## 2 DMT and PAPR  
With respect to a DMT symbol with \(N\) sub-carriers, the data stream is mapped onto \(\mathbf{X}\) based on the quadrature amplitude modulation (QAM) constellation. Then, the time domain DMT signal \(\mathbf{x}\) is generated by a \(2N\)-point inverse discrete Fourier transform (IDFT), given by [4; 5; 6; 7]:  
\[x(n)=\frac{1}{\sqrt{2N}}\sum\limits_{k=0}^{2N-1}C(k)e^{\frac{i2\pi nk}{2N}},n=0, \ 1,\ \cdots,\ 2N-1 \tag{1}\]  
where \(\mathbf{C}=\left[\mathbf{X}^{T},(\mathcal{Z}(\mathbf{X}))^{T}\right]^{T}\) conforms to Hermitian symmetry [4] to produce the real-valued time-domain signal \(\mathbf{x}\). According to the central limit theorem, for \(N\rightarrow\infty\), \(\mathbf{x}\) can be modeled as a Gaussian distribution with \(\mathcal{N}\big{(}0,\sigma_{x}^{2}\big{)}\).  
The PAPR of \(\mathbf{x}\) is defined as the ratio between the maximum peak power and the average power during one symbol period, which can be written as [5; 19]  
\[\mathrm{Y}_{d}=\frac{\max\limits_{0\leq k\leq 2N-1}\Big{\{}\left|x(n)\right|^{2} \Big{\}}}{\mathbb{E}\Big{\{}\left|x(n)\right|^{2}\Big{\}}}=\frac{\max\limits _{0\leq k\leq 2N-1}\Big{\{}\left|x(n)\right|^{2}\Big{\}}}{\sigma_{x}^{2}}. \tag{2}\]  
Based on the central limit theorem, we can obtain  
\[P_{r}\left(\frac{\left|x(n)\right|^{2}}{\sigma_{x}^{2}}\leq r_{0}\right)=P_{r }\bigg{(}-\sqrt{r_{0}}\leq\frac{x(n)}{\sigma_{x}}\leq\sqrt{r_{0}}\bigg{)}\approx erf \bigg{(}\sqrt{\frac{r_{0}}{2}}\bigg{)}, \tag{3}\]  
where \(erf(\psi)=2/\sqrt{\pi}\int_{0}^{\psi}\mathrm{e}^{-t^{2}}dt\), \(P_{r}(\cdot)\) denotes the probability, and \(r_{0}\) denotes the threshold. Therefore, the complementary cumulative distribution function (CCDF) can be calculated as  
\[CCDF_{d}=P_{r}(\mathrm{Y}_{d}>r_{0})=1-erf^{2N}\bigg{(}\sqrt{\frac{r_{0}}{2}} \bigg{)}. \tag{4}\]  
Continuous-time DMT signal \(x(t)\) can be represented approximately by the \(L_{0}\) times oversampled \(\mathbf{x}\). Based on the extreme value theory, the corresponding CCDF can be calculated by  
\[CCDF_{c}=P_{r}(\mathrm{Y}_{c}>r_{0})=1-\exp\biggl{(}\frac{-2N}{\sqrt{3}}e^{- \frac{r_{0}}{2}}\biggr{)}, \tag{5}\]  
where \(\mathrm{Y}_{c}\) is the PAPR of \(x(t)\). Study [11] demonstrates that selecting \(L_{0}\ \geq\ 4\) is sufficient for approximating the peak value of \(x(t)\).  
Taking the discrete-time DMT as an example, the instantaneous power of \(x(n)\) can be calculated by  
\[\begin{array}{rl}\left|x(n)\right|^{2}&=\frac{1}{2N}\sum\limits_{p=0}^{2N-1 }\sum\limits_{q=0}^{2N-1}C(p)C^{*}(q)e^{\jmath\pi(p-q)\eta/2N}\\ &=\frac{1}{2}+\frac{1}{N}\mathbb{R}\left\{\sum\limits_{q=1}^{2N-1}e^{j\frac{2 \pi\pi}{2N}}\underbrace{\sum\limits_{p=0}^{2N-1-q}C(p+q)C^{*}(p)}_{\rho_{C}(q) }\right\}\end{array}, \tag{6}\]where \(\rho_{C}(q)\) is defined as the aperiodic autocorrelation function (ACF) of \(\mathbf{C}\). For an arbitrary complex \(\mathcal{X}_{n}\), the following inequalities hold \(\mathbb{R}\{\mathcal{X}_{n}\}\leq\left|\mathcal{X}_{n}\right|\) and \(\left|\sum\limits_{n=0}^{N-1}\mathcal{X}_{n}\right|\leq\sum\limits_{n=0}^{N-1 }\left|\mathcal{X}_{n}\right|\). Thus, a PAPR with a unit mean power (\(\sigma_{x}^{2}=1\)) satisfies  
\[\mathrm{Y}_{d}=\frac{\max\limits_{0\leq k\leq 2N-1}\Bigl{\{}\left|x(n)\right|^{2} \Bigr{\}}}{\sigma_{x}^{2}}\leq\frac{1}{2}+\frac{1}{N}\sum\limits_{q=1}^{2N-1} \left|\rho_{C}(q)\right|=\frac{1}{2}+\frac{1}{N}\tilde{\xi}_{\mathbf{C}}, \tag{7}\]  
where \(\tilde{\xi}_{\mathbf{C}}=\sum\limits_{q=1}^{2N-1}\left|\rho_{C}(q)\right|\). Therefore, a close relationship exists between \(\mathrm{Y}_{d}\) and \(\tilde{\xi}_{\mathbf{C}}\). When the input symbols \(C(n)\) are all equal, \(\max\limits_{0\leq k\leq 2N-1}\Bigl{\{}\left|x(n)\right|^{2}\Bigr{\}}\) is able to reach its maximum value within one symbol duration.