq m\leq M-1, \tag{1}\]  
where mod is the modulo operator. It is noteworthy that the PCC variable used is a general form, such that \(d=0\) to disable the PCC mapping, \(d=1\) for conventional PCC [22], and \(d\geq 2\) for higher-order PCC. Then, the PCC-mapped symbols are linearly precoded by an \(M\)-point DFT to result in the frequency-domain data \(X_{k}\):  
\[X_{k}=\frac{1}{\sqrt{M}}\sum\limits_{m=0}^{M-1}v_{m}W_{M}^{bm},\ 0\leq k\leq M-1, \tag{2}\]  
where \(W_{M}=e^{-j(\frac{2\pi}{M})}\) is the twiddle factor. After being zero-padded and after the \(N\)-point inverse DFT (IDFT) of \(X_{k}\) is taken, the transmitted signal samples \(x_{n}\) can be written as [26]:  
\[\begin{array}{ll}x_{n}&=\frac{1}{\sqrt{N}}\sum\limits_{k=0}^{M-1}X_{k}W_{N} ^{-bm}\\ &=\frac{1}{\sqrt{MN}}\sum\limits_{k=0}^{M-1}\sum\limits_{m=0}^{M-1}v_{m}W_{M} ^{bm}W_{N}^{-bm}\\ &=\frac{1}{\sqrt{M}}\sum\limits_{m=0}^{M-1}v_{m}g\Big{[}n-\frac{N}{M}m\Big{]}, \ 0\leq n\leq N-1,\end{array} \tag{3}\]  
where  
\[g[n]=\frac{sin\Big{(}\frac{M\pi n}{N}\Big{)}}{\sqrt{N}sin\Big{(}\frac{M\pi n} {N}\Big{)}}e^{j\frac{(M-1)\pi n}{N}}. \tag{4}\]The performance of the PAPR \(\xi\) of the transmitted signal in terms of complementary cumulative density function (CCDF) can be presented as:  
\[CCDF(\xi_{0})=Pr(\xi>\xi_{0}),\ \xi\triangleq\frac{\max\limits_{0\leq n\leq N-1} \left|x_{n}\right|^{2}}{\mathbb{E}\left(\left|x_{n}\right|^{2}\right)}, \tag{5}\]  
where \(\mathbb{E}\{\bullet\}\) is the expectation operator.  
Assuming the length of cyclic prefix (CP) is longer than the multipath channel impulse response, the received signal samples over the fading channel and after removing the CP can be expressed as:  
\[y_{n}=h_{\ell}\otimes_{N}x_{n}+z_{n},\ 0\leq n\leq N-1, \tag{6}\]  
where \(\left|h_{\ell}\right|_{\ell=0}^{L-1}\) is the \(L\)-tap channel impulse response, \(\otimes_{N}\) is the circular convolution operator with length \(N\), and \(z_{n}\) is the time-domain additive white Gaussian noise (AWGN) represented in the expression PSD \(N_{0}/2\). Hence, the received data at the \(k\)-th subcarrier is given as:  
\[Y_{k}=H_{k}X_{k}+Z_{k},0\leq k\leq N-1, \tag{7}\]  
where \(Y_{k}\) and \(Z_{k}\) are the DFT of \(y_{n}\) and \(z_{n}\), respectively, and \(H_{k}=\sum\limits_{\ell=0}^{L-1}h_{\ell}W_{N}^{\ell k},\ 0\leq k\leq N-1\) is the channel frequency response. By discarding the non-user data in \(Y_{k}\), and applying \(M\)-point IDFT for the remaining components, the received symbols \(\{\hat{v}_{m}\}_{m=0}^{M-1}\) are obtained as:  
\[\hat{v}_{m}=\sum\limits_{k=0}^{M-1}Y_{k}W_{M}^{-km},0\leq m\leq M-1. \tag{8}\]  
Finally, the output PSK/QAM symbols by PCC-demapping are given as:  
\[\hat{s}_{r}=\sum\limits_{u=0}^{2^{d}-1}\hat{v}_{2^{d}+u^{\prime}}(-1)^{u},\ 0\leq r\leq P-1. \tag{9}\]  
Here, the value of \(d\) controls the growth of the symbol number in multiples of \(2^{d}\), which is positively related to the ability of PAPR reduction. Therefore, the proposed PCC-DFT-s-OFDM provides the  
Figure 1: Block diagram of the discrete Fourier transform-spread orthogonal frequency division multiplexing (DFT-s-OFDM) transceiver with polynomial cancellation coding (PCC) mapping.  
flexibility of a trade-off between spectral efficiency and energy efficiency by adjusting the value of \(d\), making the system more adaptive to the requirements of different communication scenarios.