## 2 Transmitter and Receiver of OvTDM Systems  
In general, OvTDM is a time multiplexing technique aiming at maximizing channel capacity sharing, so it differs fundamentally from the traditional time multiplexing technique (TDM). Instead of providing guard intervals for each signal, OvTDM deliberately induces overlapping by considering it as a constraint relationship. The more overlapping occurs, the higher the coding gain, which implies a longer coding constraint length. By superimposing waveforms, it is possible to transmit more symbol waveforms within the same time, thereby providing higher spectral efficiency compared to conventional modulation techniques. The structure of the transmitter side of the system is shown in Figure 1. The source bits \(U=[u_{0},u_{1},u_{2},\ldots,u_{M}-1]^{T}\), \(u_{n}\in\{0,1\}\) are first transformed into K-layer signals through a serial-parallel transformation, and then coded and modulated, respectively, while \(c_{k}\) is the codeword stream after interleaving. In order to reduce the complexity of both the transmitter and receiver, the same coding and modulation methods are used for each layer. While modulating the signal, the power is allocated to each layer of symbols as \(P_{1},P_{2},\ldots,P_{K}\). The symbol stream after mapping to K-layer can be denoted by \([x_{1},x_{2},\ldots,x_{K-1}]^{T}\), and the K-layer signals are superimposed on the transmitter to produce the final symbol stream. Therefore, \(x_{k}[n]\) after power allocation can be expressed as  
\[x[n]=\sum_{k=1}^{K}\sqrt{P_{k}T}x_{k}[n]. \tag{1}\]  
Figure 1: Block diagram of the transmitter for OvTDM. After overlapping, the symbol stream \(x[n]\) is up-sampled and pulse-formed with sampling intervals \(T\), which is orthogonal during the symbol period. During the shaping-filter process, if a further increase in spectral efficiency is required, a smaller transmission interval can be used to send the signals as \(\tau T\), (\(\tau\leq 1\)). Thus, the final OvTDM signal can be expressed as follows:  
\[S(t) = \sum_{n=0}^{N-1}x[n]h(t-n\tau T) \tag{2}\] \[= \sum_{k=0}^{K-1}\sum_{n=0}^{N-1}x_{k}[n]h(t-n\tau T),\]  
where \(N\) is the number of symbols in each layer, the default binary phase shift keying (BPSK) modulation is applied, and \(h(t)\) is the energy-normalized pulse-shaping filter waveform. And, the accelerating factor, denoted by \(\tau\), plays a crucial role in shortening the sampling interval of the sinc function. In signal processing, the sinc function is commonly used for sampling and reconstruction. The introduction of the accelerating factor \(\tau\) allows for a more rapid adjustment of the sampling intervals, facilitating quicker and more efficient signal processing. Assuming that the signal passes through a Gaussian white noise channel, the received signal \(r(t)\) can be expressed as  
\[r(t)=\sum_{n=0}^{N-1}x_{n}h(t-n\tau T)+\eta(t), \tag{3}\]  
where \(\eta(t)\) is the Gaussian white noise in the channel. The receiver model of OvTDM is shown in Figure 2, and the received signal after matched filtering and sampling can be expressed as  
\[y[n]=\int_{-\infty}^{+\infty}r(t)h^{*}(t-n\tau T)dt, \tag{4}\]  
where \(h^{*}(t)\) is the conjugate of the pulse shaping function \(h(t)\) used at the transmitter. The received signal \(y\), after the matched filtering and sampling process, \(y\) is obtained as a sufficient statistic, expressed as  
\[y=Gx+\eta, \tag{5}\]  
where \(\eta\) is the sampled value of the color noise and \(G\) is the Toplitz matrix as  
\[G=\left( \begin{array}{ccccccccc}1&g_{-1}&\cdots&g_{-L_{ISI}}&0&0&0&0&\cdots \\ g_{1}&1&g_{-1}&...&g_{-L_{ISI}}&0&0&0&\cdots\\ \vdots&\ddots&\ddots&\ddots&&\ddots&&\ddots&\\ g_{L_{ISI}}&\cdots&g_{1}&1&g_{-1}&\cdots&g_{-L_{ISI}}&0&\cdots\\ 0&g_{L_{ISI}}&\cdots&g_{1}&1&g_{-1}&\cdots&g_{-L_{ISI}}&\cdots\\ \vdots&&\ddots&&\ddots&\ddots&&\ddots&\ddots\end{array}