# Polynomial Cancellation Coded DFT-s-OFDM for Low-PAPR Uplink Signaling
## 3 Analysis of PCC-DFT-s-OFDM
### Frequency-Domain Aspect  
On the other hand, PCC-DFT-s-ODFM can be also regarded as a novel SS scheme for single-carrier waveform to reduce PAPR. Its shaping function can be derived by defining \(m=r+u\), \(0\leq r\leq P-1\), \(0\leq u\leq 2^{d}-1\) and substituting (1) into (2):  
\[\begin{array}{ll}X_{k}&=\frac{1}{\sqrt{M}}\sum\limits_{u=0}^{2^{d}-1}\sum \limits_{r=0}^{P-1}(-1)^{u}s_{r}W_{M}^{k(r+u)}\\ &=\sum\limits_{u=0}^{2^{d}-1}\left(-W_{M}^{k}\right)^{u}\cdot\frac{1}{\sqrt{M}} \sum\limits_{r=0}^{P-1}s_{r}W_{P}^{kr},0\leq k\leq M-1.\end{array} \tag{11}\]  
Using (11), the PCC-based frequency-domain data vector can be expressed by multiplying the \(P\)-point DFT of \(\{s_{r}\}_{r=0}^{P-1}\) by a weighting sequence; that is, the PCC-based SS function. As \(d=0\) (i.e., \(P=M\)) is an all-one sequence used to perform conventional DFT-s-OFDM, we consider \(d\in\{1,\ 2,\ \ldots\}\) for the derived SS sequence \(G_{k}^{d}\) (the details are provided in Appendix A):  
\[\begin{array}{ll}G_{k}^{d}&=\sum\limits_{u=0}^{2^{d}-1}\left(-W_{M}^{k} \right)^{u}\\ &=2^{d}\sin\left(\frac{\pi k}{M}\right)\mathcal{G}[k]\cdot e^{j\frac{\pi}{2}} \prod\limits_{p=1}^{d}W_{M}^{2^{d}-p-1k},0\leq k\leq M-1,\end{array} \tag{12}\]  
Figure 2: Example transforms \(g_{n,r}^{d}\) with \(N=512\) and \(M=48\) for **(a,b)**\(d=0\) (**c,d**) \(d=1\) (**e,f**) \(d=2\), and **(g,h)**\(d=3\). Since \(g_{n,r}^{d}\) is complex-valued, its real/imaginary parts and amplitude (in linear/log scale) are shown separately.  
where  
\[\mathcal{G}[k]=\left\{\begin{array}{ll}1,&d=1\\ \prod\limits_{p=1}^{d-1}\cos\Bigl{(}\frac{2^{d-p}\tau_{R}k}{M}\Bigr{)},&d>1\\ \end{array}\right.. \tag{13}\]  
Equations (12) and (13) show how PCC-DFT-s-OFDM can be regarded as a novel complex-valued SS scheme instead of a real-value filter for DFT-s-OFDM. The PCC-based and spectral-shaped frequency components \(X_{k}^{\text{PCC}}\) can be obtained as a substitute for \(X_{k}\) as follows:  
\[X_{k}^{\text{PCC}}=X_{k\text{ mod }p}^{p}G_{k}^{d},0\leq k\leq M-1, \tag{14}\]  
where \(X_{k\text{ mod }p}^{p}\) are the periodically extended coefficients of \(P\)-point DFT of \(\{v_{r}\}_{r=0}^{P-1}\). Since (14) is basically an SS process, the frequency-domain data received after inverse-shaping \(Y_{r}^{p}\) at the receiver can be obtained using the equation [21]:  
\[Y_{r}^{p}=\sum_{u=0}^{2^{d}-1}Y_{2^{d}r+u}^{\text{PCC}}G_{2^{d}r+u^{\prime}}^{ d}\,0\leq r\leq P-1, \tag{15}\]  
where \(Y_{2^{d}r+u}^{\text{PCC}}\) is the received user data at the \(\left(2^{d}r+u\right)\)-th subcarrier. Although the number of transmitted symbols decreases from \(M\) to \(P\), which shrinks spectral efficiency, the computational complexity of DFT is also decreased if we implement PCC-DFT-s-OFDM as an SS technique.  
The example impulse responses of \(G_{k}^{d}\) with \(N=512\), \(M=48\), and \(d=1,2,\ 3\) are depicted in Figure 3. As seen in Figure 3a,b. \(\left|G_{k}^{1}\right|\) is equivalent to the raised cosine (RC) spectrum-shaping vector [28] if \(d=1\) is used for the proposed method and the roll-off factor is \(\beta=1\) (\(\beta=0\)-1). This is because the half-sine curve in (12) is exactly consistent with the absolute value of the RC curve. For a higher-order SS, as shown in Figure 3c-f, shorter-period cosine curves described in (13) would be multiplied by the half-sine curve, making the signal spectrum more band-limited.  
Figure 3: _Cont._